| import Mathlib |
| import OAI.Computability.MaxCut.Machines.MachineBinaryInputReduction |
|
|
| namespace OAI |
|
|
| namespace MaxCutGames.Outer.BinaryParityReduction |
|
|
| open MaxCutGames.Foundations Target Complexity PCP |
| open MaxCutGames.Reduction ActualSource |
| open MachineFiniteAlphabet |
|
|
| noncomputable section |
|
|
| def output (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) |
| (word : List Bool) : SourceEncoding.Input := |
| SourceIncidence.input H ξ hξ (BinaryLanguage.totalRename word) |
|
|
| def formulaMachine (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) : |
| Turing.TM2ComputableInPolyTime formulaBits SourceEncoding.inputBits |
| (SourceIncidence.input H ξ hξ) := |
| MachineSequential.composeBits |
| (f := HastadSource.output H ξ hξ) (g := Clone100.clonedInput) |
| (HastadSource.computation H ξ hξ) |
| Explicit.MachineClone100.computation |
|
|
| theorem formulaMachine_finite (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) : |
| FiniteAlphabet (formulaMachine H ξ hξ).tm := |
| MachineFiniteAlphabet.composeBits |
| (f := HastadSource.output H ξ hξ) (g := Clone100.clonedInput) |
| (HastadSource.computation H ξ hξ) |
| Explicit.MachineClone100.computation |
| (HastadSource.finiteAlphabet H ξ hξ) Explicit.MachineClone100.finiteAlphabet |
|
|
| def computation (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) : |
| Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) SourceEncoding.inputBits |
| (output H ξ hξ) := |
| MachineSequential.composeBits |
| (f := BinaryLanguage.totalRename) (g := SourceIncidence.input H ξ hξ) |
| BinaryInputReduction.computation (formulaMachine H ξ hξ) |
|
|
| theorem computation_finite (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) : |
| FiniteAlphabet (computation H ξ hξ).tm := |
| MachineFiniteAlphabet.composeBits |
| (f := BinaryLanguage.totalRename) (g := SourceIncidence.input H ξ hξ) |
| BinaryInputReduction.computation (formulaMachine H ξ hξ) |
| BinaryInputReduction.computation_finiteAlphabet (formulaMachine_finite H ξ hξ) |
|
|
| theorem complete (H : RoundTables.BaseTable) (ξ : ℚ) (hξ : 0 < ξ) |
| (word : List Bool) (yes : BinaryLanguage.language word) : |
| ∃ a, 1 - ξ ≤ Clone100.success (output H ξ hξ word).equations a := by |
| obtain ⟨a, ha⟩ := HastadSource.complete H ξ hξ (BinaryLanguage.totalRename word) |
| ((BinaryLanguage.totalRename_satisfiable_iff word).mpr yes) |
| refine ⟨Clone100.lift a, ?_⟩ |
| change 1 - ξ ≤ Clone100.success |
| (Clone100.clonedInput (HastadSource.output H ξ hξ (BinaryLanguage.totalRename word))).equations |
| (Clone100.lift a) |
| rw [Clone100.completeness] |
| exact ha |
|
|
| theorem sound (H : RoundTables.BaseTable) |
| (certificate : SpectralReturn.SpectralCertificate (ExpanderTables.graph H) (1 / 100 : ℝ)) |
| (ξ : ℚ) (hξ : 0 < ξ) (small : ξ < 1 / 100) |
| (word : List Bool) (no : ¬ BinaryLanguage.language word) |
| (a : Fin (output H ξ hξ word).«variables» → Bool) : |
| Clone100.success (output H ξ hξ word).equations a ≤ 4 / 5 := by |
| have hno : ¬ (BinaryLanguage.totalRename word).Satisfiable := |
