| 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 | |