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d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Machines.ScheduledResidualProgram
import OAI.Combinatorics.ContingencyTables.Sampling.CompiledResidualCorrection
namespace OAI
/-! # Computing the exact sampler's precision from its margin-only input -/
namespace ContingencyTables.ProfilePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms Polynomial
attribute [local irreducible] marginDimension marginBinaryLength parameterWordCount
def sixthPower : Realizer (fun n : ℕ => n^6) :=
(composition (f := fun n : ℕ => n^3) (g := fun n : ℕ => n^2)
cubicPower squarePower).congr (by intro n; simp only [Function.comp_apply,←pow_mul])
lemma polynomial_sixthPower : PolynomialTime sixthPower :=
polynomial_congr _ (polynomial_composition polynomial_cubicPower polynomial_squarePower)
def exactPrecision (z : CompletionMargins) : ℕ := (marginDimension z)^6*(marginBinaryLength z)^2
def exactPrecisionRealizer : Realizer exactPrecision :=
composition (f := fun z : CompletionMargins => ((marginDimension z)^6,(marginBinaryLength z)^2))
(g := fun p : ℕ×ℕ => p.1*p.2)
(pair (composition (f := marginDimension) (g := fun n : ℕ => n^6) marginDimensionRealizer sixthPower)
(composition (f := marginBinaryLength) (g := fun n : ℕ => n^2) marginBinaryLengthRealizer squarePower))
multiply
lemma polynomial_exactPrecision : PolynomialTime exactPrecisionRealizer :=
polynomial_composition (polynomial_pair
(polynomial_composition polynomial_marginDimension polynomial_sixthPower)
(polynomial_composition polynomial_marginBinaryLength polynomial_squarePower)) polynomial_multiply
attribute [local irreducible] exactPrecision
def exactSamplingParameters (z : CompletionMargins) : SamplingParameters := (z,2*exactPrecision z)
def exactSamplingParametersRealizer : Realizer exactSamplingParameters :=
pair identity (composition (f := fun z : CompletionMargins => (2,exactPrecision z))
(g := fun p : ℕ×ℕ => p.1*p.2) (pair (constant 2) exactPrecisionRealizer) multiply)
lemma polynomial_exactSamplingParameters : PolynomialTime exactSamplingParametersRealizer :=
polynomial_pair polynomial_identity (polynomial_composition
(polynomial_pair (polynomial_constant 2) polynomial_exactPrecision) polynomial_multiply)
def exactBranchDenominator (z : CompletionMargins) : ℕ := BinaryPowerPrograms.power (2,exactPrecision z)
def exactBranchDenominatorRealizer : Realizer exactBranchDenominator :=
composition (f := fun z : CompletionMargins => (2,exactPrecision z)) (g := BinaryPowerPrograms.power)
(pair (constant 2) exactPrecisionRealizer) BinaryPowerPrograms.powerRealizer
def exactParameterMeasure (z : CompletionMargins) : ℕ := exactPrecision z+weight z
lemma work_exactBranchDenominator : WorkBound exactBranchDenominatorRealizer exactParameterMeasure := by
unfold exactBranchDenominatorRealizer
apply work_composition (work_of_polynomial (polynomial_pair (polynomial_constant 2)
polynomial_exactPrecision) X (by intro z; simp only [eval_X,exactParameterMeasure]; omega))
BinaryPowerPrograms.work_power (X+C (weight (2:ℕ)))
intro z
simp only [eval_add,eval_X,eval_C,exactParameterMeasure]
omega
def exactResidualInput (z : CompletionMargins) : SamplingParameters×ℕ :=
(exactSamplingParameters z,exactBranchDenominator z)
def exactResidualInputRealizer : Realizer exactResidualInput :=
pair exactSamplingParametersRealizer exactBranchDenominatorRealizer
lemma work_exactResidualInput : WorkBound exactResidualInputRealizer exactParameterMeasure :=
work_pair (work_of_polynomial polynomial_exactSamplingParameters X
(by intro z; simp only [eval_X,exactParameterMeasure]; omega))
work_exactBranchDenominator X (by intro z; simp only [eval_X,exactParameterMeasure]; omega)
/-- The ordinary branch needs precisely the existing public sampler's word. -/
def exactOrdinaryBits (z : CompletionMargins) : ℕ := parameterWordCount (exactSamplingParameters z)
def exactOrdinaryBitsRealizer : Realizer exactOrdinaryBits :=
composition (f := exactSamplingParameters) (g := parameterWordCount)
exactSamplingParametersRealizer parameterWordCountRealizer
lemma polynomial_exactOrdinaryBits : PolynomialTime exactOrdinaryBitsRealizer :=
polynomial_composition polynomial_exactSamplingParameters polynomial_parameterWordCount
def exactParsedMargins (as : List Symbol) : CompletionMargins := (parsedSampling as).1
def exactParsedMarginsRealizer : Realizer exactParsedMargins :=
composition (f := parsedSampling) (g := fun p : SamplingParameters => p.1) parsedSamplingRealizer first
lemma polynomial_exactParsedMargins : PolynomialTime exactParsedMarginsRealizer :=
polynomial_composition polynomial_parsedSampling polynomial_first
end ContingencyTables.ProfilePrograms
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms MatchingFPRAS.LiteralParse
open scoped BigOperators
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
lemma exactPrecision_ofFn : exactPrecision (List.ofFn r,List.ofFn c)=correctionPrecision r c := by
rw [exactPrecision,marginDimension_ofFn,marginBinaryLength_ofFn]
rfl
lemma exactSamplingParameters_ofFn :
exactSamplingParameters (List.ofFn r,List.ofFn c)=
((List.ofFn r,List.ofFn c),2*correctionPrecision r c) := by
rw [exactSamplingParameters,exactPrecision_ofFn]
lemma exactBranchDenominator_ofFn :
exactBranchDenominator (List.ofFn r,List.ofFn c)=2^(correctionPrecision r c) := by
rw [exactBranchDenominator,BinaryPowerPrograms.power_eq,exactPrecision_ofFn]
lemma exactResidualInput_ofFn :
exactResidualInput (List.ofFn r,List.ofFn c)=
(((List.ofFn r,List.ofFn c),2*correctionPrecision r c),2^(correctionPrecision r c)) := by
rw [exactResidualInput,exactSamplingParameters_ofFn,exactBranchDenominator_ofFn]
lemma exactOrdinaryBits_ofFn :
exactOrdinaryBits (List.ofFn r,List.ofFn c)=finiteSamplerBits r c (2*correctionPrecision r c) := by
rw [exactOrdinaryBits,exactSamplingParameters_ofFn,parameterWordCount_ofFn]
lemma exactParsedMargins_spec :
exactParsedMargins (Algorithms.encodeMargins r c)=(List.ofFn r,List.ofFn c) := by
unfold exactParsedMargins parsedSampling samplingInput
rw [GreedyFeasibleTable.encodeMargins_natWords,GreedyFeasibleTable.parse_natWords]
have h := GreedyFeasibleTable.marginLists_spec m n (List.ofFn r) (List.ofFn c) []
List.length_ofFn List.length_ofFn
simpa only [List.append_nil] using h
end ContingencyTables.SmallGraphProfiles
end OAI
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