The probability of a query is a ratio of two #P numbers

Lax794877.ProbabilityRatio · concepts/Lax794877/ProbabilityRatio.lean · lax-794877

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Natural Language Statement

Theorem

On a weighted instance whose positions are linearly ordered, with every uncertain fact of positive total weight, the probability of a sentence $\varphi$ is the number of weighted worlds of $\varphi$ divided by that of the sentence that always holds: a ratio of two numbers of #P, for every first-order query. On an instance whose positions are not linearly ordered, the count is $0$ .

Concept map

23 concepts

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Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

Evidence

This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

1 `weightedWorlds_of_not_isLinOrd` proven

- lem✓Lax794877.WorldsValues
→
thm✓Lax794877.ProbabilityRatio (1st statement)
⊢Lax794877Proofs.Bridge.weightedWorlds_of_not_isLinOrd

2 `worldProb_eq_ratio` proven

- lem✓Lax794877.WorldsValues
→
thm✓Lax794877.ProbabilityRatio (2nd statement)
⊢Lax794877Proofs.Bridge.worldProb_eq_ratio

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1	import Lax366625.CountingClasses
2	import Lax366625.CountingProblems
3	import Lax366625.QuantitativeLogic
4	import Lax366625.WitnessCounting
5	import Lax799700.Common
6	import Lax859101.OneCallReductions
7	import Lax904597.Machines
8	import Lax904597.SecondOrder
9	import Mathlib.Algebra.BigOperators.Field
10	import Mathlib.Algebra.BigOperators.Fin
11	import Mathlib.Algebra.BigOperators.Finprod
12	import Mathlib.Algebra.BigOperators.Group.Finset.Basic
13	import Mathlib.Algebra.BigOperators.Pi
14	import Mathlib.Algebra.BigOperators.Ring.Finset
15	import Mathlib.Algebra.Group.Action.Defs
16	import Mathlib.Algebra.Order.BigOperators.Group.Finset
17	import Mathlib.Algebra.Order.BigOperators.Ring.Finset
18	import Mathlib.Algebra.Order.Field.Basic
19	import Mathlib.Algebra.Order.Ring.Rat
20	import Mathlib.Data.Finite.Sigma
21	import Mathlib.Data.Finset.Max
22	import Mathlib.Data.Fintype.BigOperators
23	import Mathlib.Data.Fintype.Card
24	import Mathlib.Data.Fintype.EquivFin
25	import Mathlib.Data.Fintype.Lattice
26	import Mathlib.Data.Fintype.Pi
27	import Mathlib.Data.Fintype.Pigeonhole
28	import Mathlib.Data.Fintype.Sort
29	import Mathlib.Data.Nat.Bitwise
30	import Mathlib.Data.Prod.Lex
31	import Mathlib.Data.Set.Card
32	import Mathlib.Data.Set.Finite.Lemmas
33	import Mathlib.Dynamics.FixedPoints.Basic
34	import Mathlib.Logic.Equiv.Fin.Basic
35	import Mathlib.Logic.Equiv.Prod
36	import Mathlib.ModelTheory.Complexity
37	import Mathlib.ModelTheory.Graph
38	import Mathlib.ModelTheory.Order
39	import Mathlib.ModelTheory.Semantics
40	import Mathlib.ModelTheory.Syntax
41	import Mathlib.Order.Hom.Set
42	import Mathlib.Order.Lattice.Nat
43	import Mathlib.Order.PiLex
44	import Mathlib.SetTheory.Cardinal.Finite
45	import Mathlib.Tactic.FieldSimp
46	import Mathlib.Tactic.FinCases
47	import Mathlib.Tactic.Linarith
48	import Mathlib.Tactic.Ring
49	import Lax794877.PossibleWorlds
50	import Lax794877.WeightedWorlds
51	import Lax794877.Queries
52	import Lax794877.ExampleDatabase
53
54	/-!
55	---
56	title: The probability of a query is a ratio of two #P numbers
57	type: theorem
58	---
59	On a weighted instance whose positions are linearly ordered, with every
60	uncertain fact of positive total weight, the probability of a sentence
61	$\varphi$ is the number of weighted worlds of $\varphi$ divided by that of
62	the sentence that always holds: a ratio of two numbers of #P, for every
63	first-order query. On an instance whose positions are not linearly ordered,
64	the count is $0$ .
65	-/
66
67	namespace Lax794877.ProbabilityRatio
68
69	open Lax904597.SecondOrder
70	open FirstOrder
71	open Language Structure
72	open Lax794877.PossibleWorlds Lax799700.Common Lax904597.SecondOrder
73	open FirstOrder.Language
74	open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
75	Lax366625.QuantitativeLogic
76	open Lax859101.OneCallReductions Lax904597.Machines
77	open Lax794877.PossibleWorlds Lax794877.WeightedWorlds Lax794877.Queries Lax794877.ExampleDatabase
78
79	/-- The probability of a query is the ratio of two weighted world counts. -/
80	axiom worldProb_eq_ratio :
81	∀ {L : Language.{0, 0}} [Finite (Σ n, L.Relations n)] [L.IsRelational]
82	{A : Type} [(weightedLang L).Structure A] [Finite A],
83	IsLinOrd (WLe L A) → ∀ (φ : L.Sentence)
84	(hpos : ∀ x : {q : Fact L A // IsOpen q}, 0 < presWeight x + absWeight x),
85	worldProb φ hpos = (WeightedWorlds φ A : ℚ) / (WeightedWorlds (⊤ : L.Sentence) A : ℚ)
86
87	/-- On an instance whose positions are not linearly ordered, the count is `0`. -/
88	axiom weightedWorlds_of_not_isLinOrd :
89	∀ {L : Language.{0, 0}} [Finite (Σ n, L.Relations n)] [L.IsRelational] (φ : L.Sentence)
90	(A : Type) [(weightedLang L).Structure A],
91	¬IsLinOrd (WLe L A) → WeightedWorlds φ A = 0
92
93	end Lax794877.ProbabilityRatio
94

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The probability of a query is a ratio of two #P numbers

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Natural Language Statement

Theorem

1 `weightedWorlds_of_not_isLinOrd` proven

2 `worldProb_eq_ratio` proven

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The probability of a query is a ratio of two #P numbers

Natural Language Statement

Theorem

1 weightedWorlds_of_not_isLinOrd proven

2 worldProb_eq_ratio proven

Lean source view on GitHub

Discussion

1 `weightedWorlds_of_not_isLinOrd` proven

2 `worldProb_eq_ratio` proven