Notation by work
Machine Native Reasoning: wider continuation
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404 symbol spellings in 1050 mathematical passages across 13 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 404 entries are available below.
*Introduction passage · 74 passages
Search aliases: *
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
+Introduction passage · 30 passages
Search aliases: +
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
=Introduction passage · 204 passages
Search aliases: =
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
AIntroduction passage · 57 passages
Search aliases: A
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
A^X_m(x)
\mathcal AIntroduction passage · 1 passage
Search aliases: A mathcal
Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
\mathcal A
aIntroduction passage · 110 passages
Search aliases: a
Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
A^X_mIntroduction passage · 4 passages
Search aliases: A^X_m
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
A^X_m(x)
A^Z_mIntroduction passage · 3 passages
Search aliases: A^Z_m
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
A^Z_m(z)
a_1Introduction passage · 7 passages
Search aliases: a_1
Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
a_kIntroduction passage · 2 passages
Search aliases: a_k
Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
A_{\rm end}Introduction passage · 3 passages
Search aliases: A_rm end rm
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
BIntroduction passage · 42 passages
Search aliases: B
Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
B\subseteq V
\bar JIntroduction passage · 2 passages
Search aliases: bar J bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar J_0Introduction passage · 1 passage
Search aliases: bar J_0 bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar J_hIntroduction passage · 2 passages
Search aliases: bar J_h bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar J_{h-1}Introduction passage · 1 passage
Search aliases: bar J_h-1 bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar KIntroduction passage · 10 passages
Search aliases: bar K bar
Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
\bar K_{a,o}Introduction passage · 6 passages
Search aliases: bar K_a,o bar
Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
\bar uIntroduction passage · 7 passages
Search aliases: bar u bar
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
\bar u_hIntroduction passage · 5 passages
Search aliases: bar u_h bar
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
\bar u_{hao}Introduction passage · 1 passage
Search aliases: bar u_hao bar
Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
\bar zIntroduction passage · 2 passages
Search aliases: bar z bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar z_{a,o}Introduction passage · 2 passages
Search aliases: bar z_a,o bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar\PsiIntroduction passage · 2 passages
Search aliases: barΨ bar Psi
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar\Psi_{a,o}Introduction passage · 2 passages
Search aliases: barΨ_a,o bar Psi
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
CIntroduction passage · 57 passages
Search aliases: C
Introduction passage · proof_fields_chapter.tex, line 15
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
C_0=A\cup S,\qquad C_{k+1}=C_k\cup\{c_j:P_j\subseteq C_k\}.
cIntroduction passage · 13 passages
Search aliases: c
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
C_0Introduction passage · 1 passage
Search aliases: C_0
Introduction passage · proof_fields_chapter.tex, line 15
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
C_0=A\cup S,\qquad C_{k+1}=C_k\cup\{c_j:P_j\subseteq C_k\}.
c_aIntroduction passage · 2 passages
Search aliases: c_a
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
c_jIntroduction passage · 4 passages
Search aliases: c_j
Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
P_j\subseteq B\Rightarrow c_j\in B
C_kIntroduction passage · 3 passages
Search aliases: C_k
Introduction passage · proof_fields_chapter.tex, line 15
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
C_0=A\cup S,\qquad C_{k+1}=C_k\cup\{c_j:P_j\subseteq C_k\}.
C_{k+1}Introduction passage · 1 passage
Search aliases: C_k+1
Introduction passage · proof_fields_chapter.tex, line 15
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
C_0=A\cup S,\qquad C_{k+1}=C_k\cup\{c_j:P_j\subseteq C_k\}.
DIntroduction passage · 31 passages
Search aliases: D
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
dIntroduction passage · 38 passages
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Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
eIntroduction passage · 17 passages
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Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
\mathrm{end}Introduction passage · 1 passage
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Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
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m\in\{\mathrm{run},\mathrm{end}\}
FIntroduction passage · 32 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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F_a:X\to X
F_aIntroduction passage · 13 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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F_a:X\to X
F_{a_1}Introduction passage · 1 passage
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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F_w=F_{a_k}\cdots F_{a_1}
F_{a_k}Introduction passage · 1 passage
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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F_w=F_{a_k}\cdots F_{a_1}
F_wIntroduction passage · 5 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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F_w=F_{a_k}\cdots F_{a_1}
GIntroduction passage · 35 passages
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Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
HIntroduction passage · 33 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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r:X\to H
hIntroduction passage · 41 passages
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Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
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\bar u_h=\pi_*u_h
JIntroduction passage · 18 passages
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Introduction passage · core_chapters.tex, line 559
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
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J_h(p)
jIntroduction passage · 12 passages
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Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
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P_j\subseteq B\Rightarrow c_j\in B
J_hIntroduction passage · 2 passages
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Introduction passage · core_chapters.tex, line 559
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
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J_h(p)
KIntroduction passage · 27 passages
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Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
kIntroduction passage · 57 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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w=a_1\cdots a_k
K_{a,o}Introduction passage · 14 passages
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Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
mIntroduction passage · 29 passages
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Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
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A^X_m(x)
\midIntroduction passage · 21 passages
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Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
\minIntroduction passage · 8 passages
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Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\min_{a\in A(q)}Introduction passage · 1 passage
Search aliases: min_a∈ A(q) min in
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\min_{d\in D(q)}Introduction passage · 1 passage
Search aliases: min_d∈ D(q) min in
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
NIntroduction passage · 33 passages
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Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
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N
nIntroduction passage · 33 passages
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Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
oIntroduction passage · 53 passages
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Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
PIntroduction passage · 30 passages
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Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
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P_j\subseteq B\Rightarrow c_j\in B
pIntroduction passage · 34 passages
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Introduction passage · core_chapters.tex, line 559
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
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J_h(p)
P_jIntroduction passage · 4 passages
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Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
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P_j\subseteq B\Rightarrow c_j\in B
\operatorname{Post}Introduction passage · 11 passages
Search aliases: Post operatorname
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
\mathcal QIntroduction passage · 1 passage
Search aliases: Q mathcal
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
qIntroduction passage · 21 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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q:X\to Y
\mathcal Q_hIntroduction passage · 1 passage
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Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
rIntroduction passage · 25 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
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r:X\to H
\rmIntroduction passage · 10 passages
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Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
\mathrm{run}Introduction passage · 1 passage
Search aliases: run mathrm
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
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m\in\{\mathrm{run},\mathrm{end}\}
SIntroduction passage · 36 passages
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Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
TIntroduction passage · 83 passages
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Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
T_*Introduction passage · 30 passages
Search aliases: T_*
Introduction passage · ending_chapter.tex, line 21
T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_*
T_0Introduction passage · 3 passages
Search aliases: T_0
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
T_kIntroduction passage · 10 passages
Search aliases: T_k
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
T_{k+1}Introduction passage · 4 passages
Search aliases: T_k+1
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
uIntroduction passage · 46 passages
Search aliases: u
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
u_hIntroduction passage · 8 passages
Search aliases: u_h
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
u_{hao}Introduction passage · 1 passage
Search aliases: u_hao
Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
VIntroduction passage · 50 passages
Search aliases: V
Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
B\subseteq V
wIntroduction passage · 29 passages
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Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
XIntroduction passage · 46 passages
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Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
X,Z
xIntroduction passage · 82 passages
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Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
A^X_m(x)
YIntroduction passage · 13 passages
Search aliases: Y
Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
q:X\to Y
ZIntroduction passage · 53 passages
Search aliases: Z
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
X,Z
zIntroduction passage · 68 passages
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Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
A^Z_m(z)
\alphaIntroduction passage · 6 passages
Search aliases: α alpha
Introduction passage · composition_chapter.tex, line 70
horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that action. Thus π_*u_hao =π_*(α(amid h)K_a,ou_h) =α(amid h)bar K_a,oπ_*u_h =bar u_hao. Here multiplication uses only enabled columns. Values in all other …
Exact mathematical passage
\pi_*u_{hao} =\pi_*\bigl(\alpha(a\mid h)K_{a,o}u_h\bigr) =\alpha(a\mid h)\bar K_{a,o}\pi_*u_h =\bar u_{hao}.
\piIntroduction passage · 47 passages
Search aliases: π pi
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
\pi_*Introduction passage · 8 passages
Search aliases: π_* pi
Introduction passage · composition_chapter.tex, line 69
cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives th…
Exact mathematical passage
\bar u_h=\pi_*u_h
\rhoIntroduction passage · 13 passages
Search aliases: ρ rho
Introduction passage · ending_chapter.tex, line 21
T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
\rho(z)
\ellIntroduction passage · 18 passages
Search aliases: ℓ ell
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\ell_dIntroduction passage · 3 passages
Search aliases: ℓ_d ell
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\toIntroduction passage · 45 passages
Search aliases: → to
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
\pi:X\to Z
\RightarrowIntroduction passage · 1 passage
Search aliases: ⇒ Rightarrow
Introduction passage · proof_fields_chapter.tex, line 14
A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.
Exact mathematical passage
P_j\subseteq B\Rightarrow c_j\in B
\inIntroduction passage · 52 passages
Search aliases: ∈ in
Introduction passage · control_transport_chapter.tex, line 8
The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need not denote the same transition.
Exact mathematical passage
m\in\{\mathrm{run},\mathrm{end}\}
\sumIntroduction passage · 20 passages
Search aliases: ∑ sum
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\sum_{o:\bar z_{a,o}(q)>0}Introduction passage · 1 passage
Search aliases: ∑_o:bar z_a,o(q)>0 sum bar
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\cupIntroduction passage · 8 passages
Search aliases: ∪ cup
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
\subseteqIntroduction passage · 45 passages
Search aliases: ⊆ subseteq
Introduction passage · ending_chapter.tex, line 16
The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\left\{z\in S:\text{some }a\in A_{\rm end}(z) \text{ satisfies }\operatorname{Post}(z,a)\subseteq T_k\right\}.
\cdotsIntroduction passage · 2 passages
Search aliases: ⋯ cdots
Introduction passage · core_chapters.tex, line 45
One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
\langleIntroduction passage · 2 passages
Search aliases: ⟨ langle
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\rangleIntroduction passage · 2 passages
Search aliases: ⟩ rangle
Introduction passage · core_chapters.tex, line 560
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · core_chapters.tex, line 346
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\mathbf1^\topUse in source · 1 passage
Search aliases: 1^top mathbf top
Use in source · core_chapters.tex, line 346
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\mathbf1_X^\topUse in source · 1 passage
Search aliases: 1_X^top mathbf top
Use in source · core_chapters.tex, line 370
Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point masses finish that direction.
Exact mathematical passage
\mathbf1_Z^\top C=\mathbf1_X^\top
\mathbf1_Z^\topUse in source · 1 passage
Search aliases: 1_Z^top mathbf top
Use in source · core_chapters.tex, line 370
Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point masses finish that direction.
Exact mathematical passage
\mathbf1_Z^\top C=\mathbf1_X^\top
a'Use in source · 1 passage
Search aliases: a'
Use in source · core_chapters.tex, line 711
After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.
Exact mathematical passage
a'
a_*Use in source · 2 passages
Search aliases: a_*
Use in source · core_chapters.tex, line 450
For a true minimizer a_*, R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. The loss range supplies the cap one. Finally, for any other a, subtract the two error bounds to obtain R(a)-R(widehat a)≥widehat R(a)-widehat R(widehat a)-2ξ>0. source[GA], ``A common quantitative grounding certificate'' and [MU], ``A joint posterior and recipient decision certificate.'' This is a specialization to a common finite candidate set and bounded loss.
Exact mathematical passage
a_*
A_1Use in source · 1 passage
Search aliases: A_1
Use in source · core_chapters.tex, line 775
What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.
Exact mathematical passage
Av=(A_1v,\ldots,A_mv)
A_AUse in source · 1 passage
Search aliases: A_A
Use in source · core_chapters.tex, line 785
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
A_A(b,s)=b+s
A_aUse in source · 5 passages
Search aliases: A_a
Use in source · core_chapters.tex, line 132
ne quotient for several possible dynamics A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
A_a=T_a^{\theta_0}
A_BUse in source · 1 passage
Search aliases: A_B
Use in source · core_chapters.tex, line 785
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
A_B(b,s)=s
a_dUse in source · 1 passage
Search aliases: a_d
Use in source · core_chapters.tex, line 709
Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined by those components and parent claims. External assumptions are included explicitly as premises.
