File size: 3,074 Bytes
e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
|