| import OAI.Computability.BinPacking.Search.ExtensionVerifier |
| import OAI.Computability.BinPacking.Search.SearchStageMachines |
|
|
| namespace OAI |
|
|
| namespace BinPackingGap |
|
|
| open Turing BinPackingGames.Foundations.Complexity |
|
|
| namespace OptimalSearch |
|
|
| noncomputable def decider (h : PEqualsNP) : List Bool → Bool := |
| Classical.choose (h.inP packingExtension_inNP) |
|
|
| noncomputable def decisionComputation (h : PEqualsNP) : |
| TM2ComputableInPolyTime (id : List Bool → List Bool) (fun answer => [answer]) |
| (decider h) := |
| Classical.choose (Classical.choose_spec (h.inP packingExtension_inNP)) |
|
|
| theorem decisionSpec (h : PEqualsNP) : |
| MachineFiniteAlphabet.FiniteAlphabet (decisionComputation h).tm ∧ |
| ∀ input, ExtensionCertificate.language input ↔ decider h input = true := |
| Classical.choose_spec (Classical.choose_spec (h.inP packingExtension_inNP)) |
|
|
| theorem decidesExtension (h : PEqualsNP) : SearchSemantics.DecidesExtension (decider h) := |
| SearchOptimalConstruction.decidesExtension_of_language (decisionSpec h).2 |
|
|
| end OptimalSearch |
|
|
| noncomputable def absoluteAdditiveAlgorithm_of_pEqualsNP (h : PEqualsNP) : |
| AbsoluteAdditiveAlgorithm 0 := |
| SearchOptimalConstruction.ofDecision SearchStageMachines.stages |
| (OptimalSearch.decider h) (OptimalSearch.decisionComputation h) |
| (OptimalSearch.decisionSpec h).1 (OptimalSearch.decidesExtension h) |
|
|
| theorem absoluteAdditiveAlgorithm_of_pEqualsNP_optimal (h : PEqualsNP) |
| (items : RawInstance) (valid : items.Valid) : |
| ∃ packing : RawPacking, |
| (absoluteAdditiveAlgorithm_of_pEqualsNP h).run (BinaryEncoding.rawInstanceBits items) = |
| some packing ∧ packing.Feasible items ∧ packing.bins = opt (items.toInstance valid) := |
| SearchMachineComposition.algorithm_optimal (OptimalSearch.decidesExtension h) items valid |
|
|
| @[simp] theorem absoluteAdditiveAlgorithm_of_pEqualsNP_empty (h : PEqualsNP) : |
| (absoluteAdditiveAlgorithm_of_pEqualsNP h).run (BinaryEncoding.rawInstanceBits []) = |
| some ⟨0, []⟩ := |
| SearchMachineComposition.algorithm_empty (OptimalSearch.decider h) |
|
|
| theorem absoluteAdditiveAlgorithm_of_pEqualsNP_malformed (h : PEqualsNP) |
| (input : List Bool) (malformed : BinaryEncoding.decodeRawInstance input = none) : |
| (absoluteAdditiveAlgorithm_of_pEqualsNP h).run input = none := |
| SearchMachineComposition.algorithm_malformed (OptimalSearch.decider h) input malformed |
|
|
| theorem absoluteAdditiveAlgorithm_of_pEqualsNP_invalid (h : PEqualsNP) |
| (items : RawInstance) (invalid : ¬ items.Valid) : |
| (absoluteAdditiveAlgorithm_of_pEqualsNP h).run (BinaryEncoding.rawInstanceBits items) = none := |
| SearchMachineComposition.algorithm_invalid (OptimalSearch.decider h) items invalid |
|
|
| end BinPackingGap |
|
|
| end OAI |
|
|
