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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 | import OAI.Computability.PerfectCompleteness.Decoding.DecoderFamilyLawLemmas
import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedDecoderContext
import OAI.Computability.PerfectCompleteness.Foundations.SlotProjectedExperiment
import OAI.Computability.PerfectCompleteness.Reduction.FixedPhysicalStoppedComparison
import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryOutputLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedDecoderLaw
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
private abbrev familySampleFintype
{branch rows repeats : Nat → Nat} {n h t v m : Nat} [NeZero m]
{O : Type*} [Fintype O]
{upper : O → Nodes branch n}
{lower : (o : O) → HierarchicalFrozenTables.LowerNodes (upper o) (h + 1)}
(S : (o : O) → CleanDecoderRate.Setup branch rows repeats n h t v m (upper o) (lower o)) :
Fintype (DecoderFamilyLaw.Sample S) := inferInstance
attribute [local instance] linearMapFintype
variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ)
(i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool)
abbrev Context := FixedStoppedDecoderContext.Context parameters i j hij input
def labeling (strategy : FixedPreliminaryGame.Strategy parameters input) :
KeyStrategy.Strategy (TreeCanonical.locationCount (FixedParameters.branch parameters)
parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ)) :=
PreliminaryStrategy.extend (PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(FixedParameters.branch parameters) parameters.plan.depth
(FixedRows.sourceLength parameters.plan hδ) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) strategy
def flag : Fin (FixedParameters.branch parameters i.val) → FiniteDistribution Bool :=
fun _ => ProjectionPosterior.bernoulli
(FixedParameters.projectionProbability parameters i.val : ℝ)
(FixedPhysicalStoppedComparison.beta_nonneg parameters)
(FixedPhysicalStoppedComparison.beta_le_one parameters)
def markFamily (strategy : FixedPreliminaryGame.Strategy parameters input)
(pref : FixedStoppedDecoderContext.Prefixes parameters i j) :=
SlotProjectedExperiment.family
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij pref.1
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))
(flag parameters i) (labeling parameters input strategy)
abbrev setups :=
FixedDecoderFamilyRate.setups parameters
(FixedStoppedDecoderContext.geometry parameters i j hij input) input
def useful (strategy : FixedPreliminaryGame.Strategy parameters input)
(context : Context parameters i j hij input)
(record : CleanDecoderRate.Record (setups parameters i j hij input context)) :
CleanLeftDecoder.Useful
(CleanDecoderContext.leftExposed
(CleanDecoderRate.context (setups parameters i j hij input context) record))
(FixedStoppedDecoderContext.upperNode parameters i j hij input context) (i.val + 1) :=
OriginalDecoderMark.useful
(markFamily parameters i j hij input strategy context.1)
(FixedStoppedDecoderContext.upperNode parameters i j hij input context) (i.val + 1)
(InitialParameters.useful δ)
(CleanDecoderContext.leftExposed
(CleanDecoderRate.context (setups parameters i j hij input context) record))
def threshold : ℝ :=
UpperParameterScalars.tau (InitialParameters.useful δ) parameters.plan.density
parameters.plan.order (FixedRows.rows parameters.plan (j.val + 1))
abbrev Sample := DecoderFamilyLaw.Sample (setups parameters i j hij input)
local instance sampleFintype : Fintype (Sample parameters i j hij input) :=
familySampleFintype (setups parameters i j hij input)
def law (strategy : FixedPreliminaryGame.Strategy parameters input) :
FiniteDistribution (Sample parameters i j hij input) :=
DecoderFamilyLaw.law (setups parameters i j hij input)
(labeling parameters input strategy) (useful parameters i j hij input strategy)
parameters.plan.order parameters.plan.density
(FixedStoppedDecoderContext.advice parameters i j hij input) (FixedStoppedDecoderContext.direction parameters i j hij input)
(threshold parameters j) (FixedStoppedDecoderContext.law parameters i j hij input)
(parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val)
abbrev meeting : Sample parameters i j hij input → Bool :=
DecoderFamilyLaw.meeting (setups parameters i j hij input)
abbrev lowAgreement : Sample parameters i j hij input → Bool :=
DecoderFamilyLaw.lowAgreement (setups parameters i j hij input)
(FixedDecoderCleanRate.branchPositive parameters)
(FixedRows.cutoff parameters.plan hδ)
abbrev highAgreement : Sample parameters i j hij input → Bool :=
DecoderFamilyLaw.highAgreement (setups parameters i j hij input)
(FixedDecoderCleanRate.branchPositive parameters)
(FixedRows.cutoff parameters.plan hδ)
theorem probability_split (strategy : FixedPreliminaryGame.Strategy parameters input) :
(law parameters i j hij input strategy).probability (meeting parameters i j hij input) =
(law parameters i j hij input strategy).probability
(lowAgreement parameters i j hij input) +
(law parameters i j hij input strategy).probability
(highAgreement parameters i j hij input) :=
DecoderFamilyLaw.probability_split (setups parameters i j hij input)
(FixedDecoderCleanRate.branchPositive parameters)
(labeling parameters input strategy) (useful parameters i j hij input strategy)
parameters.plan.order parameters.plan.density
(FixedStoppedDecoderContext.advice parameters i j hij input) (FixedStoppedDecoderContext.direction parameters i j hij input)
(threshold parameters j) (FixedStoppedDecoderContext.law parameters i j hij input)
(FixedRows.cutoff parameters.plan hδ) (parameters.cubeSize i.val)
(FixedDecoderCleanRate.cubePositive parameters i.val)
theorem low_probability_lt_accuracy
(strategy : FixedPreliminaryGame.Strategy parameters input)
(unsat : ¬ BinaryLanguage.language input) :
(law parameters i j hij input strategy).probability
(lowAgreement parameters i j hij input) < parameters.accuracy :=
FixedDecoderFamilyRate.probability_lt_accuracy parameters
(FixedStoppedDecoderContext.geometry parameters i j hij input) input
(labeling parameters input strategy) (useful parameters i j hij input strategy)
parameters.plan.order parameters.plan.density
(FixedStoppedDecoderContext.advice parameters i j hij input) (FixedStoppedDecoderContext.direction parameters i j hij input)
(threshold parameters j) (FixedStoppedDecoderContext.law parameters i j hij input)
unsat (le_refl _) parameters.plan.density_pos
end
end PerfectCompleteness.FixedStoppedDecoderLaw
end
end OAI
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