File size: 9,396 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import OAI.Computability.PerfectCompleteness.Machines.EncodingLemmas
import OAI.Computability.PerfectCompleteness.Reduction.CanonicalNamedGame
import OAI.Computability.PerfectCompleteness.Reduction.CompletedVisitOrder
import OAI.Computability.PerfectCompleteness.Reduction.TargetGameLemmas
namespace OAI
section
namespace PerfectCompleteness.ExactPreliminaryTarget
open UniqueGamesTheorem.Foundations.Games
open CompletionSoundness CompletionSoundness.LegalProjectionGame
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n t v m : Nat} [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)) (δ : ℚ)
local notation "F" => FinitePreliminaryCompletion.blockFamily clauses branch n t rows repeats
local notation "q" => FinitePreliminaryCompletion.alphabet branch n t δ
local notation "family" => FinitePreliminaryCompletion.actualFamily
(t := t) clauses rows repeats hn hbranch hrows δ
local notation "order" => CompletedVisitOrder.completedOrder
(t := t) clauses rows repeats hn hbranch hrows δ
local notation "rawOrder" => CompletedVisitOrder.rawOrder
(t := t) clauses rows repeats hn hbranch hrows δ
local notation "coverage" => CompletedVisitOrder.mem_rawOrder
(t := t) clauses rows repeats hn hbranch hrows δ
local instance leftVertex_fintype : Fintype (CanonicalGame.LeftVertex F) := Fintype.ofFinite _
local instance rightVertex_fintype : Fintype (CanonicalGame.RightVertex F) := Fintype.ofFinite _
local instance leftLabel_fintype (x : CanonicalGame.LeftVertex F) :
Fintype (CanonicalGame.LeftLabel F x) := Fintype.ofFinite _
local instance rightLabel_fintype (y : CanonicalGame.RightVertex F) :
Fintype (CanonicalGame.RightLabel F y) := Fintype.ofFinite _
local instance rightAlphabet_nonempty : Nonempty (Fin q) :=
⟨⟨0, CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ⟩⟩
local instance leftAlphabet_nonempty : Nonempty (Fin (2 * q)) :=
⟨⟨0, Nat.mul_pos (by decide)
(CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ)⟩⟩
def namedGame := CanonicalNamedGame.rename F rawOrder coverage (family).game
def entry (e : CompletedSignedLaw.Completed (t := t) clauses rows repeats hn hbranch hrows δ) :
Edge (rawOrder).length (rawOrder).length q where
left := (namedGame (t := t) clauses rows repeats hn hbranch hrows δ).left e
right := (namedGame (t := t) clauses rows repeats hn hbranch hrows δ).right e
projection := ProjectionTable.ofMap ((family).map e.1 e.2) (by
intro b
rw [Fintype.card_eq_nat_card]
have h := (family).exact_two e.1 e.2 b
rw [Fintype.card_eq_nat_card] at h
exact h)
theorem entry_satisfied (e : CompletedSignedLaw.Completed (t := t) clauses rows repeats hn hbranch hrows δ)
(a : Fin (rawOrder).length → Fin (2 * q)) (b : Fin (rawOrder).length → Fin q) :
(entry (t := t) clauses rows repeats hn hbranch hrows δ e).satisfied a b =
(namedGame (t := t) clauses rows repeats hn hbranch hrows δ).wins (a, b) e := by
simp only [Edge.satisfied, entry, ProjectionTable.ofMap_apply]
exact decide_eq_decide.mpr Iff.rfl
def construct : Instance q where
leftVertices := (rawOrder).length
rightVertices := (rawOrder).length
edges := (order).map (entry (t := t) clauses rows repeats hn hbranch hrows δ)
nonempty := by
intro hempty
have h := List.map_eq_nil_iff.mp hempty
exact CompletedVisitOrder.completedOrder_nonempty
(t := t) clauses rows repeats hn hbranch hrows δ h
theorem edge_count :
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges.length =
SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q *
m ^ TreeCanonical.locationCount branch n t := by
change ((order).map (entry (t := t) clauses rows repeats hn hbranch hrows δ)).length = _
rw [List.length_map]
exact CompletedVisitOrder.completedOrder_length
(t := t) clauses rows repeats hn hbranch hrows δ
theorem leftVertices_eq_edges :
(construct (t := t) clauses rows repeats hn hbranch hrows δ).leftVertices =
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges.length := by
exact (CompletedVisitOrder.rawOrder_length
(t := t) clauses rows repeats hn hbranch hrows δ).trans
(edge_count (t := t) clauses rows repeats hn hbranch hrows δ).symm
theorem rightVertices_eq_edges :
(construct (t := t) clauses rows repeats hn hbranch hrows δ).rightVertices =
