| import Mathlib |
| import OAI.Computability.VertexCover.Encoding.BinaryEncoding |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace VertexCover |
|
|
| def ThreeSAT (input : List Bool) : Prop := |
| ∃ formula, UniqueGames.BinaryEncoding.decodeFormula input = some formula ∧ |
| formula.Satisfiable |
|
|
| structure ExplicitGraph where |
| n : ℕ |
| edges : List (Fin n × Fin n) |
| increasing : ∀ e ∈ edges, e.1 < e.2 |
| nodup : edges.Nodup |
|
|
| namespace ExplicitGraph |
|
|
| def Cover (G : ExplicitGraph) (S : Finset (Fin G.n)) : Prop := |
| ∀ e ∈ G.edges, e.1 ∈ S ∨ e.2 ∈ S |
| |
| instance (G : ExplicitGraph) (S : Finset (Fin G.n)) : Decidable (G.Cover S) := |
| inferInstanceAs (Decidable (∀ e ∈ G.edges, e.1 ∈ S ∨ e.2 ∈ S)) |
| |
| noncomputable def coverNumber (G : ExplicitGraph) : ℕ := |
| Nat.find (show ∃ k : ℕ, ∃ S : Finset (Fin G.n), G.Cover S ∧ S.card = k from |
| ⟨G.n, Finset.univ, by simp [Cover], by simp⟩) |
| |
| def bits (G : ExplicitGraph) : List Bool := |
| let names := [G.n] ++ List.range G.n ++ [G.edges.length] ++ |
| G.edges.flatMap (fun e => [e.1.val, e.2.val]) |
| names.flatMap UniqueGames.BinaryEncoding.nameBits |
| |
| end ExplicitGraph |
| |
| def FiniteAlphabet (M : Turing.FinTM2) : Prop := ∀ k, Finite (M.Γ k) |
| |
| structure GapReduction (m : ℕ) where |
| construct : List Bool → ExplicitGraph |
| computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) |
| ExplicitGraph.bits construct |
| finiteAlphabet : FiniteAlphabet computation.tm |
| completeness : ∀ input, ThreeSAT input → |
| ((construct input).coverNumber : ℝ) < |
| (1 / 2 + 1 / (m : ℝ)) * (construct input).n |
| soundness : ∀ input, ¬ ThreeSAT input → |
| (1 - 1 / (m : ℝ)) * (construct input).n < |
| ((construct input).coverNumber : ℝ) |
|
|
| def natListBits (L : List ℕ) : List Bool := |
| (L.length :: L).flatMap UniqueGames.BinaryEncoding.nameBits |
|
|
| structure Approximation (α : ℝ) where |
| run : ExplicitGraph → List ℕ |
| computation : Turing.TM2ComputableInPolyTime ExplicitGraph.bits natListBits run |
| finiteAlphabet : FiniteAlphabet computation.tm |
| nodup : ∀ G, (run G).Nodup |
| vertices : ∀ G, ∀ v ∈ run G, v < G.n |
| covers : ∀ G, ∀ e ∈ G.edges, e.1.val ∈ run G ∨ e.2.val ∈ run G |
| quality : ∀ G, ((run G).length : ℝ) ≤ α * (G.coverNumber : ℝ) |
|
|
| structure ThreeSATDecision where |
| run : List Bool → Bool |
| computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) |
| (fun b : Bool => [b]) run |
| finiteAlphabet : FiniteAlphabet computation.tm |
| correct : ∀ input, run input = true ↔ ThreeSAT input |
|
|
| end VertexCover |
|
|
|
|
| end |
|
|
| end OAI |
|
|
