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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | import OAI.Computability.PerfectCompleteness.Foundations.GeometricPathLemmas
import OAI.Computability.PerfectCompleteness.Reduction.CompletionAlphabetLemmas
import OAI.Computability.PerfectCompleteness.Sampling.WholeArraySampler
namespace OAI
section
namespace PerfectCompleteness.PreliminarySampler
open RecursiveSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
variable {v m n t : Nat} {branch rows repeats : Nat → Nat}
abbrev Questions (branch : Nat → Nat) (n t m : Nat) :=
Slots branch n → Fin t → Fin m
def endpoints (E : Questions branch n t m) :
Slots branch n → Fin t → SourceKeys.Endpoint v m :=
fun leaf coordinate => .clause (E leaf coordinate)
abbrev Base (branch : Nat → Nat) (n t m : Nat) :=
Questions branch n t m × Slots branch n
abbrev ArrayTape (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (b : Base branch n t m) :=
WholeArraySampler.Tape rows repeats (GeometricPath.leafPath b.2)
(sourceSlots clauses (endpoints b.1))
abbrev Choice (rows : Nat → Nat) (leaf : Slots branch n) :=
(level : Fin n) × BucketSampler.Direction
(rows (Nodes.height (GeometricPath.nodeAtLevel leaf level)))
abbrev Raw (clauses : Fin m → SourceClause.NormalizedClause v)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) :=
(b : Base branch n t m) × (ArrayTape clauses rows repeats b × Choice rows b.2)
theorem directionNonempty (width : Nat) (positive : 0 < width) :
Nonempty (BucketSampler.Direction width) := by
refine ⟨⟨fun _ => 1, ?_⟩⟩
intro hz
have h := congrFun hz ⟨0, positive⟩
change (1 : BucketSampler.F2) = 0 at h
exact one_ne_zero h
def questionsLaw [NeZero m] : FiniteDistribution (Questions branch n t m) :=
FiniteDistribution.uniform _
theorem questionsLaw_eq_product [NeZero m] :
questionsLaw (branch := branch) (n := n) (t := t) (m := m) =
FiniteProduct.law (fun _ : Slots branch n =>
FiniteProduct.law (fun _ : Fin t => FiniteDistribution.uniform (Fin m))) := by
simp only [UniformLinearImage.law_uniform]
rfl
def baseLaw [NeZero m] (hbranch : ∀ k < n, 0 < branch k) :
FiniteDistribution (Base branch n t m) :=
questionsLaw.product (GeometricPath.law n hbranch)
def choiceLaw (leaf : Slots branch n) (hn : 0 < n)
(hrows : ∀ k, 0 < rows (k + 1)) : FiniteDistribution (Choice rows leaf) := by
letI : Nonempty (Fin n) := ⟨⟨0, hn⟩⟩
exact CompletionSoundness.sigmaLaw (FiniteDistribution.uniform (Fin n)) fun level => by
letI : Nonempty (BucketSampler.Direction
(rows (Nodes.height (GeometricPath.nodeAtLevel leaf level)))) :=
directionNonempty _ (by rw [GeometricPath.nodeAtLevel_height]; exact hrows level.val)
exact FiniteDistribution.uniform _
theorem choiceLaw_level (leaf : Slots branch n) (hn : 0 < n)
(hrows : ∀ k, 0 < rows (k + 1)) (event : Fin n → Bool) :
(choiceLaw leaf hn hrows).probability (fun c => event c.1) =
(by
letI : Nonempty (Fin n) := ⟨⟨0, hn⟩⟩
exact (FiniteDistribution.uniform (Fin n)).probability event) := by
exact CompletionSoundness.sigmaLaw_probability_first _ _ event
def conditionalLaw (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hrows : ∀ k, 0 < rows (k + 1)) (b : Base branch n t m) :
FiniteDistribution (ArrayTape clauses rows repeats b × Choice rows b.2) :=
(WholeArraySampler.tapeLaw rows repeats (GeometricPath.leafPath b.2)
(sourceSlots clauses (endpoints b.1))).product (choiceLaw b.2 hn hrows)
def law [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) :
FiniteDistribution (Raw clauses branch n t rows repeats) :=
CompletionSoundness.sigmaLaw (baseLaw hbranch)
(conditionalLaw clauses rows repeats hn hrows)
theorem law_base [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1))
(event : Base branch n t m → Bool) :
(law clauses rows repeats hn hbranch hrows).probability (fun e => event e.1) =
(baseLaw hbranch).probability event :=
CompletionSoundness.sigmaLaw_probability_first _ _ event
theorem law_probability [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1))
(event : Raw clauses branch n t rows repeats → Bool) :
(law clauses rows repeats hn hbranch hrows).probability event =
(baseLaw hbranch).expectation (fun b =>
(conditionalLaw clauses rows repeats hn hrows b).probability
(fun tapeAndChoice => event ⟨b, tapeAndChoice⟩)) :=
CompletionSoundness.sigmaLaw_probability _ _ event
def family (clauses : Fin m → SourceClause.NormalizedClause v)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) :
HierarchicalGame.Family clauses branch n t rows (Raw clauses branch n t rows repeats) where
endpoints e := endpoints e.1.1
arrays e := WholeArraySampler.evaluate rows repeats (GeometricPath.leafPath e.1.2)
(sourceSlots clauses (endpoints e.1.1)) e.2.1
selected e := GeometricPath.nodeAtLevel e.1.2 e.2.2.1
direction e := e.2.2.2.val
nonzero e := e.2.2.2.property
theorem family_selected_height (clauses : Fin m → SourceClause.NormalizedClause v)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat)
(e : Raw clauses branch n t rows repeats) :
Nodes.height ((family clauses branch n t rows repeats).selected e) = e.2.2.1.val + 1 :=
GeometricPath.nodeAtLevel_height _ _
def game [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) :=
HierarchicalGame.game (family clauses branch n t rows repeats)
(law clauses rows repeats hn hbranch hrows)
theorem edge_at_most_two (clauses : Fin m → SourceClause.NormalizedClause v)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat)
(e : Raw clauses branch n t rows repeats)
(Q : HierarchicalGame.RightLabel (family clauses branch n t rows repeats)
(HierarchicalGame.rightAt (family clauses branch n t rows repeats) e)) :
Nat.card {P : HierarchicalGame.LeftLabel (family clauses branch n t rows repeats)
(HierarchicalGame.leftAt (family clauses branch n t rows repeats) e) //
HierarchicalGame.edge (family clauses branch n t rows repeats) e P = Q} ≤ 2 :=
HierarchicalGame.edge_at_most_two _ e Q
theorem completeness [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1))
(assignment : Fin v → Bool) (hsat : ∀ c, (clauses c).clause.eval assignment = true) :
(game clauses branch n t rows repeats hn hbranch hrows).success
(HierarchicalGame.satisfyingStrategy (family clauses branch n t rows repeats)
assignment hsat) = 1 :=
HierarchicalGame.success_satisfyingStrategy _ _ assignment hsat
end
end PerfectCompleteness.PreliminarySampler
end
end OAI
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