| import Mathlib |
| import OAI.Computability.MaxCut.PCP.TableIteration |
| import OAI.Computability.MaxCut.PCP.RoundComputation |
| import OAI.Computability.MaxCut.Games.Probability |
| import OAI.Computability.MaxCut.Machines.MachineLazyTableRuntime |
| import OAI.Computability.MaxCut.Machines.MachineTableIteration |
|
|
| namespace OAI |
|
|
| namespace MaxCutGames.Outer.HastadSource |
|
|
| open MaxCutGames.Foundations Target Complexity PCP |
| open MaxCutGames.Reduction ActualSource |
| open MachineFiniteAlphabet |
|
|
| def success (input : SourceEncoding.Input) (bits : Fin input.«variables» → Bool) : ℚ := |
| (input.equations.countP (fun e => CloneGap.satisfied e bits) : ℚ) / |
| input.equations.length |
|
|
| def asSource (input : SourceEncoding.Input) : Source := |
| Source.ofList input.equations input.nonempty |
|
|
| theorem failure_add_success (input : SourceEncoding.Input) |
| (bits : Fin input.«variables» → Bool) : |
| (asSource input).failure bits + success input bits = 1 := by |
| have h := SourceProbability.failure_add_satisfaction (asSource input) bits |
| rw [show (asSource input).sourceList = input.equations from |
| Source.sourceList_ofList input.equations input.nonempty] at h |
| exact h |
|
|
| def gapInput (H : RoundTables.BaseTable) (F : Formula) : |
| Hastad.SourceGeneratorContract.NonemptyFormula := |
| ⟨TableIteration.gapMap H F, TableIteration.gapMap_nonempty H F⟩ |
|
|
| def sourceMap (ξ : ℚ) (hξ : 0 < ξ) := |
| Hastad.SourceGeneratorContract.errorMap PCPIteration.finalClauseGap ξ |
| PCPIteration.finalClauseGap_positive TableGapReduction.finalClauseGap_le_one hξ |
|
|
| def output (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) |
| (F : Formula) : SourceEncoding.Input := |
| sourceMap ξ hξ (gapInput H F) |
|
|
| theorem complete (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) |
| (F : Formula) (hF : F.Satisfiable) : |
| ∃ bits, 1 - ξ ≤ success (output H ξ hξ F) bits := by |
| exact Hastad.SourceGap.forError_complete PCPIteration.finalClauseGap ξ |
| PCPIteration.finalClauseGap_positive TableGapReduction.finalClauseGap_le_one hξ |
| (TableIteration.gapMap H F) (TableIteration.gapMap_completeness H F hF) |
|
|
| theorem sound (H : RoundTables.BaseTable) |
| (certificate : SpectralReturn.SpectralCertificate (ExpanderTables.graph H) (1 / 100 : ℝ)) |
| (ξ : ℚ) (hξ : 0 < ξ) (F : Formula) (hF : ¬ F.Satisfiable) |
| (bits : Fin (output H ξ hξ F).«variables» → Bool) : |
| success (output H ξ hξ F) bits ≤ (1 + ξ) / 2 := by |
| exact Hastad.SourceGap.forError_sound PCPIteration.finalClauseGap ξ |
| PCPIteration.finalClauseGap_positive TableGapReduction.finalClauseGap_le_one hξ |
| (TableIteration.gapMap H F) (TableGapReduction.gapMap_clauseGap H certificate F hF) bits |
|
|
| noncomputable def roundMachine (H : RoundTables.BaseTable) := |
| RoundComputation.tablePolynomialTime H (PreprocessingRuntime.tablePolynomialTime H) |
|
|
| theorem roundMachine_finite (H : RoundTables.BaseTable) : |
| FiniteAlphabet (roundMachine H).tm := |
| RoundComputation.tablePolynomialTime_finite_alphabet H |
| (PreprocessingRuntime.tablePolynomialTime H) (PreprocessingRuntime.finiteAlphabet H) |
|
|
| noncomputable def gapMachine (H : RoundTables.BaseTable) : |
| Turing.TM2ComputableInPolyTime formulaBits Hastad.SourceGeneratorContract.inputEncoding |
| (gapInput H) where |
| toTM2ComputableAux := |
| (MachineTableIteration.gapMapCertificate H (roundMachine H)).toTM2ComputableAux |
| time := (MachineTableIteration.gapMapCertificate H (roundMachine H)).time |
| outputsFun F := (MachineTableIteration.gapMapCertificate H (roundMachine H)).outputsFun F |
|
|
| noncomputable def sourceMachine (ξ : ℚ) (hξ : 0 < ξ) : |
| Turing.TM2ComputableInPolyTime Hastad.SourceGeneratorContract.inputEncoding |
| SourceEncoding.inputBits (sourceMap ξ hξ) := |
| Hastad.SourceGenerator.forErrorComputableInPolyTime PCPIteration.finalClauseGap ξ |
| PCPIteration.finalClauseGap_positive TableGapReduction.finalClauseGap_le_one hξ |
|
|
| noncomputable def computation (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) : |
| Turing.TM2ComputableInPolyTime formulaBits SourceEncoding.inputBits (output H ξ hξ) := |
| MachineSequential.composeBits (f := gapInput H) (g := sourceMap ξ hξ) |
| (gapMachine H) (sourceMachine ξ hξ) |
|
|
| theorem finiteAlphabet (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) : |
| FiniteAlphabet (computation H ξ hξ).tm := |
| MachineFiniteAlphabet.composeBits (gapMachine H) (sourceMachine ξ hξ) |
| (TableIterationFiniteAlphabet.gapMap H (roundMachine H) (roundMachine_finite H)) |
| (Hastad.SourceGenerator.forError_finite_work_alphabets PCPIteration.finalClauseGap ξ |
| PCPIteration.finalClauseGap_positive TableGapReduction.finalClauseGap_le_one hξ) |
|
|
| structure Reduction (ξ : ℚ) where |
| reduce : Formula → SourceEncoding.Input |
| completeness : ∀ F, F.Satisfiable → ∃ bits, 1 - ξ ≤ success (reduce F) bits |
| soundness : ∀ F, ¬ F.Satisfiable → ∀ bits, success (reduce F) bits ≤ (1 + ξ) / 2 |
| computation : Turing.TM2ComputableInPolyTime formulaBits SourceEncoding.inputBits reduce |
| finiteAlphabet : FiniteAlphabet computation.tm |
|
|
| theorem exists_reduction (ξ : ℚ) (hξ : 0 < ξ) : Nonempty (Reduction ξ) := by |
| obtain ⟨H, certificate⟩ := ExpanderTables.exists_base_table |
| exact ⟨⟨output H ξ hξ, complete H ξ hξ, sound H certificate ξ hξ, |
| computation H ξ hξ, finiteAlphabet H ξ hξ⟩⟩ |
|
|
| end MaxCutGames.Outer.HastadSource |
|
|
| end OAI |
|
|
