File size: 2,575 Bytes
92416a2 | 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 | 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
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