| import Mathlib |
| import OAI.Computability.MaxCut.PCP.PoweringTableSemantics |
| import OAI.Computability.MaxCut.PCP.InitialGraph |
|
|
| namespace OAI |
|
|
| /-! |
| Pull a graph predicate back through a surjective label decoder. This preserves |
| every rejection count after decoding. A concrete section embeds the initial |
| eight-label graph into the fixed 64-label alphabet used throughout amplification. |
| -/ |
|
|
| namespace MaxCutGames.Foundations.PCP.AlphabetRetraction |
|
|
| variable {V E A B : Type*} |
|
|
| def pullback (G : ConstraintGraph V E A) (decode : B → A) : ConstraintGraph V E B where |
| reverse := G.reverse |
| reverse_involutive := G.reverse_involutive |
| tail := G.tail |
| accepts := fun e b c => G.accepts e (decode b) (decode c) |
| reverse_accepts := fun e b c => G.reverse_accepts e (decode b) (decode c) |
|
|
| @[simp] theorem edgeSatisfied_pullback (G : ConstraintGraph V E A) |
| (decode : B → A) (labeling : V → B) (e : E) : |
| (pullback G decode).edgeSatisfied labeling e = |
| G.edgeSatisfied (decode ∘ labeling) e := rfl |
|
|
| @[simp] theorem rejectionCount_pullback [Fintype E] (G : ConstraintGraph V E A) |
| (decode : B → A) (labeling : V → B) : |
| (pullback G decode).rejectionCount labeling = |
| G.rejectionCount (decode ∘ labeling) := rfl |
|
|
| theorem satisfiable_pullback_iff (G : ConstraintGraph V E A) |
| (decode : B → A) (encode : A → B) (sectionLaw : ∀ a, decode (encode a) = a) : |
| (pullback G decode).Satisfiable ↔ G.Satisfiable := by |
| constructor |
| · rintro ⟨labeling, h⟩ |
| exact ⟨decode ∘ labeling, h⟩ |
| · rintro ⟨labeling, h⟩ |
| refine ⟨encode ∘ labeling, fun e => ?_⟩ |
| change G.accepts e (decode (encode (labeling (G.tail e)))) |
| (decode (encode (labeling (G.head e)))) = true |
| simpa only [ConstraintGraph.edgeSatisfied, sectionLaw] using h e |
|
|
| abbrev Label64 := QueryIncidence.Label 6 |
|
|
| def decode64 (label : Label64) : InitialGraph.Label := (label 0, label 1, label 2) |
|
|
| def encode64 (label : InitialGraph.Label) : Label64 := |
| fun i => if i = 0 then label.1 else if i = 1 then label.2.1 else label.2.2 |
|
|
| @[simp] theorem decode_encode64 (label : InitialGraph.Label) : |
| decode64 (encode64 label) = label := by |
| rcases label with ⟨a, b, c⟩ |
| rfl |
|
|
| def initial64 (F : MaxCutGames.Foundations.Target.Formula) : |
| ConstraintGraph (InitialGraph.CompactVertex F) (InitialGraph.CompactDart F) Label64 := |
| pullback (InitialGraph.build F) decode64 |
|
|
| theorem initial64_satisfiable_iff (F : MaxCutGames.Foundations.Target.Formula) : |
| (initial64 F).Satisfiable ↔ F.Satisfiable := |
| (satisfiable_pullback_iff (InitialGraph.build F) decode64 encode64 decode_encode64).trans |
| (InitialGraph.build_satisfiable_iff F) |
|
|
| theorem initial64_alphabet : Fintype.card Label64 = 64 := |
| QueryIncidence.six_query_alphabet |
|
|
| theorem initial64_inverse_size_gap (F : MaxCutGames.Foundations.Target.Formula) |
| (unsat : ¬ F.Satisfiable) (labeling : InitialGraph.CompactVertex F → Label64) : |
| Fintype.card (InitialGraph.CompactDart F) ≤ |
| (6 * F.clauses.length + 1) * (initial64 F).rejectionCount labeling := |
| InitialGraph.build_inverse_size_gap F unsat (decode64 ∘ labeling) |
|
|
| end MaxCutGames.Foundations.PCP.AlphabetRetraction |
|
|
| end OAI |
|
|