Exact mathematical passage
a=(a_1,\ldots,a_d)
A_iUse in source · 2 passages
Search aliases: A_i
Use in source · core_chapters.tex, line 775
What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.
Exact mathematical passage
A_i v
a_iUse in source · 1 passage
Search aliases: a_i
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
a_{i-1}Use in source · 1 passage
Search aliases: a_i-1
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
A_jUse in source · 3 passages
Search aliases: A_j
Use in source · core_chapters.tex, line 753
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
A_j
A_mUse in source · 1 passage
Search aliases: A_m
Use in source · core_chapters.tex, line 775
What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.
Exact mathematical passage
Av=(A_1v,\ldots,A_mv)
A_{\rm run}Use in source · 3 passages
Search aliases: A_rm run rm
Use in source · core_chapters.tex, line 816
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
A_tUse in source · 1 passage
Search aliases: A_t
Use in source · core_chapters.tex, line 340
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
a_tUse in source · 2 passages
Search aliases: a_t
Use in source · composition_chapter.tex, line 42
Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include it in the observation interface of both models.
Exact mathematical passage
h=(a_1,o_1,\ldots,a_t,o_t).
A_wUse in source · 3 passages
Search aliases: A_w
Use in source · core_chapters.tex, line 135
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
E=\Span\{A_wD_a^\theta v: v\in V,\ a,\theta,\ w\text{ finite}\}.
bUse in source · 47 passages
Search aliases: b
Use in source · composition_chapter.tex, line 100
An exact sensor realization. Return to x=(b,s,n)∈{0,1}^3 and retain z=(b,s). The actions hold and flip are available everywhere. A repair action is available only where b=0 and leaves that target value fixed. The nuisance bit is retained by every fine transition but ignored by the coarse state.
Exact mathematical passage
x=(b,s,n)\in\{0,1\}^3
\bar DUse in source · 4 passages
Search aliases: bar D bar
Use in source · composition_chapter.tex, line 19
For a nonnegative vector u on X, write (π_*u)(z)=∑_π x=zu(x). This notation applies to probability distributions and to unnormalized history masses. A coarse model has labels barℓ, enabled sets bar D, and joint instruments bar K on its enabled actions.
Exact mathematical passage
\bar D
\bar FUse in source · 5 passages
Search aliases: bar F bar
Use in source · core_chapters.tex, line 58
[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its readout is bar q([x])=q(x). If two initial records are equal, a deterministic predictor has the same initialized state. Applying t…
Exact mathematical passage
\bar F_a([x])=[F_ax]
\bar F_aUse in source · 4 passages
Search aliases: bar F_a bar
Use in source · core_chapters.tex, line 58
[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its readout is bar q([x])=q(x). If two initial records are equal, a deterministic predictor has the same initialized state. Applying t…
Exact mathematical passage
\bar F_a([x])=[F_ax]
\bar K_a^\topUse in source · 1 passage
Search aliases: bar K_a^top bar top
Use in source · core_chapters.tex, line 320
nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse transition and yield C_π K_a^top=bar K_a^top C_π, first on point masses and then by linearity. Likelihoods are constant within final blocks. Grouping the …
Exact mathematical passage
C_\pi K_a^\top=\bar K_a^\top C_\pi
\bar K_{a_i,o_i}Use in source · 1 passage
Search aliases: bar K_a_i,o_i bar
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
\bar pUse in source · 2 passages
Search aliases: bar p bar
Use in source · composition_chapter.tex, line 47
Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is empty, no policy can make another admitted action at that history. A stopping convention can be included as a shared terminal action, but is not silen…
Exact mathematical passage
\bar p_0=\pi_*p_0
\bar p_0Use in source · 2 passages
Search aliases: bar p_0 bar
Use in source · composition_chapter.tex, line 47
Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is empty, no policy can make another admitted action at that history. A stopping convention can be included as a shared terminal action, but is not silen…
Exact mathematical passage
\bar p_0=\pi_*p_0
\bar QUse in source · 2 passages
Search aliases: bar Q bar
Use in source · core_chapters.tex, line 145
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For ev…
Exact mathematical passage
\bar Q:Z\to Y
\bar qUse in source · 6 passages
Search aliases: bar q bar
Use in source · core_chapters.tex, line 58
havioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its readout is bar q([x])=q(x). If two initial records are equal, a deterministic predictor has the same initialized state. Applying the same subsequent opera…
Exact mathematical passage
\bar q([x])=q(x)
\bar u_\varnothingUse in source · 1 passage
Search aliases: bar u_∅ bar varnothing
Use in source · core_chapters.tex, line 372
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
\bar u_\varnothing=Cu_\varnothing
\bar\ellUse in source · 6 passages
Search aliases: barℓ bar ell
Use in source · composition_chapter.tex, line 19
For a nonnegative vector u on X, write (π_*u)(z)=∑_π x=zu(x). This notation applies to probability distributions and to unnormalized history masses. A coarse model has labels barℓ, enabled sets bar D, and joint instruments bar K on its enabled actions.
Exact mathematical passage
\bar\ell
\bigwedgeUse in source · 1 passage
Search aliases: bigwedge
Use in source · proof_fields_chapter.tex, line 85
Logical interpretation is an additional map. Let μ(v) be the proposition represented by claim address v. Suppose every admitted leaf and positive observation is true under μ, and each admitted rule satisfies (bigwedge_p∈ P_jμ(p))Longrightarrowμ(c_j). Then every node in C is true under μ. Indeed, induction over the closure stages propagates truth from the seeds through the admitted implications.
Exact mathematical passage
\left(\bigwedge_{p\in P_j}\mu(p)\right)\Longrightarrow\mu(c_j).
\bigwedge_{p\in P_j}Use in source · 1 passage
Search aliases: bigwedge_p∈ P_j bigwedge in
Use in source · proof_fields_chapter.tex, line 85
Logical interpretation is an additional map. Let μ(v) be the proposition represented by claim address v. Suppose every admitted leaf and positive observation is true under μ, and each admitted rule satisfies (bigwedge_p∈ P_jμ(p))Longrightarrowμ(c_j). Then every node in C is true under μ. Indeed, induction over the closure stages propagates truth from the seeds through the admitted implications.
Exact mathematical passage
\left(\bigwedge_{p\in P_j}\mu(p)\right)\Longrightarrow\mu(c_j).
\bmodUse in source · 1 passage
Search aliases: bmod
Use in source · core_chapters.tex, line 29
An agent given only y can predict every future display under repeated flips: after k flips it is y⊕(kbmod2). It cannot determine the current physical bit. Worlds (0,0,0) and (1,1,0) share display zero and have different targets. The obstruction is the fibre of the available record, not a shortage of computational effort.
Exact mathematical passage
y\oplus(k\bmod2)
\mathcal CUse in source · 7 passages
Search aliases: C mathcal
Use in source · core_chapters.tex, line 465
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\mathcal C
c_0Use in source · 2 passages
Search aliases: c_0
Use in source · core_chapters.tex, line 485
[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0. In the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.
Exact mathematical passage
c_0
C_1Use in source · 1 passage
Search aliases: C_1
Use in source · core_chapters.tex, line 236
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
C_1,\ldots,C_m
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · core_chapters.tex, line 485
[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0. In the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.
Exact mathematical passage
c_1\ne c_0
c_eUse in source · 2 passages
Search aliases: c_e
Use in source · core_chapters.tex, line 494
tion, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
c_e
\mathcal C_hUse in source · 5 passages
Search aliases: C_h mathcal
Use in source · core_chapters.tex, line 468
Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).
Exact mathematical passage
\mathcal C_h=\bigcap_{x:\alpha(x)=h}\mathcal C(x).
C_iUse in source · 1 passage
Search aliases: C_i
Use in source · core_chapters.tex, line 237
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
C_mUse in source · 1 passage
Search aliases: C_m
Use in source · core_chapters.tex, line 236
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
C_1,\ldots,C_m
c_{\max}Use in source · 2 passages
Search aliases: c_max max
Use in source · core_chapters.tex, line 612
ng must not share one unqualified guarantee. Fix a finite horizon H and one nonempty finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and at most H charges bounded by c_max give B=1+Hc_max.
Exact mathematical passage
c_{\max}
C_\piUse in source · 2 passages
Search aliases: C_π pi
Use in source · core_chapters.tex, line 308
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
C_\pi(z,x)=1_{\{\pi(x)=z\}}
\operatorname{CPre}Use in source · 4 passages
Search aliases: CPre operatorname
Use in source · control_transport_chapter.tex, line 17
For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.
Exact mathematical passage
\operatorname{CPre}^X_m(B) =\{x:\exists a\in A^X_m(x),\ \operatorname{Post}_X(x,a)\subseteq B\}.
\operatorname{CPre}^X_mUse in source · 2 passages
Search aliases: CPre^X_m operatorname
Use in source · control_transport_chapter.tex, line 17
For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.
Exact mathematical passage
\operatorname{CPre}^X_m(B) =\{x:\exists a\in A^X_m(x),\ \operatorname{Post}_X(x,a)\subseteq B\}.
\operatorname{CPre}^X_{\rm end}Use in source · 1 passage
Search aliases: CPre^X_rm end operatorname rm
Use in source · control_transport_chapter.tex, line 65
licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X_k+1 · =T^X_k∪(S_X∩CPre^X_rm end(T^X_k)) · =π^-1 (T^Z_k∪ (S_Z∩CPre^Z_rm end(T^Z_k))). The finite increasing sequences stabilize, so their stable sets …
Exact mathematical passage
T^X_{k+1} &=T^X_k\cup\bigl(S_X\cap\operatorname{CPre}^X_{\rm end}(T^X_k)\bigr)\\ &=\pi^{-1}\!\left(T^Z_k\cup (S_Z\cap\operatorname{CPre}^Z_{\rm end}(T^Z_k))\right).
\operatorname{CPre}^Z_mUse in source · 2 passages
Search aliases: CPre^Z_m operatorname
Use in source · control_transport_chapter.tex, line 21
For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.
Exact mathematical passage
\operatorname{CPre}^Z_m
\operatorname{CPre}^Z_{\rm end}Use in source · 1 passage
Search aliases: CPre^Z_rm end operatorname rm
Use in source · control_transport_chapter.tex, line 65
licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X_k+1 · =T^X_k∪(S_X∩CPre^X_rm end(T^X_k)) · =π^-1 (T^Z_k∪ (S_Z∩CPre^Z_rm end(T^Z_k))). The finite increasing sequences stabilize, so their stable sets …
Exact mathematical passage
T^X_{k+1} &=T^X_k\cup\bigl(S_X\cap\operatorname{CPre}^X_{\rm end}(T^X_k)\bigr)\\ &=\pi^{-1}\!\left(T^Z_k\cup (S_Z\cap\operatorname{CPre}^Z_{\rm end}(T^Z_k))\right).
d_0Use in source · 3 passages
Search aliases: d_0
Use in source · core_chapters.tex, line 430
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\TV(p,\widehat p)\le e,\qquad \sup_\omega d(v_\omega,\widehat v_\omega)\le d_0.
D_a^\thetaUse in source · 5 passages
Search aliases: D_a^θ theta
Use in source · core_chapters.tex, line 134
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
D_a^\theta=T_a^\theta-A_a
D_eUse in source · 4 passages
Search aliases: D_e
Use in source · core_chapters.tex, line 494
One investigation, one resource budget. questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
D_e\subseteq X
D_hUse in source · 1 passage
Search aliases: D_h
Use in source · composition_chapter.tex, line 48
Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is empty, no policy can make another admitted action at that history. A stopping convention can be included as a shared terminal action, but is not silen…
Exact mathematical passage
D_h=\bigcap_{x:u_h(x)>0}D(x).
d_HUse in source · 11 passages
Search aliases: d_H
Use in source · core_chapters.tex, line 392
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p…
Exact mathematical passage
\TV(P_H,\widetilde P_H)\le d_H:=\min\left\{1,\delta_0+\sum_{t=0}^{H-1}\epsilon_t\right\}.