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges.length :=
leftVertices_eq_edges (t := t) clauses rows repeats hn hbranch hrows δ
theorem rate_eq (a : Fin (rawOrder).length → Fin (2 * q)) (b : Fin (rawOrder).length → Fin q) :
(countSatisfied a b (construct (t := t) clauses rows repeats hn hbranch hrows δ).edges : ℝ) /
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges.length =
(namedGame (t := t) clauses rows repeats hn hbranch hrows δ).success (a, b) := by
change (countSatisfied a b
((order).map (entry (t := t) clauses rows repeats hn hbranch hrows δ)) : ℝ) /
((order).map (entry (t := t) clauses rows repeats hn hbranch hrows δ)).length = _
rw [RoundedTarget.countSatisfied_eq_filter]
simp only [List.filter_map, List.length_map, Function.comp_def]
have h := CompletedVisitOrder.completedOrder_probability
(t := t) clauses rows repeats hn hbranch hrows δ
(fun e => (entry (t := t) clauses rows repeats hn hbranch hrows δ e).satisfied a b)
simpa only [Function.comp_def, entry_satisfied, OccurrenceGame.success,
namedGame, CanonicalNamedGame.rename_occurrences] using h
theorem value_eq :
(construct (t := t) clauses rows repeats hn hbranch hrows δ).value =
FinitePreliminaryCompletion.actualValue
(t := t) clauses rows repeats hn hbranch hrows δ := by
have hq : 0 < q :=
CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ
have hnamed : (construct (t := t) clauses rows repeats hn hbranch hrows δ).value =
(namedGame (t := t) clauses rows repeats hn hbranch hrows δ).value := by
apply le_antisymm
· obtain ⟨s, hs⟩ :=
(construct (t := t) clauses rows repeats hn hbranch hrows δ).maxSatisfied_attained hq
unfold Instance.value
rw [← hs]
exact (rate_eq (t := t) clauses rows repeats hn hbranch hrows δ s.1 s.2).le.trans
((namedGame (t := t) clauses rows repeats hn hbranch hrows δ).success_le_value s)
· obtain ⟨s, hs⟩ :=
(namedGame (t := t) clauses rows repeats hn hbranch hrows δ).exists_optimal_strategy
calc
(namedGame (t := t) clauses rows repeats hn hbranch hrows δ).value =
(namedGame (t := t) clauses rows repeats hn hbranch hrows δ).success s := hs.symm
_ = (countSatisfied s.1 s.2
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges : ℝ) /
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges.length :=
(rate_eq (t := t) clauses rows repeats hn hbranch hrows δ s.1 s.2).symm
_ ≤ (construct (t := t) clauses rows repeats hn hbranch hrows δ).value :=
(construct (t := t) clauses rows repeats hn hbranch hrows δ).rate_le_value s
exact hnamed.trans (CanonicalNamedGame.rename_value_eq F rawOrder coverage (family).game)
theorem value_le (hδ : 0 < δ) :
(construct (t := t) clauses rows repeats hn hbranch hrows δ).value ≤
(PreliminarySampler.game clauses branch n t rows repeats hn hbranch hrows).value +
(δ : ℝ) / 3 := by
rw [value_eq]
exact FinitePreliminaryCompletion.actualValue_le
(t := t) clauses rows repeats hn hbranch hrows δ hδ
theorem perfectlyComplete (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true) :
PerfectlyComplete (construct (t := t) clauses rows repeats hn hbranch hrows δ) := by
apply ((construct (t := t) clauses rows repeats hn hbranch hrows δ).perfectlyComplete_iff_value_eq_one
(CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ)).2
rw [value_eq]
exact FinitePreliminaryCompletion.actualValue_of_satisfying
(t := t) clauses rows repeats hn hbranch hrows δ assignment hsat
theorem encoding_length_le :
let target := construct (t := t) clauses rows repeats hn hbranch hrows δ
let N := target.edges.length
(Encoding.gameBits target).length ≤
2 * N + q + N + 4 + N * (2 * N + 2 * q * q + 2 * q + 2) := by
dsimp only
have h := Encoding.gameBits_length_le_of_edge_count
(construct (t := t) clauses rows repeats hn hbranch hrows δ)
(construct (t := t) clauses rows repeats hn hbranch hrows δ).edges.length le_rfl
have arithmetic (l r N : Nat) (hl : l = N) (hr : r = N) :
l + r + q + N + 4 + N * (l + r + 2 * q * q + 2 * q + 2) =
2 * N + q + N + 4 + N * (2 * N + 2 * q * q + 2 * q + 2) := by
simp only [hl, hr, two_mul]
exact h.trans_eq (arithmetic _ _ _
(leftVertices_eq_edges (t := t) clauses rows repeats hn hbranch hrows δ)
(rightVertices_eq_edges (t := t) clauses rows repeats hn hbranch hrows δ))
end
end PerfectCompleteness.ExactPreliminaryTarget
end
end OAI
|