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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 | import OAI.Computability.PerfectCompleteness.Machines.CircuitProducerStagesLemmas
import OAI.Computability.PerfectCompleteness.Machines.ProducerTimeLemmas
import OAI.Computability.UniqueGames.Machines.GraphCounterStartLemmas
import OAI.Computability.UniqueGames.Machines.MachineFiniteAlphabet
namespace OAI
section
noncomputable section
namespace UniqueGamesTheorem.Foundations.Complexity.CookLevin.Completeness
open Turing
theorem circuitProducerFinite :
MachineFiniteAlphabet.FiniteAlphabet CircuitProducer.computableInPolyTime.tm := by
intro k
change Finite Bool
infer_instance
def verifierReduction (V : NPVerifier) : PolynomialThreeSATReduction V.Accepts where
reduce input := (VerifierCircuit.circuitOfVerifier V input).toFormula
computation := MachineSequential.composeBits (ProducerMachine.computableInPolyTime V)
CircuitProducer.computableInPolyTime
correct input := (VerifierCircuit.circuitOfVerifier_correct V input).symm
theorem verifierReduction_finiteAlphabet (V : NPVerifier) :
MachineFiniteAlphabet.FiniteAlphabet (verifierReduction V).computation.tm :=
MachineFiniteAlphabet.composeBits (ProducerMachine.computableInPolyTime V)
CircuitProducer.computableInPolyTime (ProducerMachine.finiteAlphabet V) circuitProducerFinite
theorem threeSATReduction (language : List Bool → Prop) (membership : InNP language) :
∃ source : PolynomialThreeSATReduction language,
MachineFiniteAlphabet.FiniteAlphabet source.computation.tm := by
obtain ⟨V, recognized⟩ := membership
let source : PolynomialThreeSATReduction language := {
reduce := (verifierReduction V).reduce
computation := (verifierReduction V).computation
correct := fun input => (recognized input).trans ((verifierReduction V).correct input) }
exact ⟨source, verifierReduction_finiteAlphabet V⟩
end UniqueGamesTheorem.Foundations.Complexity.CookLevin.Completeness
end
end
end OAI
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