D_tUse in source · 2 passages
Search aliases: D_t
Use in source · core_chapters.tex, line 237
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
D_vUse in source · 1 passage
Search aliases: D_v
Use in source · core_chapters.tex, line 709
Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined by those components and parent claims. External assumptions are included explicitly as premises.
Exact mathematical passage
D_v
EUse in source · 37 passages
Search aliases: E
Use in source · core_chapters.tex, line 135
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
E=\Span\{A_wD_a^\theta v: v\in V,\ a,\theta,\ w\text{ finite}\}.
\mathcal EUse in source · 5 passages
Search aliases: E mathcal
Use in source · book.tex, line 99
E,Σ · An evidence family and its correction signature.
Exact mathematical passage
\mathcal E,\Sigma
E_0Use in source · 3 passages
Search aliases: E_0
Use in source · core_chapters.tex, line 222
[Minimal exact memory for a correction contract] Equality of Σ is a congruence for every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct signatures. Thus the image of Σ has the least possible cardinality for this contract. If Σ(V)=Σ(W), then for each E, t((V∩ E_0)∩ E)=t(V∩(E_0∩ E))=t(W∩(E_…
Exact mathematical passage
V\mapsto V\cap E_0
\operatorname{ev}Use in source · 4 passages
Search aliases: ev operatorname
Use in source · realization_prologue.tex, line 18
The evaluation ev_v:Γ(F)→ F_v records the information visible in one view. When Γ(F) is nonempty and a target q:Γ(F)→ Y is specified, that view is sufficient exactly when the target is constant on its fibres. [Target factorization through a realization view] Suppose Γ(F) and Y are nonempty. There is a function d:ev_v(Γ(F))→ Y such that q=d∘ev_v if and only if ev_v(x)=ev_v(x') Longrightarrow q(x)=q(x') for all coherent realizations x,x'∈Γ(F). A function of one attained record must assign the same…
Exact mathematical passage
\operatorname{ev}_v:\Gamma(F)\to F_v
\operatorname{ev}_vUse in source · 4 passages
Search aliases: ev_v operatorname
Use in source · realization_prologue.tex, line 18
The evaluation ev_v:Γ(F)→ F_v records the information visible in one view. When Γ(F) is nonempty and a target q:Γ(F)→ Y is specified, that view is sufficient exactly when the target is constant on its fibres. [Target factorization through a realization view] Suppose Γ(F) and Y are nonempty. There is a function d:ev_v(Γ(F))→ Y such that q=d∘ev_v if and only if ev_v(x)=ev_v(x') Longrightarrow q(x)=q(x') for all coherent realizations x,x'∈Γ(F). A function of one attained record must assign the same…
Exact mathematical passage
\operatorname{ev}_v:\Gamma(F)\to F_v
\mathbb FUse in source · 1 passage
Search aliases: F mathbb
Use in source · core_chapters.tex, line 785
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
\mathbb F_2
fUse in source · 9 passages
Search aliases: f
Use in source · core_chapters.tex, line 27
The sensor used throughout the book. Write a world as ω=(b,s,n)∈{0,1}^3. The physical target bit is b. The sensor convention is s. The nuisance bit n never affects the operations or outputs of the initial contract. The display is y=b⊕ s, where ⊕ is addition modulo two. A permitted control f flips the physical bit and leaves s,n unchanged. Consequently the next display is 1-y.
Exact mathematical passage
f
f^*Use in source · 3 passages
Search aliases: f^*
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
f^*:F_w\to F_v
\mathbb F_2Use in source · 1 passage
Search aliases: F_2 mathbb
Use in source · core_chapters.tex, line 785
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
\mathbb F_2
f_eUse in source · 2 passages
Search aliases: f_e
Use in source · core_chapters.tex, line 494
, one resource budget questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
f_e:D_e\to X
F_vUse in source · 6 passages
Search aliases: F_v
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
F_v
\mathcal GUse in source · 1 passage
Search aliases: G mathcal
Use in source · core_chapters.tex, line 641
Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that agrees everywhere or returns a context where it disagrees. The teacher's work is part of the development account.
Exact mathematical passage
\mathcal G=(p_0,\ldots,p_{N-1})
gUse in source · 17 passages
Search aliases: g
Use in source · control_transport_chapter.tex, line 131
Why a small probability error is not a sure guarantee. Take three states s,g,b, with safe set {s,g} and goal {g}. Running holds s or g in place. The nominal ending action at s moves to g with probability one. The perturbed action instead moves to g with probability 1-ϵ and to unsafe state b with probability ϵ, where 0<ϵ<1. A hold action is available at g.
Exact mathematical passage
s,g,b
g^*Use in source · 1 passage
Search aliases: g^*
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
(g\circ f)^*=f^*\circ g^*
G_aUse in source · 3 passages
Search aliases: G_a
Use in source · core_chapters.tex, line 145
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For ev…
Exact mathematical passage
G_a:Z\to Z
g_aUse in source · 2 passages
Search aliases: g_a
Use in source · core_chapters.tex, line 502
The experiment catalogue and the terminal-artifact catalogue are finite, and the resource vector has a fixed finite dimension. A terminal artifact a is acceptable only if every fixed recipient succeeds in every state of V. Its worst-case terminal cost is g_a(V), computed componentwise. A controller is a literal finite tree with one record per experiment or terminal node. Stored memory counts all branches, including those not visited in one execution.
Exact mathematical passage
g_a(V)
G_XUse in source · 1 passage
Search aliases: G_X
Use in source · control_transport_chapter.tex, line 11
Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.
Exact mathematical passage
S_X=\pi^{-1}(S_Z),\qquad G_X=\pi^{-1}(G_Z),\qquad G_Z\subseteq S_Z.
G_ZUse in source · 1 passage
Search aliases: G_Z
Use in source · control_transport_chapter.tex, line 11
Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.
Exact mathematical passage
S_X=\pi^{-1}(S_Z),\qquad G_X=\pi^{-1}(G_Z),\qquad G_Z\subseteq S_Z.
G_{\tau(a)}Use in source · 4 passages
Search aliases: G_τ(a) tau
Use in source · core_chapters.tex, line 109
A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.
Exact mathematical passage
G_{\tau(a)}
G_{\tau(w)}Use in source · 2 passages
Search aliases: G_τ(w) tau
Use in source · core_chapters.tex, line 117
[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ(w)T. Induction proves the first claim. Composition with the matched readout proves the second. source[MU], Theorem 6.1. The same local-to-continuatio…
Exact mathematical passage
TF_w=G_{\tau(w)}T,\qquad qF_w=\bar qG_{\tau(w)}T.
\mathcal HUse in source · 2 passages
Search aliases: H mathcal
Use in source · core_chapters.tex, line 753
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\mathcal H_{j-1}
h'Use in source · 2 passages
Search aliases: h'
Use in source · migration_chapter.tex, line 16
[Exact migration criterion] There is a map J:U_0→ V_0 satisfying JM=N if and only if M(h)=M(h') Longrightarrow N(h)=N(h') (h,h'∈ H). When it exists, J is unique on attained old records. Necessity follows by applying J to the equal old records. Conversely, define J(M(h))=N(h). The fibre condition makes the definition independent of the representative. Every u∈ U_0 has such a representative, so the same equation determines J uniquely.
Exact mathematical passage
M(h)=M(h')\quad\Longrightarrow\quad N(h)=N(h') \qquad(h,h'\in H).
\mathcal H_{j-1}Use in source · 2 passages
Search aliases: H_j-1 mathcal
Use in source · core_chapters.tex, line 753
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\mathcal H_{j-1}
IUse in source · 13 passages
Search aliases: I
Use in source · core_chapters.tex, line 82
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
SA=I_V
iUse in source · 26 passages
Search aliases: i
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
i_1Use in source · 3 passages
Search aliases: i_1
Use in source · core_chapters.tex, line 545
Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of the frontier, leaving that pair as the whole nondominated frontier. source[SII], Theorem 11.1. The assumptions exclude destructive resets and repairs …
Exact mathematical passage
i_1,i_2
i_2Use in source · 3 passages
Search aliases: i_2
Use in source · core_chapters.tex, line 545
Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of the frontier, leaving that pair as the whole nondominated frontier. source[SII], Theorem 11.1. The assumptions exclude destructive resets and repairs …
Exact mathematical passage
i_1,i_2
I_UUse in source · 1 passage
Search aliases: I_U
Use in source · core_chapters.tex, line 122
[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ(w)T. Induction proves the first claim. Composition with the matched readout proves the second. source[MU], Theorem 6.1. The same local-to-continuatio…
Exact mathematical passage
TI_Z=I_UT
I_VUse in source · 4 passages
Search aliases: I_V
Use in source · core_chapters.tex, line 82
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
SA=I_V
I_ZUse in source · 4 passages
Search aliases: I_Z
Use in source · core_chapters.tex, line 86
be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
I_Z
\mathrm{id}Use in source · 1 passage
Search aliases: id mathrm
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
(1_v)^*=\mathrm{id}
\imUse in source · 1 passage
Search aliases: im
Use in source · core_chapters.tex, line 781
:reconstruction There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then v-w∈ker A⊆ker Q, so this definition is independent of the representative. It is linear and extends to the full record space by extending a basis. For…
Exact mathematical passage
\im A
j_1Use in source · 1 passage
Search aliases: j_1
Use in source · proof_fields_chapter.tex, line 31
Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.
Exact mathematical passage
\sigma=(j_1,\ldots,j_m)
j_mUse in source · 1 passage
Search aliases: j_m
Use in source · proof_fields_chapter.tex, line 31
Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.
Exact mathematical passage
\sigma=(j_1,\ldots,j_m)
\mathcal KUse in source · 8 passages
Search aliases: K mathcal
Use in source · core_chapters.tex, line 172
Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state need not itself be valid.
Exact mathematical passage
\mathcal K
K'Use in source · 2 passages
Search aliases: K'
Use in source · proof_fields_chapter.tex, line 36
[Replay is sound and compositional] If replay accepts a trace from a set K⊆ C and returns K', then K⊆ K'⊆ C. For either fixed replay mode, replay of a concatenated trace is replay of its first part followed by replay of its second part from the returned set. Failure in either part is failure of the concatenated replay. The empty trace changes nothing. At a successful next step, all premises of an admitted rule already lie in C. Closedness of C places its conclusion in C. Adjoining it preserves t…
Exact mathematical passage
K'
k_*Use in source · 5 passages
Search aliases: k_*
Use in source · migration_chapter.tex, line 64
[Minimum fixed-length supplementary information] For u∈ U_0, put k(u)=|N(M^-1{u})|, k_* =max_u∈ U_0k(u). Suppose an auxiliary encoder may inspect the complete source history and sends one fixed-length binary word in addition to the old record. Exact recovery of N is possible with b supplementary bits if and only if 2^b≥ k_*. Thus the minimum is lceillog_2 k_*rceil. For a fixed old record u, distinct required new values must receive distinct supplementary words. Otherwise the decoder would receiv…
Exact mathematical passage
k(u)=|N(M^{-1}\{u\})|,\qquad k_* =\max_{u\in U_0}k(u).
K_0Use in source · 1 passage
Search aliases: K_0
Use in source · proof_fields_chapter.tex, line 31
Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.
Exact mathematical passage
K_0=A\cup S
K_aUse in source · 2 passages
Search aliases: K_a
Use in source · core_chapters.tex, line 308
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
K_a(x,x')
K_a^\topUse in source · 2 passages
Search aliases: K_a^top top
Use in source · core_chapters.tex, line 308
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
p\mapsto K_a^\top p
\kerUse in source · 6 passages
Search aliases: ker
Use in source · core_chapters.tex, line 144
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For ev…
Exact mathematical passage
E\subseteq\ker Q
LUse in source · 45 passages
Search aliases: L
Use in source · book.tex, line 100
T_*,L_*,ρ · Ending region, safe-running kernel, and first-entry rank.
