The probability of h₀ is #P-hard

Lax794877.H0Complete · concepts/Lax794877/H0Complete.lean · lax-794877

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Natural Language Statement

Theorem

Counting the possible worlds of $h_0$ is one-call #P-complete, after Dalvi and Suciu, by a parsimonious reduction from #PP2DNF: the left vertices of a bipartite graph become uncertain $R$ -facts, its edges certain $S$ -facts, and its right vertices uncertain $T$ -facts, so that a world is a set of vertices and $h_0$ holds in it when some edge has both ends chosen. The numerator of the probability of $h_0$ with weights in the instance is one-call #P-complete too, already when every weight is $1$ , by an ordered parsimonious reduction from the uniform case.

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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 `possibleWorlds_h0_sharpP_oneCallComplete` proven

- lem✓Lax794877.WorldsValues
- thm✓Lax859101.BipartiteComplete
- lem✓Lax859101.BipartiteValues
→
thm✓Lax794877.H0Complete (1st statement)
⊢Lax794877Proofs.Bridge.possibleWorlds_h0_sharpP_oneCallComplete

2 `weightedWorlds_h0_sharpP_oneCallComplete` proven

- thm✓Lax794877.H0Complete
- lem✓Lax794877.WorldsValues
→
thm✓Lax794877.H0Complete (2nd statement)
⊢Lax794877Proofs.Bridge.weightedWorlds_h0_sharpP_oneCallComplete

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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 h₀ is #P-hard
57	type: theorem
58	---
59	Counting the possible worlds of $h_0$ is one-call #P-complete, after Dalvi
60	and Suciu, by a parsimonious reduction from #PP2DNF: the left vertices of a
61	bipartite graph become uncertain $R$ -facts, its edges certain $S$ -facts, and
62	its right vertices uncertain $T$ -facts, so that a world is a set of vertices
63	and $h_0$ holds in it when some edge has both ends chosen. The numerator of
64	the probability of $h_0$ with weights in the instance is one-call
65	#P-complete too, already when every weight is $1$ , by an ordered
66	parsimonious reduction from the uniform case.
67	-/
68
69	namespace Lax794877.H0Complete
70
71	open Lax904597.SecondOrder
72	open FirstOrder
73	open Language Structure
74	open Lax794877.PossibleWorlds Lax799700.Common Lax904597.SecondOrder
75	open FirstOrder.Language
76	open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
77	Lax366625.QuantitativeLogic
78	open Lax859101.OneCallReductions Lax904597.Machines
79	open Lax794877.PossibleWorlds Lax794877.WeightedWorlds Lax794877.Queries Lax794877.ExampleDatabase
80
81	/-- Counting the possible worlds of h₀ is one-call #P-complete. -/
82	axiom possibleWorlds_h0_sharpP_oneCallComplete :
83	OneCallComplete SharpP (PossibleWorlds h0)
84
85	/-- The numerator of the probability of h₀ is one-call #P-complete. -/
86	axiom weightedWorlds_h0_sharpP_oneCallComplete :
87	OneCallComplete SharpP (WeightedWorlds h0)
88
89	end Lax794877.H0Complete
90

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The probability of h₀ is #P-hard

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1 `possibleWorlds_h0_sharpP_oneCallComplete` proven

2 `weightedWorlds_h0_sharpP_oneCallComplete` proven

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The probability of h₀ is #P-hard

Natural Language Statement

Theorem

1 possibleWorlds_h0_sharpP_oneCallComplete proven

2 weightedWorlds_h0_sharpP_oneCallComplete proven

Lean source view on GitHub

Discussion

1 `possibleWorlds_h0_sharpP_oneCallComplete` proven

2 `weightedWorlds_h0_sharpP_oneCallComplete` proven