| import Mathlib |
| import OAI.Computability.MaxCut.PCP.PoweringTableSemantics |
|
|
| namespace OAI |
|
|
| /-! |
| The explicit assignment tester applied to the encoded satisfying pairs of an |
| actual finite edge predicate. The resulting soundness loss is independent of |
| the alphabet size. Uniform padding of the finite sample space and assembly of |
| the global incidence graph are separate constructions. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace MaxCutGames.Foundations.PCP.AlphabetReduction |
|
|
| open MaxCutGames.Foundations.Hastad |
| open CodeComposition |
|
|
| variable {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A] |
|
|
| abbrev LegalPair (accepts : A → A → Bool) := |
| {p : A × A // accepts p.1 p.2 = true} |
|
|
| abbrev InputCoordinate (A : Type*) := Bool × Cube A |
|
|
| def pairEncoding (accepts : A → A → Bool) (p : LegalPair accepts) : |
| InputCoordinate A → Bool := |
| pairWord (codeword p.1.1) (codeword p.1.2) |
|
|
| def edgeTesterReject (accepts : A → A → Bool) (left right : Cube A → Bool) |
| (F : Cube (LegalPair accepts) → Bool) : ℝ := |
| AssignmentTester.explicitTesterReject (pairEncoding accepts) (pairWord left right) F |
|
|
| theorem rejected_edge_tester_bound (accepts : A → A → Bool) |
| (left right : Cube A → Bool) |
| (F : Cube (LegalPair accepts) → Bool) |
| (hbad : accepts (nearest left) (nearest right) = false) : |
| 1 / 2048 ≤ edgeTesterReject accepts left right F := by |
| have hfar (p : LegalPair accepts) : |
| (1 / 8 : ℝ) ≤ AssignmentTester.distance (pairWord left right) (pairEncoding accepts p) := by |
| have h := rejected_pair_far accepts left right hbad p.1.1 p.1.2 p.2 |
| convert h using 1 |
| rfl |
| have h := AssignmentTester.explicit_proximity_soundness (pairEncoding accepts) |
| (pairWord left right) F (δ := 1 / 8) (by norm_num) (by norm_num) hfar |
| norm_num at h |
| exact h |
|
|
| omit [Nonempty A] in |
| theorem honest_edge_tester (accepts : A → A → Bool) (a b : A) |
| (hab : accepts a b = true) : |
| edgeTesterReject accepts (codeword a) (codeword b) |
| (fun f => f ⟨(a, b), hab⟩) = 0 := by |
| unfold edgeTesterReject |
| rw [AssignmentTester.explicitTesterReject_eq] |
| exact AssignmentTester.tester_perfect_completeness |
| (pairEncoding accepts) ⟨(a, b), hab⟩ |
|
|
| end MaxCutGames.Foundations.PCP.AlphabetReduction |
| end |
|
|
| end OAI |
|
|