Exact mathematical passage
T_*,L_*,\rho
lUse in source · 4 passages
Search aliases: l
Use in source · core_chapters.tex, line 545
Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of the frontier, leaving that pair as the whole nondominated frontier. source[SII], Theorem 11.1. The assumptions exclude destructive resets and repairs …
Exact mathematical passage
l
L^X_*Use in source · 4 passages
Search aliases: L^X_*
Use in source · control_transport_chapter.tex, line 51
[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine policy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coo…
Exact mathematical passage
T^X_k=\pi^{-1}(T^Z_k),\qquad T^X_*=\pi^{-1}(T^Z_*),\qquad L^X_k=\pi^{-1}(L^Z_k),\qquad L^X_*=\pi^{-1}(L^Z_*)
L^X_kUse in source · 2 passages
Search aliases: L^X_k
Use in source · control_transport_chapter.tex, line 47
Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.
Exact mathematical passage
T^X_k,L^X_k
L^Z_*Use in source · 6 passages
Search aliases: L^Z_*
Use in source · control_transport_chapter.tex, line 51
[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine policy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coo…
Exact mathematical passage
T^X_k=\pi^{-1}(T^Z_k),\qquad T^X_*=\pi^{-1}(T^Z_*),\qquad L^X_k=\pi^{-1}(L^Z_k),\qquad L^X_*=\pi^{-1}(L^Z_*)
L^Z_kUse in source · 2 passages
Search aliases: L^Z_k
Use in source · control_transport_chapter.tex, line 47
Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.
Exact mathematical passage
T^Z_k,L^Z_k
L_*Use in source · 19 passages
Search aliases: L_*
Use in source · book.tex, line 100
T_*,L_*,ρ · Ending region, safe-running kernel, and first-entry rank.
Exact mathematical passage
T_*,L_*,\rho
L_0Use in source · 2 passages
Search aliases: L_0
Use in source · core_chapters.tex, line 816
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
L_iUse in source · 1 passage
Search aliases: L_i
Use in source · core_chapters.tex, line 465
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
L_i
L_kUse in source · 9 passages
Search aliases: L_k
Use in source · core_chapters.tex, line 816
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
L_{k+1}Use in source · 4 passages
Search aliases: L_k+1
Use in source · core_chapters.tex, line 816
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
L_mUse in source · 5 passages
Search aliases: L_m
Use in source · core_chapters.tex, line 265
Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability…
Exact mathematical passage
L_m
\lceilUse in source · 3 passages
Search aliases: lceil
Use in source · core_chapters.tex, line 540
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\logUse in source · 3 passages
Search aliases: log
Use in source · core_chapters.tex, line 540
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\log_2Use in source · 3 passages
Search aliases: log_2 log
Use in source · core_chapters.tex, line 540
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\LongleftrightarrowUse in source · 2 passages
Search aliases: Longleftrightarrow
Use in source · control_transport_chapter.tex, line 38
π(Post_X(x,a)) · =Post_Z(π x,a) (a∈ A^X_m(x)). Then for every B⊆ Z, CPre^X_m(π^-1B) =π^-1(CPre^Z_m(B)). For an admitted action, Post_X(x,a)⊆π^-1B Longleftrightarrow π(Post_X(x,a))⊆ B. Replace the projected successor set by its assumed coarse value. The action ranges in the two existential quantifiers are identical, so existence of a witnessing action is equivalent on both sides.
Exact mathematical passage
\operatorname{Post}_X(x,a)\subseteq\pi^{-1}B \quad\Longleftrightarrow\quad \pi(\operatorname{Post}_X(x,a))\subseteq B.
\LongrightarrowUse in source · 8 passages
Search aliases: Longrightarrow
Use in source · core_chapters.tex, line 7
[A record determines a target exactly on its fibres] There exists d:r(Ω)→ Y with d(r(ω))=t(ω) for every world if and only if r(ω)=r(ω') Longrightarrow t(ω)=t(ω'). Necessity follows because a function assigns one value to one input. For sufficiency, define d(h) to be t(ω) for any world with r(ω)=h. Such a world exists for attained h, and the hypothesis makes the choice immaterial. The resulting decoder satisfies the required equation. sourceGrounded Abstraction [GA], Proposition 3.2 and Theorem 3…
Exact mathematical passage
r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
MUse in source · 23 passages
Search aliases: M
Use in source · book.tex, line 101
M,N,J · Old record, required new record, and migration map.
Exact mathematical passage
M,N,J
\mathcal MUse in source · 1 passage
Search aliases: M mathcal
Use in source · core_chapters.tex, line 265
Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability…
Exact mathematical passage
\mathcal M
M^{-1}Use in source · 1 passage
Search aliases: M^-1
Use in source · migration_chapter.tex, line 64
[Minimum fixed-length supplementary information] For u∈ U_0, put k(u)=|N(M^-1{u})|, k_* =max_u∈ U_0k(u). Suppose an auxiliary encoder may inspect the complete source history and sends one fixed-length binary word in addition to the old record. Exact recovery of N is possible with b supplementary bits if and only if 2^b≥ k_*. Thus the minimum is lceillog_2 k_*rceil. For a fixed old record u, distinct required new values must receive distinct supplementary words. Otherwise the decoder would receiv…
Exact mathematical passage
k(u)=|N(M^{-1}\{u\})|,\qquad k_* =\max_{u\in U_0}k(u).
m_oUse in source · 2 passages
Search aliases: m_o
Use in source · core_chapters.tex, line 509
athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it is no larger in every resou…
Exact mathematical passage
(v_o,m_o)
\maxUse in source · 8 passages
Search aliases: max
Use in source · core_chapters.tex, line 396
e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either event probability exceeds d_H, the other is necessarily pos…
Exact mathematical passage
\TV\bigl(P_H(Z_H\in\cdot\mid E),\widetilde P_H(Z_H\in\cdot\mid E)\bigr) \le\min\left\{1,\frac{d_H}{\max(p_E,\widetilde p_E)}\right\}.
\max_oUse in source · 1 passage
Search aliases: max_o max
Use in source · core_chapters.tex, line 510
m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it is no larger in every resource coordinate and strictly smaller in at l…
Exact mathematical passage
\left(c_e+\max_o v_o,\quad1+\sum_o m_o\right).
\max_{u\in U_0}Use in source · 1 passage
Search aliases: max_u∈ U_0 max in
Use in source · migration_chapter.tex, line 64
[Minimum fixed-length supplementary information] For u∈ U_0, put k(u)=|N(M^-1{u})|, k_* =max_u∈ U_0k(u). Suppose an auxiliary encoder may inspect the complete source history and sends one fixed-length binary word in addition to the old record. Exact recovery of N is possible with b supplementary bits if and only if 2^b≥ k_*. Thus the minimum is lceillog_2 k_*rceil. For a fixed old record u, distinct required new values must receive distinct supplementary words. Otherwise the decoder would receiv…
Exact mathematical passage
k(u)=|N(M^{-1}\{u\})|,\qquad k_* =\max_{u\in U_0}k(u).
\min_aUse in source · 1 passage
Search aliases: min_a min
Use in source · core_chapters.tex, line 442
[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add and subtract ∑_ω p(ω)ℓ(a,widehat v_ω). The response term has absolute value at most Kd_0. The expectation of a function in [0,1] changes by at most tot…
Exact mathematical passage
R(\widehat a)-\min_a R(a)\le\min\{1,2\xi\}.
\min_{w\in W}Use in source · 1 passage
Search aliases: min_w∈ W min in
Use in source · core_chapters.tex, line 740
[Which gains survive every admitted price?] The saving w·Δ is nonnegative for every nonnegative price vector if and only if every coordinate of Δ is nonnegative. If prices lie in the convex hull of a finite nonempty set W, their minimum saving equals min_w∈ Ww·Δ. A componentwise nonnegative vector has nonnegative inner product with every nonnegative vector. If coordinate j is negative, assigning price one only to that coordinate gives a negative result. This proves the first assertion. A point i…
Exact mathematical passage
\min_{w\in W}w\cdot\Delta
N'Use in source · 2 passages
Search aliases: N'
Use in source · migration_chapter.tex, line 110
Now require the nuisance bit as well. The new record N'=(b⊕ s,b,n) has four possible values inside each old fibre, giving a two-bit supplementary lower bound. Calibration alone no longer suffices. Two histories with equal b,s and different n collide under (M,E) but have different N'. The lower bound identifies a missing distinction rather than a computational bottleneck.
Exact mathematical passage
N'=(b\oplus s,b,n)
n_iUse in source · 2 passages
Search aliases: n_i
Use in source · core_chapters.tex, line 733
Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive measure. Storage capacity, peak memory, and elapsed time need their own models unless a justified conversion is specified.
Exact mathematical passage
n_i
OUse in source · 5 passages
Search aliases: O
Use in source · composition_chapter.tex, line 8
The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)=1 specify the joint distribution of the next state and report. No transition is specified for a disabled action. An implementation may fill unused en…
Exact mathematical passage
O
\mathcal OUse in source · 1 passage
Search aliases: O mathcal
Use in source · core_chapters.tex, line 265
Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability…
Exact mathematical passage
\mathcal O
o_1Use in source · 3 passages
Search aliases: o_1
Use in source · composition_chapter.tex, line 42
Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include it in the observation interface of both models.
Exact mathematical passage
h=(a_1,o_1,\ldots,a_t,o_t).
O_eUse in source · 1 passage
Search aliases: O_e
Use in source · core_chapters.tex, line 494
belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
o_e:D_e\to O_e
o_eUse in source · 2 passages
Search aliases: o_e
Use in source · core_chapters.tex, line 494
belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
o_e:D_e\to O_e
o_iUse in source · 1 passage
Search aliases: o_i
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
o_{i-1}Use in source · 1 passage
Search aliases: o_i-1
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
o_tUse in source · 2 passages
Search aliases: o_t
Use in source · composition_chapter.tex, line 42
Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include it in the observation interface of both models.
Exact mathematical passage
h=(a_1,o_1,\ldots,a_t,o_t).
O_{t+1}Use in source · 1 passage
Search aliases: O_t+1
Use in source · core_chapters.tex, line 340
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
\mathcal PUse in source · 1 passage
Search aliases: P mathcal
Use in source · core_chapters.tex, line 505
The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it …
Exact mathematical passage
\mathcal P_h(V)
\mathfrak PUse in source · 2 passages
Search aliases: P mathfrak
Use in source · core_chapters.tex, line 612
An approximate model supports a different, quantitative claim. Exact policy transport and approximate planning must not share one unqualified guarantee. Fix a finite horizon H and one nonempty finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and at most H charges bounded by c_max give B=…
Exact mathematical passage
\mathfrak P
P'Use in source · 2 passages
Search aliases: P'
Use in source · core_chapters.tex, line 296
The theorem permits removal of an independent nuisance flag. It does not permit removal of every association. Consider two priors on (b,s), with rows indexed by b and columns by s: P=1/2 · 0 0 · 1/2, P'=0 · 1/2 1/2 · 0. Both individual marginals are uniform. A noisy calibration reports s correctly with probability 3/4, independently conditional on the fixed world. The probability of report one is 1/2 under both priors. After that report, however, the target posteriors are 3/4 and 1/4. Separate t…
Exact mathematical passage
P=\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix},\qquad P'=\begin{pmatrix}0&1/2\\1/2&0\end{pmatrix}.
p_0Use in source · 5 passages
Search aliases: p_0
Use in source · composition_chapter.tex, line 47
Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is empty, no policy can make another admitted action at that history. A stopping convention can be included as a shared terminal action, but is not silen…
Exact mathematical passage
p_0
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · core_chapters.tex, line 175
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
P_1,\ldots,P_m
P_aUse in source · 4 passages
Search aliases: P_a
Use in source · migration_chapter.tex, line 79
Migration must commute with subsequent operations. Recovering one new record is not yet preserving a method. Let a common finite action alphabet act on H through total maps F_a:H→ H. Suppose both representations close under those same source operations: M F_a=P_a M, N F_a=Q_a N. The maps P_a and Q_a preserve the attained record sets. These equations refer to the same source action, not merely to operations sharing a name.
