File size: 13,687 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 | import OAI.Computability.PerfectCompleteness.Foundations.DualQuotientRowsLemmas
import OAI.Computability.PerfectCompleteness.Foundations.LegalRationalOutputLemmas
import OAI.Computability.PerfectCompleteness.Reduction.FixedCallBudgetLemmas
import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryLevelPair
import OAI.Computability.PerfectCompleteness.Sampling.SamplerCardinalityLemmas
namespace OAI
namespace PerfectCompleteness.PreliminaryOutput
open HierarchicalArrays
open scoped Classical
noncomputable section
variable {v m : Nat} [NeZero m]
def value (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)) : ℝ := by
let F := PreliminarySampler.family clauses branch n t rows repeats
let : Fintype (HierarchicalGame.LeftVertex F) := Fintype.ofFinite _
let : Fintype (HierarchicalGame.RightVertex F) := Fintype.ofFinite _
let : ∀ x, Fintype (HierarchicalGame.LeftLabel F x) := fun _ => Fintype.ofFinite _
let : ∀ y, Fintype (HierarchicalGame.RightLabel F y) := fun _ => Fintype.ofFinite _
exact (PreliminarySampler.game clauses branch n t rows repeats hn hbranch hrows).value
theorem exists_target (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))
(δ : ℚ) (hδ : 0 < δ) :
let F := PreliminarySampler.family clauses branch n t rows repeats
let q := CompletionAlphabet.size (Nat.card (Output branch n rows)) δ
let M := ⌈(3 : ℚ) / δ⌉₊ *
Nat.card (PreliminarySampler.Raw clauses branch n t rows repeats) * (2 * q)
∃ target : Instance q,
2 ≤ q ∧
target.leftVertices = Nat.card (HierarchicalGame.LeftVertex F) ∧
target.rightVertices = Nat.card (HierarchicalGame.RightVertex F) ∧
target.edges.length ≤ M ∧
(Encoding.gameBits target).length ≤
Nat.card (HierarchicalGame.LeftVertex F) + Nat.card (HierarchicalGame.RightVertex F) +
q + M + 4 + M *
(Nat.card (HierarchicalGame.LeftVertex F) +
Nat.card (HierarchicalGame.RightVertex F) + 2 * q * q + 2 * q + 2) ∧
((∃ assignment : Fin v → Bool,
∀ c, (clauses c).clause.eval assignment = true) → PerfectlyComplete target) ∧
(value clauses branch n t rows repeats hn hbranch hrows ≤ (δ : ℝ) / 3 →
target.value ≤ (δ : ℝ)) := by
let F := PreliminarySampler.family clauses branch n t rows repeats
let : Nonempty (PreliminarySampler.Raw clauses branch n t rows repeats) :=
SamplerRationality.raw_nonempty (t := t) clauses rows repeats hn hbranch hrows
let : Fintype (HierarchicalGame.LeftVertex F) := Fintype.ofFinite _
let : Fintype (HierarchicalGame.RightVertex F) := Fintype.ofFinite _
let : ∀ x, Fintype (HierarchicalGame.LeftLabel F x) := fun _ => Fintype.ofFinite _
let : ∀ y, Fintype (HierarchicalGame.RightLabel F y) := fun _ => Fintype.ofFinite _
let μ := PreliminarySampler.law (t := t) clauses rows repeats hn hbranch hrows
let rational := SamplerRationality.preliminaryLaw (t := t)
clauses rows repeats hn hbranch hrows
have h := CanonicalRationalOutput.exists_target (HierarchicalGame.toBlockFamily F)
μ rational.weights rational.positive rational.total rational.cast_weight δ hδ
simpa only [← Nat.card_eq_fintype_card, value, PreliminarySampler.game,
HierarchicalGame.game, F, μ] using h
theorem exists_target_size_bound (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))
(δ : ℚ) (hδ : 0 < δ) :
let q := CompletionAlphabet.size (Nat.card (Output branch n rows)) δ
let R := SamplerCardinality.occurrenceConstant branch n t rows repeats *
m ^ (t * Fintype.card (RecursiveSpaces.Slots branch n))
let M := ⌈(3 : ℚ) / δ⌉₊ * R * (2 * q)
∃ target : Instance q,
2 ≤ q ∧ target.leftVertices ≤ R ∧ target.rightVertices ≤ R ∧
target.edges.length ≤ M ∧
(Encoding.gameBits target).length ≤
2 * R + q + M + 4 + M * (2 * R + 2 * q * q + 2 * q + 2) ∧
((∃ assignment : Fin v → Bool,
∀ c, (clauses c).clause.eval assignment = true) → PerfectlyComplete target) ∧