| fun h => no ((BinaryLanguage.totalRename_satisfiable_iff word).mp h) |
| exact Clone100.soundness_four_fifths |
| (HastadSource.output H ξ hξ (BinaryLanguage.totalRename word)) ξ small |
| (HastadSource.sound H certificate ξ hξ (BinaryLanguage.totalRename word) hno) a |
|
|
| structure Reduction (ξ : ℚ) where |
| reduce : List Bool → SourceEncoding.Input |
| distinct : ∀ word, (HastadSource.asSource (reduce word)).DistinctNames |
| completeness : ∀ word, BinaryLanguage.language word → |
| ∃ a, 1 - ξ ≤ Clone100.success (reduce word).equations a |
| soundness : ∀ word, ¬ BinaryLanguage.language word → |
| ∀ a, Clone100.success (reduce word).equations a ≤ 4 / 5 |
| computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) |
| SourceEncoding.inputBits reduce |
| finiteAlphabet : FiniteAlphabet computation.tm |
|
|
| theorem exists_reduction (ξ : ℚ) (positive : 0 < ξ) (small : ξ < 1 / 100) : |
| Nonempty (Reduction ξ) := by |
| obtain ⟨H, certificate⟩ := ExpanderTables.exists_base_table |
| exact ⟨⟨output H ξ positive, |
| (fun word => SourceIncidence.source_distinct H ξ positive (BinaryLanguage.totalRename word)), |
| complete H ξ positive, sound H certificate ξ positive small, |
| computation H ξ positive, computation_finite H ξ positive⟩⟩ |
|
|
| theorem failure_completeness {ξ : ℚ} (R : Reduction ξ) (word : List Bool) |
| (yes : BinaryLanguage.language word) : |
| ∃ a, (HastadSource.asSource (R.reduce word)).failure a ≤ ξ := by |
| obtain ⟨a, ha⟩ := R.completeness word yes |
| refine ⟨a, ?_⟩ |
| have hsum := HastadSource.failure_add_success (R.reduce word) a |
| change _ + Clone100.success (R.reduce word).equations a = 1 at hsum |
| linarith |
|
|
| theorem parity_soundness {ξ : ℚ} (R : Reduction ξ) (word : List Bool) |
| (no : ¬ BinaryLanguage.language word) : |
| MaxCutGames.Clean.IncidenceGap.parityValue |
| (SourceIncidence.incidence (HastadSource.asSource (R.reduce word))) |
| (MaxCutGames.Foundations.Games.FiniteDistribution.uniform |
| (Fin (HastadSource.asSource (R.reduce word)).occurrences)) ≤ 4 / 5 := by |
| unfold MaxCutGames.Clean.IncidenceGap.parityValue |
| apply Finset.sup'_le |
| intro a _ |
| have hq := R.soundness word no a |
| have hreal : (Clone100.success (R.reduce word).equations a : ℝ) ≤ 4 / 5 := by |
| simpa only [Rat.cast_div, Rat.cast_ofNat] using ((Rat.cast_le (K := ℝ)).mpr hq) |
| exact (SourceIncidence.paritySuccess_input (R.reduce word) a).trans_le hreal |
|
|
| theorem ordinaryGame_soundness {ξ : ℚ} (R : Reduction ξ) (word : List Bool) |
| (no : ¬ BinaryLanguage.language word) : |
| (MaxCutGames.Clean.IncidenceGap.fourAnswerGame |
| (SourceIncidence.incidence (HastadSource.asSource (R.reduce word))) |
| (MaxCutGames.Foundations.Games.FiniteDistribution.uniform |
| (Fin (HastadSource.asSource (R.reduce word)).occurrences))).value ≤ 14 / 15 := |
| MaxCutGames.Clean.IncidenceGap.fourAnswerGame_value_le_fourteen_fifteenths _ _ |
| (SourceIncidence.names_distinct _ (R.distinct word)) (parity_soundness R word no) |
|
|
| end |
| end MaxCutGames.Outer.BinaryParityReduction |
|
|
| end OAI |
|
|