Exact mathematical passage
M F_a=P_a M,\qquad N F_a=Q_a N.
p_EUse in source · 7 passages
Search aliases: p_E
Use in source · core_chapters.tex, line 395
d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either event…
Exact mathematical passage
p_E,\widetilde p_E>0
P_HUse in source · 3 passages
Search aliases: P_H
Use in source · core_chapters.tex, line 391
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p…
Exact mathematical passage
P_H,\widetilde P_H
\mathcal P_hUse in source · 1 passage
Search aliases: P_h mathcal
Use in source · core_chapters.tex, line 505
The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it …
Exact mathematical passage
\mathcal P_h(V)
P_iUse in source · 5 passages
Search aliases: P_i
Use in source · core_chapters.tex, line 175
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
\sum_iP_i=I
P_mUse in source · 1 passage
Search aliases: P_m
Use in source · core_chapters.tex, line 175
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
P_1,\ldots,P_m
p_{N-1}Use in source · 1 passage
Search aliases: p_N-1
Use in source · core_chapters.tex, line 641
Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that agrees everywhere or returns a context where it disagrees. The teacher's work is part of the development account.
Exact mathematical passage
\mathcal G=(p_0,\ldots,p_{N-1})
P_{\rm av}Use in source · 1 passage
Search aliases: P_rm av rm
Use in source · core_chapters.tex, line 191
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero. Normalizing it as a belief is undefined. Clipping its negative entry and normalizing the resulting sum would instead produce (3/5,2/5)^top, a differe…
Exact mathematical passage
P_{\rm av}=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad P_{\rm det}=\tfrac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
P_{\rm det}Use in source · 1 passage
Search aliases: P_rm det rm
Use in source · core_chapters.tex, line 191
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero. Normalizing it as a belief is undefined. Clipping its negative entry and normalizing the resulting sum would instead produce (3/5,2/5)^top, a differe…
Exact mathematical passage
P_{\rm av}=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad P_{\rm det}=\tfrac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
P_wUse in source · 1 passage
Search aliases: P_w
Use in source · migration_chapter.tex, line 89
[A valid migration intertwines the common operations] If JM=N, then on U_0, JP_a=Q_aJ. Consequently JP_w=Q_wJ for every finite action word w, with the execution-order convention of Chapter~[source reference]. Any protected readout that factors through N agrees along the two migrated executions. Take u=M(h). Then JP_a(u)=JM(F_a h)=N(F_a h)=Q_aN(h)=Q_aJ(u). Every attained u has a representative, so the equality holds throughout U_0. Closure keeps later values within that domain. Induction on the w…
Exact mathematical passage
JP_w=Q_wJ
\operatorname{pa}Use in source · 1 passage
Search aliases: pa operatorname
Use in source · core_chapters.tex, line 709
Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined by those components and parent claims. External assumptions are included explicitly as premises.
Exact mathematical passage
\operatorname{pa}(v)
\operatorname{Pareto}Use in source · 1 passage
Search aliases: Pareto operatorname
Use in source · core_chapters.tex, line 506
The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it …
Exact mathematical passage
\operatorname{Pareto}\{(g_a(V),1):a\text{ succeeds throughout }V\}.
\operatorname{Post}_XUse in source · 6 passages
Search aliases: Post_X operatorname
Use in source · control_transport_chapter.tex, line 17
For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.
Exact mathematical passage
\operatorname{CPre}^X_m(B) =\{x:\exists a\in A^X_m(x),\ \operatorname{Post}_X(x,a)\subseteq B\}.
\operatorname{Post}_ZUse in source · 3 passages
Search aliases: Post_Z operatorname
Use in source · control_transport_chapter.tex, line 25
[Predecessors commute with an exact controlled quotient] Suppose, for each mode and fine state, A^X_m(x) · =A^Z_m(π x), π(Post_X(x,a)) · =Post_Z(π x,a) (a∈ A^X_m(x)). Then for every B⊆ Z, CPre^X_m(π^-1B) =π^-1(CPre^Z_m(B)). For an admitted action, Post_X(x,a)⊆π^-1B Longleftrightarrow π(Post_X(x,a))⊆ B. Replace the projected successor set by its assumed coarse value. The action ranges in the two existential quantifiers are identical, so existence of a witnessing action is equivalent on both sides…
Exact mathematical passage
A^X_m(x)&=A^Z_m(\pi x),\\ \pi(\operatorname{Post}_X(x,a))&=\operatorname{Post}_Z(\pi x,a) \qquad(a\in A^X_m(x)).
\PowUse in source · 1 passage
Search aliases: Pow
Use in source · core_chapters.tex, line 212
Let E⊆Pow(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].
Exact mathematical passage
\mathcal E\subseteq\Pow(\Omega)
\PrUse in source · 6 passages
Search aliases: Pr
Use in source · core_chapters.tex, line 266
el set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\Pr(m,o,z)=p(m)L_m(o)B(o,z),\qquad Q_m(z)=\sum_oL_m(o)B(o,z).
QUse in source · 26 passages
Search aliases: Q
Use in source · core_chapters.tex, line 132
itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
Q:V\to Y
Q_aUse in source · 4 passages
Search aliases: Q_a
Use in source · migration_chapter.tex, line 79
Migration must commute with subsequent operations. Recovering one new record is not yet preserving a method. Let a common finite action alphabet act on H through total maps F_a:H→ H. Suppose both representations close under those same source operations: M F_a=P_a M, N F_a=Q_a N. The maps P_a and Q_a preserve the attained record sets. These equations refer to the same source action, not merely to operations sharing a name.
Exact mathematical passage
M F_a=P_a M,\qquad N F_a=Q_a N.
Q_mUse in source · 2 passages
Search aliases: Q_m
Use in source · core_chapters.tex, line 266
el set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\Pr(m,o,z)=p(m)L_m(o)B(o,z),\qquad Q_m(z)=\sum_oL_m(o)B(o,z).
Q_wUse in source · 1 passage
Search aliases: Q_w
Use in source · migration_chapter.tex, line 89
[A valid migration intertwines the common operations] If JM=N, then on U_0, JP_a=Q_aJ. Consequently JP_w=Q_wJ for every finite action word w, with the execution-order convention of Chapter~[source reference]. Any protected readout that factors through N agrees along the two migrated executions. Take u=M(h). Then JP_a(u)=JM(F_a h)=N(F_a h)=Q_aN(h)=Q_aJ(u). Every attained u has a representative, so the equality holds throughout U_0. Closure keeps later values within that domain. Induction on the w…
Exact mathematical passage
JP_w=Q_wJ
RUse in source · 22 passages
Search aliases: R
Use in source · core_chapters.tex, line 88
For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.
Exact mathematical passage
R:Z\to V
\mathbb RUse in source · 2 passages
Search aliases: R mathbb
Use in source · core_chapters.tex, line 103
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
V=\mathbb R^2
\mathbb R^2Use in source · 1 passage
Search aliases: R^2 mathbb
Use in source · core_chapters.tex, line 103
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
V=\mathbb R^2
\mathbb R^d_{\ge0}Use in source · 1 passage
Search aliases: R^d_≥0 mathbb ge
Use in source · core_chapters.tex, line 180
ition[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its m…
Exact mathematical passage
\mathcal K=\mathbb R^d_{\ge0}
r_jUse in source · 1 passage
Search aliases: r_j
Use in source · proof_fields_chapter.tex, line 9
A finite field with explicitly admitted rules. Let V={0,…,n-1} be a finite set of claim addresses. Let A⊆ V be admitted leaves, S⊆ V positive observations, and N⊆ V forbidden consequences. These sets need not be disjoint. A finite rule list R has rows r_j=(P_j,c_j), P_j⊆ V, c_j∈ V. Only admitted rules are included. Identical rows may occupy different addresses. A conjectural rule elsewhere in the discovery archive is not admitted merely because it has a convenient conclusion.
Exact mathematical passage
r_j=(P_j,c_j),\qquad P_j\subseteq V,\quad c_j\in V.
\rceilUse in source · 3 passages
Search aliases: rceil
Use in source · core_chapters.tex, line 540
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
sUse in source · 49 passages
Search aliases: s
Use in source · composition_chapter.tex, line 100
An exact sensor realization. Return to x=(b,s,n)∈{0,1}^3 and retain z=(b,s). The actions hold and flip are available everywhere. A repair action is available only where b=0 and leaves that target value fixed. The nuisance bit is retained by every fine transition but ignored by the coarse state.
Exact mathematical passage
x=(b,s,n)\in\{0,1\}^3
s_iUse in source · 2 passages
Search aliases: s_i
Use in source · core_chapters.tex, line 733
Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive measure. Storage capacity, peak memory, and elapsed time need their own models unless a justified conversion is specified.
Exact mathematical passage
s_i
S_XUse in source · 2 passages
Search aliases: S_X
Use in source · control_transport_chapter.tex, line 11
Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.
Exact mathematical passage
S_X=\pi^{-1}(S_Z),\qquad G_X=\pi^{-1}(G_Z),\qquad G_Z\subseteq S_Z.
S_ZUse in source · 2 passages
Search aliases: S_Z
Use in source · control_transport_chapter.tex, line 11
Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.
Exact mathematical passage
S_X=\pi^{-1}(S_Z),\qquad G_X=\pi^{-1}(G_Z),\qquad G_Z\subseteq S_Z.
\SpanUse in source · 1 passage
Search aliases: Span
Use in source · core_chapters.tex, line 135
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
E=\Span\{A_wD_a^\theta v: v\in V,\ a,\theta,\ w\text{ finite}\}.
\supUse in source · 1 passage
Search aliases: sup
Use in source · core_chapters.tex, line 430
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\TV(p,\widehat p)\le e,\qquad \sup_\omega d(v_\omega,\widehat v_\omega)\le d_0.
\sup_\omegaUse in source · 1 passage
Search aliases: sup_ω sup omega
Use in source · core_chapters.tex, line 430
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\TV(p,\widehat p)\le e,\qquad \sup_\omega d(v_\omega,\widehat v_\omega)\le d_0.
\operatorname{supp}Use in source · 1 passage
Search aliases: supp operatorname
Use in source · composition_chapter.tex, line 55
[Support and admissible actions agree] If bar u_h=π_*u_h, then suppbar u_h=π(suppu_h), ⋂_x:u_h(x)>0D(x)=⋂_z:bar u_h(z)>0bar D(z) whenever the history has positive mass. A finite sum of nonnegative terms is positive exactly when at least one summand is positive. This gives the support identity. The enabled sets are constant on each fibre, so repeating a fibre's action set in an intersection changes nothing. The two intersections are therefore equal.
Exact mathematical passage
\operatorname{supp}\bar u_h=\pi(\operatorname{supp}u_h),\qquad \bigcap_{x:u_h(x)>0}D(x)=\bigcap_{z:\bar u_h(z)>0}\bar D(z)
tUse in source · 25 passages
Search aliases: t
Use in source · composition_chapter.tex, line 42
Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include it in the observation interface of both models.
Exact mathematical passage
h=(a_1,o_1,\ldots,a_t,o_t).
t^{-1}Use in source · 1 passage
Search aliases: t^-1
Use in source · core_chapters.tex, line 237
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
T^2Use in source · 2 passages
Search aliases: T^2
Use in source · core_chapters.tex, line 103
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
T^2=I_V
T^X_*Use in source · 4 passages
Search aliases: T^X_*
Use in source · control_transport_chapter.tex, line 51
[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine policy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coo…
Exact mathematical passage
T^X_k=\pi^{-1}(T^Z_k),\qquad T^X_*=\pi^{-1}(T^Z_*),\qquad L^X_k=\pi^{-1}(L^Z_k),\qquad L^X_*=\pi^{-1}(L^Z_*)
T^X_kUse in source · 4 passages
Search aliases: T^X_k
Use in source · control_transport_chapter.tex, line 47
Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.
Exact mathematical passage
T^X_k,L^X_k
T^X_{k+1}Use in source · 1 passage
Search aliases: T^X_k+1
Use in source · control_transport_chapter.tex, line 65
licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X_k+1 · =T^X_k∪(S_X∩CPre^X_rm end(T^X_k)) · =π^-1 (T^Z_k∪ (S_Z∩CPre^Z_rm end(T^Z_k))). The finite increasing sequences stabilize, so their stable sets …
Exact mathematical passage
T^X_{k+1} &=T^X_k\cup\bigl(S_X\cap\operatorname{CPre}^X_{\rm end}(T^X_k)\bigr)\\ &=\pi^{-1}\!\left(T^Z_k\cup (S_Z\cap\operatorname{CPre}^Z_{\rm end}(T^Z_k))\right).