(value clauses branch n t rows repeats hn hbranch hrows ≤ (δ : ℝ) / 3 →
target.value ≤ (δ : ℝ)) := by
let F := PreliminarySampler.family clauses branch n t rows repeats
let q := CompletionAlphabet.size (Nat.card (Output branch n rows)) δ
let R := SamplerCardinality.occurrenceConstant branch n t rows repeats *
m ^ (t * Fintype.card (RecursiveSpaces.Slots branch n))
let M := ⌈(3 : ℚ) / δ⌉₊ * R * (2 * q)
obtain ⟨target, hq, htL, htR, htE, _htBits, hc, hs⟩ :=
exists_target clauses branch n t rows repeats hn hbranch hrows δ hδ
have hraw : Nat.card (PreliminarySampler.Raw clauses branch n t rows repeats) ≤ R :=
SamplerCardinality.raw_card_le clauses branch n t rows repeats
have hlraw : Nat.card (HierarchicalGame.LeftVertex F) ≤
Nat.card (PreliminarySampler.Raw clauses branch n t rows repeats) := by
apply Nat.card_le_card_of_surjective (HierarchicalGame.leftAt F)
rintro ⟨key, e, rfl⟩
exact ⟨e, rfl⟩
have hrraw : Nat.card (HierarchicalGame.RightVertex F) ≤
Nat.card (PreliminarySampler.Raw clauses branch n t rows repeats) := by
apply Nat.card_le_card_of_surjective (HierarchicalGame.rightAt F)
rintro ⟨key, e, rfl⟩
exact ⟨e, rfl⟩
have hl : target.leftVertices ≤ R := htL.le.trans (hlraw.trans hraw)
have hr : target.rightVertices ≤ R := htR.le.trans (hrraw.trans hraw)
have he : target.edges.length ≤ M :=
htE.trans (Nat.mul_le_mul_right (2 * q)
(Nat.mul_le_mul_left ⌈(3 : ℚ) / δ⌉₊ hraw))
refine ⟨target, hq, hl, hr, he, ?_, hc, hs⟩
apply (Encoding.gameBits_length_le_of_edge_count target M he).trans
have hvertices : target.leftVertices + target.rightVertices ≤ 2 * R := by omega
have hheader := Nat.add_le_add_right
(Nat.add_le_add_right (Nat.add_le_add_right hvertices q) M) 4
have hbody := Nat.add_le_add_right
(Nat.add_le_add_right (Nat.add_le_add_right hvertices (2 * q * q)) (2 * q)) 2
exact Nat.add_le_add hheader (Nat.mul_le_mul_left M hbody)
end
end PerfectCompleteness.PreliminaryOutput
namespace PerfectCompleteness.FixedPreliminaryGame
open FixedParameters FixedRows
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem depth_pos : 0 < p.plan.depth := by have := p.plan.depth_ge; omega
theorem rows_pos (height : Nat) : 0 < rows p.plan height :=
lt_of_lt_of_le Nat.zero_lt_one (p.plan.rows_pos (p.plan.depth - height))
def game (input : List Bool) :=
PreliminarySampler.game (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1))
abbrev Strategy (input : List Bool) :=
PreliminaryStrategy.Strategy (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
def value (input : List Bool) : ℝ :=
PreliminaryOutput.value (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1))
theorem completeness (input : List Bool) (accepted : BinaryLanguage.language input) :
value p input = 1 := by
obtain ⟨assignment, satisfies⟩ :=
(PCPSourceMachine.rawSource_satisfiable_iff input).mpr accepted
let F := PreliminarySampler.family (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
let : Fintype (HierarchicalGame.LeftVertex F) := Fintype.ofFinite _
let : Fintype (HierarchicalGame.RightVertex F) := Fintype.ofFinite _
let : ∀ x, Fintype (HierarchicalGame.LeftLabel F x) := fun _ => Fintype.ofFinite _
let : ∀ y, Fintype (HierarchicalGame.RightLabel F y) := fun _ => Fintype.ofFinite _
unfold value PreliminaryOutput.value
apply HierarchicalGame.value_eq_one F _ assignment
intro occurrence
exact satisfies _ (List.getElem_mem occurrence.isLt)
theorem exists_level_pair (input : List Bool) (strategy : Strategy p input)
(hsuccess : InitialParameters.epsilon δ ≤ (game p input).success strategy) :
∃ i j : Fin p.plan.depth, i < j ∧ InitialParameters.simultaneous δ ≤
(CandidateCoupling.sharedLaw
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(rows p.plan) (repeats p.plan)
(fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1))).probability
