File size: 3,187 Bytes
ea76aac | 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 68 69 70 71 72 73 74 75 76 77 78 79 | 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
|