| import OAI.Computability.BinPacking.Hardness.LiteralSlotGraphComputation |
| import OAI.Computability.BinPacking.Hardness.PCPPackingGap |
| import OAI.Computability.BinPacking.Machines.GraphPackingMachine |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace BinPackingGap |
|
|
| open Turing BinPackingGames.Foundations.Complexity BinPackingCompleteness |
|
|
| theorem packingGapNPHard (c : Nat) : PackingGapNPHard c := by |
| intro L hL |
| obtain ⟨source, runtime, hfinite, hsource⟩ := PCPFoundation.normalizedSource L hL |
| let normalized : List Bool → LiteralSlotGraphMachine.NonemptyFormula := fun input => |
| ⟨PCPSourceHardness.formula source input, (hsource input).2.1⟩ |
| let first : TM2ComputableInPolyTime (id : List Bool → List Bool) |
| LiteralSlotGraphMachine.inputBits normalized := { |
| tm := runtime.tm |
| inputAlphabet := runtime.inputAlphabet |
| outputAlphabet := runtime.outputAlphabet |
| time := runtime.time |
| outputsFun input := runtime.outputsFun input } |
| let graphRun := MachineSequential.composeBits first LiteralSlotGraphMachine.computation |
| let packingRun := MachineSequential.composeBits graphRun |
| (graphPackingComputation c PCPFoundation.rho) |
| refine ⟨{ |
| reduce := fun input => PCPFoundation.packingOutput c (source.reduce input) |
| computation := packingRun |
| finiteAlphabet := ?_ |
| valid := fun input => PCPFoundation.packingOutput_valid c (source.reduce input) |
| itemCount := fun input => PCPFoundation.packingOutput_count c (source.reduce input) |
| itemLower := fun input i => PCPFoundation.packingOutput_size_lower c (source.reduce input) i |
| completeness := ?_ |
| soundness := ?_ }⟩ |
| · exact MachineFiniteAlphabet.composeBits graphRun |
| (graphPackingComputation c PCPFoundation.rho) |
| (MachineFiniteAlphabet.composeBits first LiteralSlotGraphMachine.computation |
| hfinite LiteralSlotGraphMachine.computation_finiteAlphabet) |
| (graphPackingComputation_finiteAlphabet c PCPFoundation.rho) |
| · intro input accepted |
| exact PCPFoundation.packingOutput_complete c (source.reduce input) |
| ((source.correct input).mp accepted) |
| · intro input rejected |
| exact PCPFoundation.packingOutput_sound c (source.reduce input) |
| (fun satisfiable => rejected ((source.correct input).mpr satisfiable)) |
|
|
| theorem packingPromiseNPHard (c : Nat) : |
| PromiseNPHard BinaryEncoding.rawReductionOutputBits (packingPromise c) := |
| (packingGapNPHard c).promiseNPHard |
|
|
| end BinPackingGap |
|
|
| end |
|
|
| end OAI |
|
|