T^Z_*Use in source · 2 passages
Search aliases: T^Z_*
Use in source · control_transport_chapter.tex, line 51
[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine policy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coo…
Exact mathematical passage
T^X_k=\pi^{-1}(T^Z_k),\qquad T^X_*=\pi^{-1}(T^Z_*),\qquad L^X_k=\pi^{-1}(L^Z_k),\qquad L^X_*=\pi^{-1}(L^Z_*)
T^Z_kUse in source · 4 passages
Search aliases: T^Z_k
Use in source · control_transport_chapter.tex, line 47
Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.
Exact mathematical passage
T^Z_k,L^Z_k
T^\thetaUse in source · 1 passage
Search aliases: T^θ theta
Use in source · core_chapters.tex, line 161
For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.
Exact mathematical passage
T^\theta(x_1,x_2,x_3)=(x_1+x_2,x_2,\theta x_3).
T_a^\thetaUse in source · 5 passages
Search aliases: T_a^θ theta
Use in source · core_chapters.tex, line 132
One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
T_a^\theta:V\to V
T_a^{\theta_0}Use in source · 1 passage
Search aliases: T_a^θ_0 theta
Use in source · core_chapters.tex, line 132
ne quotient for several possible dynamics A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
A_a=T_a^{\theta_0}
t_iUse in source · 1 passage
Search aliases: t_i
Use in source · core_chapters.tex, line 465
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
t_i(x)
T_w^\thetaUse in source · 2 passages
Search aliases: T_w^θ theta
Use in source · core_chapters.tex, line 143
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For ev…
Exact mathematical passage
QT_w^\theta=QA_w
T_{\rho(z)-1}Use in source · 1 passage
Search aliases: T_ρ(z)-1 rho
Use in source · ending_chapter.tex, line 27
r strategy. Conversely, a policy that reaches G in at most k+1 steps must either start in G, or select some first ending action such that the continuation from every possible successor reaches G within k steps. By induction every such successor belongs to T_k, hence the initial state lies in T_k+1. At the first-entry stage ρ(z), the selected action sends every successor into T_ρ(z)-1, establishing strict rank decrease.
Exact mathematical passage
T_{\rho(z)-1}
\topUse in source · 8 passages
Search aliases: top
Use in source · core_chapters.tex, line 190
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1
Exact mathematical passage
p=(3/4,1/4)^\top
\TVUse in source · 5 passages
Search aliases: TV
Use in source · core_chapters.tex, line 384
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative …
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
UUse in source · 19 passages
Search aliases: U
Use in source · core_chapters.tex, line 83
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
\mathcal E(T)\mathcal E(U)=ATSAUS=ATUS=\mathcal E(TU).
U_0Use in source · 8 passages
Search aliases: U_0
Use in source · migration_chapter.tex, line 12
A migration map exists exactly on constant fibres. Let H be a finite nonempty set of possible source histories. A history may include a model tag or a past observation whose value must remain available for later correction. Let M:H→ U, N:H→ V be the old record and the required new record. Work on their attained images U_0=M(H) and V_0=N(H). The new record must be specified before sufficiency is claimed. It can be an answer, a correction signature, or the complete profile of future outputs in a d…
Exact mathematical passage
U_0=M(H)
u_\varnothingUse in source · 1 passage
Search aliases: u_∅ varnothing
Use in source · core_chapters.tex, line 372
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
\bar u_\varnothing=Cu_\varnothing
\mathcal VUse in source · 1 passage
Search aliases: V mathcal
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
\mathcal V
vUse in source · 55 passages
Search aliases: v
Use in source · core_chapters.tex, line 135
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
E=\Span\{A_wD_a^\theta v: v\in V,\ a,\theta,\ w\text{ finite}\}.
V_0Use in source · 3 passages
Search aliases: V_0
Use in source · migration_chapter.tex, line 12
A migration map exists exactly on constant fibres. Let H be a finite nonempty set of possible source histories. A history may include a model tag or a past observation whose value must remain available for later correction. Let M:H→ U, N:H→ V be the old record and the required new record. Work on their attained images U_0=M(H) and V_0=N(H). The new record must be specified before sufficiency is claimed. It can be an answer, a correction signature, or the complete profile of future outputs in a d…
Exact mathematical passage
V_0=N(H)
V_{e,o}Use in source · 2 passages
Search aliases: V_e,o
Use in source · core_chapters.tex, line 497
A current support V≠∅ records possible states. The experiment is permitted only if V⊆ D_e. After report o, the possible post-experiment states are V_e,o={f_e(x):x∈ V, o_e(x)=o}. Only nonempty report branches are retained. If the target concerns a past value, that value must survive as an immutable tag. Resetting the current state is not recovering its history.
Exact mathematical passage
V_{e,o}=\{f_e(x):x\in V,\ o_e(x)=o\}.
v_iUse in source · 1 passage
Search aliases: v_i
Use in source · core_chapters.tex, line 176
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i\quad(w_i>0).
v_oUse in source · 2 passages
Search aliases: v_o
Use in source · core_chapters.tex, line 509
athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it is no larger in every resou…
Exact mathematical passage
(v_o,m_o)
v_\omegaUse in source · 3 passages
Search aliases: v_ω omega
Use in source · core_chapters.tex, line 429
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
v_\omega
WUse in source · 7 passages
Search aliases: W
Use in source · core_chapters.tex, line 225
every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct signatures. Thus the image of Σ has the least possible cardinality for this contract. If Σ(V)=Σ(W), then for each E, t((V∩ E_0)∩ E)=t(V∩(E_0∩ E))=t(W∩(E_0∩ E)). Intersection closure supplies the coordinate E_0∩ E, so the updated signatu…
Exact mathematical passage
\Sigma(V)=\Sigma(W)
w_iUse in source · 2 passages
Search aliases: w_i
Use in source · core_chapters.tex, line 176
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i\quad(w_i>0).
\widehat aUse in source · 4 passages
Search aliases: widehat a widehat
Use in source · core_chapters.tex, line 441
[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add and subtract ∑_ω p(ω)ℓ(a,widehat v_ω). The response term has absolute value at most Kd_0. The expectation of a function in [0,1] changes by at most tot…
Exact mathematical passage
\widehat a
\widehat pUse in source · 3 passages
Search aliases: widehat p widehat
Use in source · core_chapters.tex, line 429
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\widehat p
\widehat RUse in source · 4 passages
Search aliases: widehat R widehat
Use in source · core_chapters.tex, line 435
widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
R(a)=\sum_\omega p(\omega)\ell(a,v_\omega),\qquad \widehat R(a)=\sum_\omega\widehat p(\omega)\ell(a,\widehat v_\omega).
\widehat vUse in source · 4 passages
Search aliases: widehat v widehat
Use in source · core_chapters.tex, line 429
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\widehat v_\omega
\widehat v_\omegaUse in source · 4 passages
Search aliases: widehat v_ω widehat omega
Use in source · core_chapters.tex, line 429
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\widehat v_\omega
\widetilde JUse in source · 3 passages
Search aliases: widetilde J widetilde
Use in source · core_chapters.tex, line 617
[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certify a zero-risk constraint or preservation of admissibility outside the common class. The expectation of a function in [0,B] changes by at most B times…
Exact mathematical passage
\widetilde J(\pi)
\widetilde KUse in source · 2 passages
Search aliases: widetilde K widetilde
Use in source · core_chapters.tex, line 383
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative …
Exact mathematical passage
\widetilde K
\widetilde K_{a,o}Use in source · 1 passage
Search aliases: widetilde K_a,o widetilde
Use in source · core_chapters.tex, line 384
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative …
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\widetilde PUse in source · 3 passages
Search aliases: widetilde P widetilde
Use in source · core_chapters.tex, line 391
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p…
Exact mathematical passage
P_H,\widetilde P_H
\widetilde pUse in source · 7 passages
Search aliases: widetilde p widetilde
Use in source · core_chapters.tex, line 388
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordinates,…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\widetilde p_0Use in source · 1 passage
Search aliases: widetilde p_0 widetilde
Use in source · core_chapters.tex, line 388
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordinates,…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\widetilde p_EUse in source · 6 passages
Search aliases: widetilde p_E widetilde
Use in source · core_chapters.tex, line 395
d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either event…
Exact mathematical passage
p_E,\widetilde p_E>0
\widetilde P_HUse in source · 3 passages
Search aliases: widetilde P_H widetilde
Use in source · core_chapters.tex, line 391
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p…
Exact mathematical passage
P_H,\widetilde P_H
\widetilde\piUse in source · 3 passages
Search aliases: widetildeπ widetilde pi
Use in source · core_chapters.tex, line 621
[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certify a zero-risk constraint or preservation of admissibility outside the common class. The expectation of a function in [0,B] changes by at most B times…
Exact mathematical passage
\widetilde\pi
x'Use in source · 14 passages
Search aliases: x'
Use in source · composition_chapter.tex, line 9
The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)=1 specify the joint distribution of the next state and report. No transition is specified for a disabled action. An implementation may fill unused en…
Exact mathematical passage
K_{a,o}(x'\mid x),\qquad \sum_{o\in O}\sum_{x'\in X}K_{a,o}(x'\mid x)=1
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · core_chapters.tex, line 185
On the nonnegative coordinate cone, preservation is equivalent to nonnegative matrix entries, as is seen by applying a matrix to every standard basis vector. Because the matrices sum to I, their nonnegative off-diagonal entries must all vanish. An idempotent diagonal entry satisfies x^2=x, hence is zero or one. The diagonal sums are one, so each coordinate is selected by exactly one map. Finally, completeness and orthogonality without positivity do not give the first premise, as the following ex…
Exact mathematical passage
x^2=x
x_1Use in source · 5 passages
Search aliases: x_1
Use in source · core_chapters.tex, line 103
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
C(x_1,x_2)=x_1
x_2Use in source · 5 passages
Search aliases: x_2
Use in source · core_chapters.tex, line 103
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
C(x_1,x_2)=x_1
x_3Use in source · 2 passages
Search aliases: x_3
Use in source · core_chapters.tex, line 160
For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.
Exact mathematical passage
Q(x_1,x_2,x_3)=x_1
X_tUse in source · 1 passage
Search aliases: X_t
Use in source · core_chapters.tex, line 340
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
X_{t+1}Use in source · 1 passage
Search aliases: X_t+1
Use in source · core_chapters.tex, line 340
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
x_vUse in source · 2 passages
Search aliases: x_v
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
x_v\in F_v
x_wUse in source · 1 passage
Search aliases: x_w
Use in source · realization_prologue.tex, line 11
realization Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. A nonempty fibre at every individual …
Exact mathematical passage
\Gamma(F)=\left\{(x_v)_v\in\prod_v F_v:\ f^*(x_w)=x_v\text{ for every }f:v\to w\right\}.
yUse in source · 24 passages
Search aliases: y
Use in source · composition_chapter.tex, line 27
ic quotient criterion] There are coarse data satisfying D(x)=bar D(π x), ℓ(x)=barℓ(π x), ∑_π x'=z'K_a,o(x'mid x)=bar K_a,o(z'midπ x) for every x, every a∈ D(x), and every o,z', if and only if states in the same fibre have identical labels and enabled sets and identical projected joint successor/report rows. Explicitly, π x=π y must imply D(x) · =D(y), ℓ(x)=ℓ(y),
Exact mathematical passage
\pi x=\pi y
\mathcal ZUse in source · 2 passages
Search aliases: Z mathcal
Use in source · core_chapters.tex, line 265
Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability…
Exact mathematical passage
\mathcal Z
z'Use in source · 10 passages
Search aliases: z'
Use in source · composition_chapter.tex, line 23
[Joint legal and stochastic quotient criterion] There are coarse data satisfying D(x)=bar D(π x), ℓ(x)=barℓ(π x), ∑_π x'=z'K_a,o(x'mid x)=bar K_a,o(z'midπ x) for every x, every a∈ D(x), and every o,z', if and only if states in the same fibre have identical labels and enabled sets and identical projected joint successor/report rows. Explicitly, π x=π y must imply D(x) · =D(y), ℓ(x)=ℓ(y),
Exact mathematical passage
D(x)=\bar D(\pi x),\quad \ell(x)=\bar\ell(\pi x),\quad \sum_{\pi x'=z'}K_{a,o}(x'\mid x)=\bar K_{a,o}(z'\mid\pi x)
z_0Use in source · 2 passages
Search aliases: z_0
Use in source · composition_chapter.tex, line 79
This induction alone records only the current state and the observed history. To retain the entire state path, fix a coarse prefix (z_0,a_1,o_1,z_1,…,a_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state…
Exact mathematical passage
(z_0,a_1,o_1,z_1,\ldots,a_t,o_t,z_t).