(fun e => PreliminaryLevelPair.win
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy i e &&
PreliminaryLevelPair.win
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy j e) :=
PreliminaryLevelPair.exists_pair
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1))
strategy (InitialParameters.epsilon_pos hδ) p.plan.depth_mass hsuccess
end
end PerfectCompleteness.FixedPreliminaryGame
namespace PerfectCompleteness.FixedPreliminaryValue
open FixedParameters FixedRows FixedPreliminaryGame
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem exists_optimal_strategy (input : List Bool) :
∃ strategy : Strategy p input,
(game p input).success strategy = value p input := by
let F := PreliminarySampler.family (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
let : Fintype (HierarchicalGame.LeftVertex F) := Fintype.ofFinite _
let : Fintype (HierarchicalGame.RightVertex F) := Fintype.ofFinite _
let : ∀ x, Fintype (HierarchicalGame.LeftLabel F x) := fun _ => Fintype.ofFinite _
let : ∀ y, Fintype (HierarchicalGame.RightLabel F y) := fun _ => Fintype.ofFinite _
exact (game p input).exists_optimal_strategy
theorem value_lt_of_strategy_lt (input : List Bool) (bound : ℝ)
(hbound : ∀ strategy : Strategy p input, (game p input).success strategy < bound) :
value p input < bound := by
obtain ⟨strategy, hs⟩ := exists_optimal_strategy p input
rw [← hs]
exact hbound strategy
theorem value_le_of_no_large_strategy (input : List Bool) (bound : ℝ)
(hbound : ∀ strategy : Strategy p input, ¬bound ≤ (game p input).success strategy) :
value p input ≤ bound :=
le_of_lt (value_lt_of_strategy_lt p input bound (fun strategy =>
lt_of_not_ge (hbound strategy)))
end
end PerfectCompleteness.FixedPreliminaryValue
namespace PerfectCompleteness.PreliminaryPairReserve
open FixedParameters FixedRows
open scoped BigOperators
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem exists_pair (input : List Bool) (strategy : FixedPreliminaryGame.Strategy p input)
(hsuccess : InitialParameters.epsilon δ ≤
(FixedPreliminaryGame.game p input).success strategy) :
∃ i j : Fin p.plan.depth, i < j ∧
InitialParameters.simultaneous δ + 3 * p.accuracy ≤
(CandidateCoupling.sharedLaw
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(rows p.plan) (repeats p.plan)
(fun k _ => branch_pos p k)
(fun k => FixedPreliminaryGame.rows_pos p (k + 1))).probability
(fun e => PreliminaryLevelPair.win
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy i e &&
PreliminaryLevelPair.win
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy j e) := by
have hmean : InitialParameters.epsilon δ ≤
𝔼 level : Fin p.plan.depth,
(CandidateCoupling.sharedLaw
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(rows p.plan) (repeats p.plan)
(fun k _ => branch_pos p k)
(fun k => FixedPreliminaryGame.rows_pos p (k + 1))).probability
(PreliminaryLevelPair.win
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy level) := by
rw [← PreliminaryLevelPair.success_eq_mean
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(FixedPreliminaryGame.depth_pos p) (fun k _ => branch_pos p k)
(fun k => FixedPreliminaryGame.rows_pos p (k + 1)) strategy]
exact hsuccess
obtain ⟨i, j, hij, hprob⟩ := LevelPairReserve.exists_pair _ _
(InitialParameters.epsilon δ) (InitialParameters.epsilon_pos hδ)
hmean p.plan.depth_mass
refine ⟨i, j, hij, ?_⟩
have hsmall := p.accuracy_small 0 (Nat.zero_le _)
simp only [CommonAccuracy.tolerance, lt_min_iff] at hsmall
have hε := InitialParameters.epsilon_pos hδ
have hreserved : InitialParameters.simultaneous δ + 3 * p.accuracy ≤
2 * InitialParameters.epsilon δ ^ 2 / 3 := by
unfold InitialParameters.simultaneous
nlinarith [hsmall.1, sq_pos_of_pos hε]
exact hreserved.trans hprob
end
end PerfectCompleteness.PreliminaryPairReserve
end OAI
|