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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 | import OAI.Computability.PerfectCompleteness.Decoding.CleanDecoderRateLemmas
import OAI.Computability.PerfectCompleteness.Repetition.FixedCleanRate
namespace OAI
section
namespace PerfectCompleteness.FixedDecoderCleanRate
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
abbrev F2 := ZMod 2
structure Geometry {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ)
(n h : Nat) (upper : Nodes (FixedParameters.branch p) n)
(d : HierarchicalFrozenTables.LowerNodes upper (h + 1)) where
designated : Fin (FixedParameters.branch p h) → Slots (FixedParameters.branch p) h
cut : OwnInputReference.Cut upper (HierarchicalLeftDecoder.LowerNode upper (h + 1) d)
outside : Slots (FixedParameters.branch p) n →
Fin (FixedRows.sourceLength p.plan hδ) → MixedSupport.Slot
placeholder : Slots (FixedParameters.branch p) (h + 1) →
Fin (FixedRows.sourceLength p.plan hδ) → MixedSupport.Slot
exterior : CleanPhysicalReplay.Exterior (FixedRows.rows p.plan) (FixedRows.repeats p.plan)
(CleanRecordDecoderInput.path upper d cut) outside placeholder
variable {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ)
{n h : Nat} {upper : Nodes (FixedParameters.branch p) n}
{d : HierarchicalFrozenTables.LowerNodes upper (h + 1)}
(G : Geometry p n h upper d)
def setup (input : List Bool) :
CleanDecoderRate.Setup (FixedParameters.branch p) (FixedRows.rows p.plan)
(FixedRows.repeats p.plan) n h (FixedRows.sourceLength p.plan hδ)
(PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).variables
(PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length
upper d where
clauses := PCPSource.clauseFamily (BinaryLanguage.totalRename input)
designated := G.designated
cut := G.cut
outside := G.outside
placeholder := G.placeholder
exterior := G.exterior
theorem calls_le (input : List Bool) (hn : n ≤ p.plan.depth) :
Fintype.card (CleanDecoderRate.Calls (setup p G input)) ≤ FixedParameters.calls p.plan := by
have hactual := FixedCallBudget.modified_calls_le (FixedRows.rows p.plan)
(FixedRows.repeats p.plan) (CleanDecoderRate.path (setup p G input)) hn
apply hactual.trans
unfold FixedParameters.calls
omega
theorem cubePositive (height : Nat) : 0 < p.cubeSize height := by
have hsize := (p.cube_errors height).1
omega
theorem branchPositive : ∀ k < n, 0 < FixedParameters.branch p k :=
fun k _ => FixedParameters.branch_pos p k
variable (input : List Bool)
(σ : KeyStrategy.Strategy
(TreeCanonical.locationCount (FixedParameters.branch p) n
(FixedRows.sourceLength p.plan hδ)))
(useful : (record : CleanDecoderRate.Record (setup p G input)) →
CleanLeftDecoder.Useful
(CleanDecoderContext.leftExposed (CleanDecoderRate.context (setup p G input) record))
upper (h + 1))
(r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block (FixedRows.rows p.plan) upper) r)
(W : Submodule F2 (OwnInputReference.UpperVector (FixedRows.rows p.plan) upper))
(a : OwnInputReference.LowerVector (FixedRows.rows p.plan)
(HierarchicalLeftDecoder.LowerNode upper (h + 1) d))
(threshold : ℝ)
theorem probability_lt_accuracy (unsat : ¬ BinaryLanguage.language input)
(hn : n ≤ p.plan.depth) (hρ : 0 < ρ) :
(CleanDecoderRate.law (setup p G input) σ useful r ρ A W a threshold
(p.cubeSize h) (cubePositive p h)).probability
(fun z => CleanDecoderRate.agrees (setup p G input) (branchPositive p)
(FixedRows.cutoff p.plan hδ) z.1 z.2) < p.accuracy := by
have hbound := CleanDecoderRate.bound (setup p G input) σ useful r ρ A W a threshold
(branchPositive p) (cubePositive p h) hρ (FixedParameters.source_rate p input unsat)
apply hbound.trans_lt
simpa only [FixedParameters.branch] using
FixedCleanRate.actual_clean_error p (calls_le p G input hn) h
end
end PerfectCompleteness.FixedDecoderCleanRate
end
section
namespace PerfectCompleteness.FixedDecoderPairRate
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays FixedDecoderCleanRate
abbrev F2 := ZMod 2
variable {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ)
{n h : Nat} {upper : Nodes (FixedParameters.branch p) n}
{d : HierarchicalFrozenTables.LowerNodes upper (h + 1)}
(G : Geometry p n h upper d)
(input : List Bool)
(σ : KeyStrategy.Strategy
(TreeCanonical.locationCount (FixedParameters.branch p) n
(FixedRows.sourceLength p.plan hδ)))
(useful : (record : CleanDecoderRate.Record (setup p G input)) →
CleanLeftDecoder.Useful
(CleanDecoderContext.leftExposed (CleanDecoderRate.context (setup p G input) record))
upper (h + 1))
(r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block (FixedRows.rows p.plan) upper) r)
(W : Submodule F2 (OwnInputReference.UpperVector (FixedRows.rows p.plan) upper))
(a : OwnInputReference.LowerVector (FixedRows.rows p.plan)
(HierarchicalLeftDecoder.LowerNode upper (h + 1) d))
(threshold : ℝ)
theorem probability_lt_accuracy (unsat : ¬ BinaryLanguage.language input)
(hn : n ≤ p.plan.depth) (hρ : 0 < ρ) :
(CleanDecoderPairLaw.pairLaw (setup p G input) σ useful r ρ A W a threshold
(p.cubeSize h) (cubePositive p h)).probability
(fun z => CleanDecoderPairLaw.pairAgrees (setup p G input) (branchPositive p)
(FixedRows.cutoff p.plan hδ) z.1 z.2) < p.accuracy := by
rw [CleanDecoderPairLaw.pair_probability_eq]
exact FixedDecoderCleanRate.probability_lt_accuracy p G input σ useful r ρ A W a threshold
unsat hn hρ
end
end PerfectCompleteness.FixedDecoderPairRate
end
end OAI
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