z_1Use in source · 1 passage
Search aliases: z_1
Use in source · composition_chapter.tex, line 79
This induction alone records only the current state and the observed history. To retain the entire state path, fix a coarse prefix (z_0,a_1,o_1,z_1,…,a_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state…
Exact mathematical passage
(z_0,a_1,o_1,z_1,\ldots,a_t,o_t,z_t).
z_{a,o}Use in source · 2 passages
Search aliases: z_a,o
Use in source · core_chapters.tex, line 346
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
Z_HUse in source · 1 passage
Search aliases: Z_H
Use in source · core_chapters.tex, line 396
e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either event probability exceeds d_H, the other is necessarily pos…
Exact mathematical passage
\TV\bigl(P_H(Z_H\in\cdot\mid E),\widetilde P_H(Z_H\in\cdot\mid E)\bigr) \le\min\left\{1,\frac{d_H}{\max(p_E,\widetilde p_E)}\right\}.
z_iUse in source · 2 passages
Search aliases: z_i
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
z_{i-1}Use in source · 1 passage
Search aliases: z_i-1
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
z_tUse in source · 1 passage
Search aliases: z_t
Use in source · composition_chapter.tex, line 79
This induction alone records only the current state and the observed history. To retain the entire state path, fix a coarse prefix (z_0,a_1,o_1,z_1,…,a_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state…
Exact mathematical passage
(z_0,a_1,o_1,z_1,\ldots,a_t,o_t,z_t).
|Use in source · 10 passages
Search aliases: |
Use in source · core_chapters.tex, line 237
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
\cdotUse in source · 4 passages
Search aliases: · cdot
Use in source · core_chapters.tex, line 289
probability. For a positive pair, posterior equality yields p(m)L_m(o)B(o,z)=Pr(o,z)p(mmid o) =ρ(omid z)Pr(z)p(mmid z). Substituting Bayes' formula and cancelling positive p(m) gives the factorization. A zero-probability pair under a positive prior has L_m(o)B(o,z)=0 for all m. Its conditional weight is zero. If z is impossible for every model, assign any probability row to ρ(·mid z) because both sides vanish.
Exact mathematical passage
\rho(\cdot\mid z)
\betaUse in source · 1 passage
Search aliases: β beta
Use in source · core_chapters.tex, line 465
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\beta_i(x)
\beta_iUse in source · 1 passage
Search aliases: β_i beta
Use in source · core_chapters.tex, line 465
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\beta_i(x)
\GammaUse in source · 9 passages
Search aliases: Γ Gamma
Use in source · book.tex, line 96
midrule Γ(F) · Compatible families in a specified system of views.
Exact mathematical passage
\Gamma(F)
\gammaUse in source · 3 passages
Search aliases: γ gamma
Use in source · core_chapters.tex, line 372
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
\gamma(a\mid h)
\DeltaUse in source · 7 passages
Search aliases: Δ Delta
Use in source · core_chapters.tex, line 711
After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.
Exact mathematical passage
\Delta
\deltaUse in source · 10 passages
Search aliases: δ delta
Use in source · core_chapters.tex, line 388
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordinates,…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\delta_0Use in source · 3 passages
Search aliases: δ_0 delta
Use in source · core_chapters.tex, line 388
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordinates,…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\delta_jUse in source · 3 passages
Search aliases: δ_j delta
Use in source · core_chapters.tex, line 754
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\Pr(A_j\mid\mathcal H_{j-1})\le\delta_j\quad\text{almost surely}, \qquad\sum_{j\ge1}\delta_j\le\delta,
\epsilonUse in source · 11 passages
Search aliases: ϵ epsilon
Use in source · control_transport_chapter.tex, line 131
Why a small probability error is not a sure guarantee. Take three states s,g,b, with safe set {s,g} and goal {g}. Running holds s or g in place. The nominal ending action at s moves to g with probability one. The perturbed action instead moves to g with probability 1-ϵ and to unsafe state b with probability ϵ, where 0<ϵ<1. A hold action is available at g.
Exact mathematical passage
1-\epsilon
\epsilon_tUse in source · 6 passages
Search aliases: ϵ_t epsilon
Use in source · core_chapters.tex, line 384
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative …
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\etaUse in source · 8 passages
Search aliases: η eta
Use in source · core_chapters.tex, line 322
For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equality of the posterior odds gives ℓ(x)/ℓ(z)=barℓ(η(x))/barℓ(η(z))=ℓ(y)/ℓ(z). Thus ℓ(x)=ℓ(y), and η refines the first likelihood split. Coarse likelihoo…
Exact mathematical passage
\eta
\thetaUse in source · 18 passages
Search aliases: θ theta
Use in source · core_chapters.tex, line 132
One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
T_a^\theta:V\to V
\theta_0Use in source · 2 passages
Search aliases: θ_0 theta
Use in source · core_chapters.tex, line 132
One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
\theta_0
\lambdaUse in source · 5 passages
Search aliases: λ lambda
Use in source · core_chapters.tex, line 172
Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state need not itself be valid.
Exact mathematical passage
\lambda
\muUse in source · 4 passages
Search aliases: μ mu
Use in source · proof_fields_chapter.tex, line 84
Logical interpretation is an additional map. Let μ(v) be the proposition represented by claim address v. Suppose every admitted leaf and positive observation is true under μ, and each admitted rule satisfies (bigwedge_p∈ P_jμ(p))Longrightarrowμ(c_j). Then every node in C is true under μ. Indeed, induction over the closure stages propagates truth from the seeds through the admitted implications.
Exact mathematical passage
\mu(v)
\xiUse in source · 6 passages
Search aliases: ξ xi
Use in source · core_chapters.tex, line 441
[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add and subtract ∑_ω p(ω)ℓ(a,widehat v_ω). The response term has absolute value at most Kd_0. The expectation of a function in [0,1] changes by at most tot…
Exact mathematical passage
\xi=\min\{1,e+Kd_0\}
\pi^{-1}Use in source · 8 passages
Search aliases: π^-1 pi
Use in source · control_transport_chapter.tex, line 11
Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.
Exact mathematical passage
S_X=\pi^{-1}(S_Z),\qquad G_X=\pi^{-1}(G_Z),\qquad G_Z\subseteq S_Z.
\pi_0Use in source · 4 passages
Search aliases: π_0 pi
Use in source · core_chapters.tex, line 310
A mandatory partition π_0 with at least two blocks protects the target distinctions. We seek a refinement that supports every prediction and each separate Bayesian update for every prior. The observation is an operation available in its own right. This is stronger than inspecting only a chosen combined transition-and-report schedule.
Exact mathematical passage
\pi_0
\pi_{\rm init}Use in source · 2 passages
Search aliases: π_rm init pi rm
Use in source · core_chapters.tex, line 313
[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly posit…
Exact mathematical passage
\pi_{\rm init}
\rho_XUse in source · 1 passage
Search aliases: ρ_X rho
Use in source · control_transport_chapter.tex, line 58
[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine policy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coo…
Exact mathematical passage
\rho_X(x)=\rho_Z(\pi x).
\rho_ZUse in source · 1 passage
Search aliases: ρ_Z rho
Use in source · control_transport_chapter.tex, line 58
[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine policy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coo…
Exact mathematical passage
\rho_X(x)=\rho_Z(\pi x).
\SigmaUse in source · 5 passages
Search aliases: Σ Sigma
Use in source · book.tex, line 99
E,Σ · An evidence family and its correction signature.
Exact mathematical passage
\mathcal E,\Sigma
\sigmaUse in source · 1 passage
Search aliases: σ sigma
Use in source · proof_fields_chapter.tex, line 31
Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.
Exact mathematical passage
\sigma=(j_1,\ldots,j_m)
\tauUse in source · 7 passages
Search aliases: τ tau
Use in source · core_chapters.tex, line 109
A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.
Exact mathematical passage
G_{\tau(a)}
\PhiUse in source · 5 passages
Search aliases: Φ Phi
Use in source · core_chapters.tex, line 88
For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.
Exact mathematical passage
\Phi(T)=CTR
\varphiUse in source · 3 passages
Search aliases: φ varphi
Use in source · migration_chapter.tex, line 118
Certificates after a change of contract. A migration map transports data. A certificate authorizes a conclusion only relative to its premises. To reuse a certificate across a revision, specify an old predicate φ on U_0, a new predicate ψ on V_0, and a proved implication φ(u) Longrightarrow ψ(Ju). An old witness for φ(u) and this implication then yield the new conclusion. The implication may express semantic preservation, a stronger guard, or a deliberately weaker postcondition. It cannot be repl…
Exact mathematical passage
\varphi
\PsiUse in source · 2 passages
Search aliases: Ψ Psi
Use in source · core_chapters.tex, line 346
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\psiUse in source · 2 passages
Search aliases: ψ psi
Use in source · migration_chapter.tex, line 118
Certificates after a change of contract. A migration map transports data. A certificate authorizes a conclusion only relative to its premises. To reuse a certificate across a revision, specify an old predicate φ on U_0, a new predicate ψ on V_0, and a proved implication φ(u) Longrightarrow ψ(Ju). An old witness for φ(u) and this implication then yield the new conclusion. The implication may express semantic preservation, a stronger guard, or a deliberately weaker postcondition. It cannot be repl…
Exact mathematical passage
\psi
\Psi_{a,o}Use in source · 2 passages
Search aliases: Ψ_a,o Psi
Use in source · core_chapters.tex, line 346
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\OmegaUse in source · 13 passages
Search aliases: Ω Omega
Use in source · core_chapters.tex, line 3
From evidence to a reusable state. questionThe first question is not how much state to store. It is which distinctions are accessible. Let Ω be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function r:Ω→ H. A target is a function t:Ω→ Y. All codomains used below are nonempty. We work on attained records r(Ω), so no arbitrary interpretation of an impossible record is needed.
Exact mathematical passage
\Omega
\omegaUse in source · 11 passages
Search aliases: ω omega
Use in source · core_chapters.tex, line 6
[A record determines a target exactly on its fibres] There exists d:r(Ω)→ Y with d(r(ω))=t(ω) for every world if and only if r(ω)=r(ω') Longrightarrow t(ω)=t(ω'). Necessity follows because a function assigns one value to one input. For sufficiency, define d(h) to be t(ω) for any world with r(ω)=h. Such a world exists for attained h, and the hypothesis makes the choice immaterial. The resulting decoder satisfies the required equation. sourceGrounded Abstraction [GA], Proposition 3.2 and Theorem 3…
Exact mathematical passage
d(r(\omega))=t(\omega)
\omega'Use in source · 1 passage
Search aliases: ω' omega
Use in source · core_chapters.tex, line 7
[A record determines a target exactly on its fibres] There exists d:r(Ω)→ Y with d(r(ω))=t(ω) for every world if and only if r(ω)=r(ω') Longrightarrow t(ω)=t(ω'). Necessity follows because a function assigns one value to one input. For sufficiency, define d(h) to be t(ω) for any world with r(ω)=h. Such a world exists for attained h, and the hypothesis makes the choice immaterial. The resulting decoder satisfies the required equation. sourceGrounded Abstraction [GA], Proposition 3.2 and Theorem 3…
Exact mathematical passage
r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\ldotsUse in source · 10 passages
Search aliases: … ldots
Use in source · composition_chapter.tex, line 42
Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include it in the observation interface of both models.
Exact mathematical passage
h=(a_1,o_1,\ldots,a_t,o_t).
