| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Reduction.Simple |
| import OAI.Combinatorics.IndependentSets.PCP.RoundTableGap |
| import OAI.Combinatorics.IndependentSets.Reduction.FrameNonempty |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets |
| open IndependentSetsGames.Foundations |
| open scoped Classical |
| noncomputable section |
|
|
| namespace SemanticReduction |
| open Target PCP Hastad.SourceContexts Hastad.SourceGame |
|
|
| def gapFormula (H : RoundTables.BaseTable) (F : LargeIndependentSets.Formula) : Target.Formula := |
| TableIteration.gapMap H (Normalization.normalize F) |
| |
| instance eventsNonempty (H : RoundTables.BaseTable) (F : LargeIndependentSets.Formula) : |
| Nonempty (RandomEvent (gapFormula H F)) := |
| event_nonempty _ (TableIteration.gapMap_nonempty H _) |
| |
| def reduce (H : RoundTables.BaseTable) (p : SamplerParameters) (u q : ℕ) (hq : 0 < q) |
| (F : LargeIndependentSets.Formula) : Graph := |
| p.output q hq (LCInput.labelCover (gapFormula H F) u) |
| |
| lemma complete (H : RoundTables.BaseTable) (p : SamplerParameters) (u q : ℕ) (hq : 0 < q) |
| (F : LargeIndependentSets.Formula) (hs : F.Satisfiable) : |
| (reduce H p u q hq F).ThreeColorable := by |
| obtain ⟨l,r,hr⟩ := LCInput.complete (gapFormula H F) u |
| (TableIteration.gapMap_completeness H _ ((Normalization.normalize_satisfiable_iff F).mpr hs)) |
| exact p.output_complete q hq _ l r hr |
| |
| lemma vertex_bound (H : RoundTables.BaseTable) (p : SamplerParameters) (u q : ℕ) (hq : 0 < q) |
| (F : LargeIndependentSets.Formula) : |
| (reduce H p u q hq F).vertices ≤ |
| max q ((((gapFormula H F).clauses.length * 3)^u)^p.n * (p.law (J u) (I u)).denominator) := by |
| simpa only [reduce, RandomEvent, Hastad.SourceContexts.card_slot, Fintype.card_fun, Fintype.card_prod, |
| Fintype.card_fin] using p.output_size q hq (LCInput.labelCover (gapFormula H F) u) |
| |
| end SemanticReduction |
| |
| theorem semantic_reduction {δ : ℚ} (hδ : 0 < δ) : |
| ∃ H : PCP.RoundTables.BaseTable, ∃ p : SamplerParameters, ∃ u q : ℕ, ∃ hq : 0 < q, |
| (∀ F : Formula, F.Satisfiable → (SemanticReduction.reduce H p u q hq F).ThreeColorable) ∧ |
| (∀ F : Formula, ¬F.Satisfiable → |
| ((SemanticReduction.reduce H p u q hq F).independenceNumber : ℚ) < |
| δ*(SemanticReduction.reduce H p u q hq F).vertices) := by |
| obtain ⟨H,hH⟩ := PCP.ExpanderTables.exists_base_table |
| obtain ⟨p,σ,q,hq,hσ,hσ1,hp⟩ := finite_graph_reduction (show (0:ℝ) < δ by exact_mod_cast hδ) |
| obtain ⟨u,hu,hrep⟩ := LCInput.uniform_repetition |
| PCP.PCPIteration.finalClauseGap_positive PCP.TableGapReduction.finalClauseGap_le_one hσ |
| refine ⟨H,p,u,q,hq,SemanticReduction.complete H p u q hq,?_⟩ |
| intro F hF |
| have hg : Hastad.SourceGap.ClauseGap (SemanticReduction.gapFormula H F) |
| PCP.PCPIteration.finalClauseGap := |
| PCP.TableGapReduction.gapMap_clauseGap H hH _ |
| (fun hs => hF ((Normalization.normalize_satisfiable_iff F).mp hs)) |
| have hs := (hp _ _ _ _ _ (LCInput.labelCover (SemanticReduction.gapFormula H F) u)).2 |
| (hrep _ hg) |
| exact_mod_cast hs |
| |
| end |
| end LargeIndependentSets |
| |
| end OAI |
| |
