| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Bridge |
| import OAI.Computability.MinUncut.Machines.MachineBinaryRenameMachine |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.SourceBridge |
| open MinUncutGames MinUncutGames.Reduction MinUncutGames.Foundations |
| open MinUncutGames.Foundations.Complexity |
| open Outer MinUncut.Inner |
|
|
| structure BinaryReduction (ξ : ℚ) where |
| reduce : BinaryFormula.Formula → SourceEncoding.Input |
| complete : ∀ F, F.Satisfiable → ∃ bits, 1-ξ≤Outer.HastadSource.success (reduce F) bits |
| sound : ∀ F, ¬F.Satisfiable → ∀ bits, Outer.HastadSource.success (reduce F) bits≤(1+ξ)/2 |
| computation : Turing.TM2ComputableInPolyTime BinaryEncoding.formulaBits SourceEncoding.inputBits reduce |
| finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm |
|
|
| def binaryReduction {ξ : ℚ} (R : Outer.HastadSource.Reduction ξ) : BinaryReduction ξ where |
| reduce F := R.reduce (BinaryOccurrenceRename.renamed F) |
| complete F hF := R.completeness _ ((BinaryOccurrenceRename.renamed_satisfiable_iff F).mpr hF) |
| sound F hF := R.soundness _ (fun h => hF ((BinaryOccurrenceRename.renamed_satisfiable_iff F).mp h)) |
| computation := MachineSequential.composeBits BinaryRenameMachine.computableInPolyTime R.computation |
| finiteAlphabet := MachineFiniteAlphabet.composeBits _ _ |
| BinaryRenameMachine.computation_finiteAlphabet R.finiteAlphabet |
|
|
| lemma binaryReduction_exists {ξ : ℚ} (hξ : 0<ξ) : Nonempty (BinaryReduction ξ) := by |
| obtain ⟨R⟩ := Outer.HastadSource.exists_reduction ξ hξ |
| exact ⟨binaryReduction R⟩ |
|
|
| lemma binaryReduction_sound {ξ : ℚ} (hξ : ξ≤1/2) (R : BinaryReduction ξ) |
| (F : BinaryFormula.Formula) (hF : ¬F.Satisfiable) (s : Fin (R.reduce F).«variables» → F₂) : |
| equationFraction (equations (R.reduce F)) s≤3/4 := by |
| rw [fraction_eq_assignment] |
| have hh : (Outer.HastadSource.success (R.reduce F) (fun x => fieldBool (s x)):ℝ)≤ |
| ((1+ξ)/2:ℚ) := Rat.cast_le.mpr (R.sound F hF _) |
| have hξ' : (ξ:ℝ)≤1/2 := by |
| simpa only [Rat.cast_div,Rat.cast_one,Rat.cast_ofNat] using |
| (Rat.cast_le.mpr hξ : (ξ:ℝ)≤((1/2:ℚ):ℝ)) |
| push_cast at hh |
| linarith |
|
|
| lemma binaryReduction_complete {ξ : ℚ} (R : BinaryReduction ξ) |
| (F : BinaryFormula.Formula) (hF : F.Satisfiable) (t : ℕ) (b : ℝ) |
| (hξ : (t:ℝ)*(ξ:ℝ)≤b) : |
| ∃ s : Fin (R.reduce F).«variables» → F₂, |
| (t:ℝ)*(1-equationFraction (equations (R.reduce F)) s)≤b := by |
| obtain ⟨bits,hb⟩ := R.complete F hF |
| refine ⟨fun x => boolField (bits x),?_⟩ |
| rw [fraction_eq] |
| have hb' : 1-(ξ:ℝ)≤(Outer.HastadSource.success (R.reduce F) bits:ℝ) := by exact_mod_cast hb |
| exact (mul_le_mul_of_nonneg_left (by linarith : |
| 1-(Outer.HastadSource.success (R.reduce F) bits:ℝ)≤(ξ:ℝ)) (Nat.cast_nonneg t)).trans hξ |
| end MinUncut.SourceBridge |
|
|
| end |
| end |
|
|
| end OAI |
|
|
