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ea76aac | 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 | import Mathlib
import OAI.Computability.MaxCut.Machines.MachineClone100
import OAI.Computability.MaxCut.Machines.MachineBinaryRenameWords
import OAI.Computability.MaxCut.Games.NP
namespace OAI
/-!
The complete ordinary-binary input bridge. Arbitrary raw input bits are first
validated and retained, or replaced by a fixed contradictory encoded formula.
The checked occurrence-index renamer then emits the inherited dense 3CNF
codec. Both stages and their physical handoff have actual polynomial-time
finite-alphabet machine certificates.
This is a bridge for the explicit ordinary binary 3SAT language. It is not the
PCP construction, a proof of that construction's runtime, or an NP-hardness
assertion about the eventual 2-to-1 target.
-/
namespace MaxCutGames.BinaryInputReduction
open Turing MaxCutGames.Foundations.Complexity
/-- The actual sequential machine handles every bitstring, including malformed
and noncanonical encodings, and outputs the fixed dense formula serialization. -/
noncomputable def computation :
TM2ComputableInPolyTime (id : List Bool → List Bool) formulaBits BinaryLanguage.totalRename := by
change TM2ComputableInPolyTime (id : List Bool → List Bool) formulaBits
(fun input => BinaryOccurrenceRename.renamed (BinaryLanguage.totalParsed input))
exact MachineSequential.composeBits BinaryTotalInputMachine.computableInPolyTime
BinaryRenameMachine.computableInPolyTime
theorem computation_finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm :=
MachineFiniteAlphabet.composeBits BinaryTotalInputMachine.computableInPolyTime
BinaryRenameMachine.computableInPolyTime
BinaryTotalInputMachine.computation_finiteAlphabet
BinaryRenameMachine.computation_finiteAlphabet
/-- Satisfiability equivalence plus an actual raw-bit machine; no external
verifier, parsing-time assertion, or recoding hypothesis is an input. -/
noncomputable def reduction : CookLevin.PolynomialThreeSATReduction BinaryLanguage.language where
reduce := BinaryLanguage.totalRename
computation := computation
correct input := (BinaryLanguage.totalRename_satisfiable_iff input).symm
theorem reduction_finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet reduction.computation.tm :=
computation_finiteAlphabet
end MaxCutGames.BinaryInputReduction
end OAI
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