\ell^1Use in source · 1 passage
Search aliases: ℓ^1 ell
Use in source · core_chapters.tex, line 405
For conditioning, let u,v be the restrictions of the two history laws to E, with masses p_E,widetilde p_E. The difference on the complement has ℓ^1 norm at least |p_E-widetilde p_E|. Thus u-v _1+|p_E-widetilde p_E|≤2d_H. Adding and subtracting v/p_E bounds the conditional distance by d_H/p_E. Interchanging the laws gives d_H/widetilde p_E, so the smaller of these bounds has the displayed maximum denominator. Projection to the final state cannot increase the distance. Finally, the probability of …
Exact mathematical passage
\ell^1
\ell_oUse in source · 3 passages
Search aliases: ℓ_o ell
Use in source · core_chapters.tex, line 308
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
\ell_o(x)>0
\mapstoUse in source · 7 passages
Search aliases: ↦ mapsto
Use in source · core_chapters.tex, line 95
[Exact accounting of the discarded excursion] For any endomorphisms T,U of V, Φ(TU)-Φ(T)Φ(U)=CT(I-P)UR. Thus compression preserves this composition precisely when the discarded excursion on the right vanishes. Full split encoding T↦ ATS preserves composition without that extra condition. Expand the product Φ(T)Φ(U)=CTRCUR=CTPUR and subtract it from CTUR. The identity for full split encoding follows from SA=I_V as computed above. These computations make no approximation and require no orthogonali…
Exact mathematical passage
T\mapsto ATS
\existsUse in source · 5 passages
Search aliases: ∃ exists
Use in source · control_transport_chapter.tex, line 17
For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.
Exact mathematical passage
\operatorname{CPre}^X_m(B) =\{x:\exists a\in A^X_m(x),\ \operatorname{Post}_X(x,a)\subseteq B\}.
\varnothingUse in source · 6 passages
Search aliases: ∅ varnothing
Use in source · core_chapters.tex, line 237
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
\prodUse in source · 2 passages
Search aliases: ∏ prod
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
\prod_{i=1}^{t}Use in source · 1 passage
Search aliases: ∏_i=1^t prod
Use in source · composition_chapter.tex, line 83
_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state sum. Induction gives the exact path mass bar p_0(z_0)∏_i=1^t α(a_imid a_1,o_1,…,a_i-1,o_i-1) bar K_a_i,o_i(z_imid z_i-1). An inadmissible or zero-mass…
Exact mathematical passage
\bar p_0(z_0)\prod_{i=1}^{t} \alpha(a_i\mid a_1,o_1,\ldots,a_{i-1},o_{i-1}) \bar K_{a_i,o_i}(z_i\mid z_{i-1}).
\prod_vUse in source · 1 passage
Search aliases: ∏_v prod
Use in source · realization_prologue.tex, line 11
realization Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. A nonempty fibre at every individual …
Exact mathematical passage
\Gamma(F)=\left\{(x_v)_v\in\prod_v F_v:\ f^*(x_w)=x_v\text{ for every }f:v\to w\right\}.
\sum_iUse in source · 4 passages
Search aliases: ∑_i sum
Use in source · core_chapters.tex, line 175
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
\sum_iP_i=I
\sum_{i:w_i>0}Use in source · 1 passage
Search aliases: ∑_i:w_i>0 sum
Use in source · core_chapters.tex, line 176
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i\quad(w_i>0).
\sum_{j\ge1}Use in source · 1 passage
Search aliases: ∑_j≥1 sum ge
Use in source · core_chapters.tex, line 754
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\Pr(A_j\mid\mathcal H_{j-1})\le\delta_j\quad\text{almost surely}, \qquad\sum_{j\ge1}\delta_j\le\delta,
\sum_oUse in source · 3 passages
Search aliases: ∑_o sum
Use in source · core_chapters.tex, line 266
el set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\Pr(m,o,z)=p(m)L_m(o)B(o,z),\qquad Q_m(z)=\sum_oL_m(o)B(o,z).
\sum_{o,x'}Use in source · 1 passage
Search aliases: ∑_o,x' sum
Use in source · core_chapters.tex, line 343
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
\sum_{o,x'}K_{a,o}(x',x)=1
\sum_{o\in O}Use in source · 1 passage
Search aliases: ∑_o∈ O sum in
Use in source · composition_chapter.tex, line 9
The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)=1 specify the joint distribution of the next state and report. No transition is specified for a disabled action. An implementation may fill unused en…
Exact mathematical passage
K_{a,o}(x'\mid x),\qquad \sum_{o\in O}\sum_{x'\in X}K_{a,o}(x'\mid x)=1
\sum_{t=0}^{H-1}Use in source · 1 passage
Search aliases: ∑_t=0^H-1 sum
Use in source · core_chapters.tex, line 392
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p…
Exact mathematical passage
\TV(P_H,\widetilde P_H)\le d_H:=\min\left\{1,\delta_0+\sum_{t=0}^{H-1}\epsilon_t\right\}.
\sum_{x':\pi(x')=z'}Use in source · 2 passages
Search aliases: ∑_x':π(x')=z' sum pi
Use in source · core_chapters.tex, line 358
[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0, these identiti…
Exact mathematical passage
\sum_{x':\pi(x')=z'}K_{a,o}(x',x)
\sum_{x'\in B}Use in source · 1 passage
Search aliases: ∑_x'∈ B sum in
Use in source · core_chapters.tex, line 314
[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly posit…
Exact mathematical passage
\left(\sum_{x'\in B}K_a(x,x')\right)_{a,B},
\sum_{x'\in X}Use in source · 1 passage
Search aliases: ∑_x'∈ X sum in
Use in source · composition_chapter.tex, line 9
The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)=1 specify the joint distribution of the next state and report. No transition is specified for a disabled action. An implementation may fill unused en…
Exact mathematical passage
K_{a,o}(x'\mid x),\qquad \sum_{o\in O}\sum_{x'\in X}K_{a,o}(x'\mid x)=1
\sum_{\pi x'=z'}Use in source · 2 passages
Search aliases: ∑_π x'=z' sum pi
Use in source · composition_chapter.tex, line 23
[Joint legal and stochastic quotient criterion] There are coarse data satisfying D(x)=bar D(π x), ℓ(x)=barℓ(π x), ∑_π x'=z'K_a,o(x'mid x)=bar K_a,o(z'midπ x) for every x, every a∈ D(x), and every o,z', if and only if states in the same fibre have identical labels and enabled sets and identical projected joint successor/report rows. Explicitly, π x=π y must imply D(x) · =D(y), ℓ(x)=ℓ(y),
Exact mathematical passage
D(x)=\bar D(\pi x),\quad \ell(x)=\bar\ell(\pi x),\quad \sum_{\pi x'=z'}K_{a,o}(x'\mid x)=\bar K_{a,o}(z'\mid\pi x)
\sum_{\pi x=z}Use in source · 1 passage
Search aliases: ∑_π x=z sum pi
Use in source · composition_chapter.tex, line 16
For a nonnegative vector u on X, write (π_*u)(z)=∑_π x=zu(x). This notation applies to probability distributions and to unnormalized history masses. A coarse model has labels barℓ, enabled sets bar D, and joint instruments bar K on its enabled actions.
Exact mathematical passage
(\pi_*u)(z)=\sum_{\pi x=z}u(x).
\sum_\omegaUse in source · 2 passages
Search aliases: ∑_ω sum omega
Use in source · core_chapters.tex, line 435
widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
R(a)=\sum_\omega p(\omega)\ell(a,v_\omega),\qquad \widehat R(a)=\sum_\omega\widehat p(\omega)\ell(a,\widehat v_\omega).
\setminusUse in source · 3 passages
Search aliases: ∖ setminus
Use in source · ending_chapter.tex, line 27
xactly when there exists an ending-only strategy that, whatever the successors, reaches G within at most k transitions while remaining in S. In particular, if ρ(z)>0, there is an ending action whose every successor has strictly smaller first-entry rank. The assertion at k=0 says precisely that the initial state is already in G. Assume it holds for k. If z∈ T_k+1∖ T_k, the construction supplies an ending action with all successors in T_k. After the action the controller uses the strategy given by…
Exact mathematical passage
z\in T_{k+1}\setminus T_k
\circUse in source · 2 passages
Search aliases: ∘ circ
Use in source · realization_prologue.tex, line 10
Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F)={(x_v)_v∈∏_v F_v: f^*(x_w)=x_v for every f:v→ w}. The set Γ(F) is the strict realization set. It may be empty. …
Exact mathematical passage
(g\circ f)^*=f^*\circ g^*
\capUse in source · 11 passages
Search aliases: ∩ cap
Use in source · control_transport_chapter.tex, line 65
licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X_k+1 · =T^X_k∪(S_X∩CPre^X_rm end(T^X_k)) · =π^-1 (T^Z_k∪ (S_Z∩CPre^Z_rm end(T^Z_k))). The finite increasing sequences stabilize, so their stable sets …
Exact mathematical passage
T^X_{k+1} &=T^X_k\cup\bigl(S_X\cap\operatorname{CPre}^X_{\rm end}(T^X_k)\bigr)\\ &=\pi^{-1}\!\left(T^Z_k\cup (S_Z\cap\operatorname{CPre}^Z_{\rm end}(T^Z_k))\right).
\simUse in source · 3 passages
Search aliases: ∼ sim
Use in source · core_chapters.tex, line 48
Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.
Exact mathematical passage
x\sim x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\text{ for every finite word }w.
\neUse in source · 4 passages
Search aliases: ≠ ne
Use in source · core_chapters.tex, line 237
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
\leUse in source · 17 passages
Search aliases: ≤ le
Use in source · core_chapters.tex, line 384
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative …
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\geUse in source · 12 passages
Search aliases: ≥ ge
Use in source · core_chapters.tex, line 176
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass i…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i\quad(w_i>0).
\oplusUse in source · 11 passages
Search aliases: ⊕ oplus
Use in source · composition_chapter.tex, line 102
After an action, report the new display b⊕ s correctly with probability 3/4 and incorrectly with probability 1/4. Both the label and the joint instrument ignore n. The ten coarse state/action rows satisfy the quotient conditions exactly.
Exact mathematical passage
b\oplus s
\bigcapUse in source · 4 passages
Search aliases: ⋂ bigcap
Use in source · composition_chapter.tex, line 48
Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is empty, no policy can make another admitted action at that history. A stopping convention can be included as a shared terminal action, but is not silen…
Exact mathematical passage
D_h=\bigcap_{x:u_h(x)>0}D(x).
\bigcap_iUse in source · 1 passage
Search aliases: ⋂_i bigcap
Use in source · core_chapters.tex, line 778
[Target-relative joint reconstruction] There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then v-w∈ker A⊆ker Q, so this definition is independent of the representative. It is linear and extends to the full record space by…
Exact mathematical passage
\ker A=\bigcap_i\ker A_i
\bigcap_{x:u_h(x)>0}Use in source · 2 passages
Search aliases: ⋂_x:u_h(x)>0 bigcap
Use in source · composition_chapter.tex, line 48
Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is empty, no policy can make another admitted action at that history. A stopping convention can be included as a shared terminal action, but is not silen…
Exact mathematical passage
D_h=\bigcap_{x:u_h(x)>0}D(x).
\bigcap_{x:\alpha(x)=h}Use in source · 1 passage
Search aliases: ⋂_x:α(x)=h bigcap alpha
Use in source · core_chapters.tex, line 468
Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).
Exact mathematical passage
\mathcal C_h=\bigcap_{x:\alpha(x)=h}\mathcal C(x).
\bigcap_{z:\bar u_h(z)>0}Use in source · 1 passage
Search aliases: ⋂_z:bar u_h(z)>0 bigcap bar
Use in source · composition_chapter.tex, line 55
[Support and admissible actions agree] If bar u_h=π_*u_h, then suppbar u_h=π(suppu_h), ⋂_x:u_h(x)>0D(x)=⋂_z:bar u_h(z)>0bar D(z) whenever the history has positive mass. A finite sum of nonnegative terms is positive exactly when at least one summand is positive. This gives the support identity. The enabled sets are constant on each fibre, so repeating a fibre's action set in an intersection changes nothing. The two intersections are therefore equal.
Exact mathematical passage
\operatorname{supp}\bar u_h=\pi(\operatorname{supp}u_h),\qquad \bigcap_{x:u_h(x)>0}D(x)=\bigcap_{z:\bar u_h(z)>0}\bar D(z)
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