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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
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