File size: 2,748 Bytes
e320e82 | 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 80 | import Mathlib
namespace OAI
namespace LargeIndependentSets
structure Literal where
name : ℕ
positive : Bool
deriving DecidableEq
structure Formula where
clauses : List (List Literal)
width : ∀ C ∈ clauses, C.length ≤ 3
def Formula.Satisfiable (φ : Formula) : Prop :=
∃ assignment : ℕ → Bool, ∀ C ∈ φ.clauses,
∃ l ∈ C, assignment l.name = l.positive
/-- Binary framing: `true,b` for a payload bit and `false` as terminator. -/
def frame : List Bool → List Bool
| [] => [false]
| b :: bs => true :: b :: frame bs
def nameBits (n : ℕ) : List Bool := frame n.bits
def literalBits (l : Literal) : List Bool := l.positive :: nameBits l.name
def clauseBits (C : List Literal) : List Bool :=
nameBits C.length ++ C.flatMap literalBits
def formulaBits (φ : Formula) : List Bool :=
nameBits φ.clauses.length ++ φ.clauses.flatMap clauseBits
/-- A nonempty, finite, simple, undirected, unweighted graph. -/
structure Graph where
vertices : ℕ
nonempty : 0 < vertices
adj : Fin vertices → Fin vertices → Bool
symm : ∀ u v, adj u v = adj v u
loopless : ∀ u, adj u u = false
def Graph.Independent (G : Graph) (S : Finset (Fin G.vertices)) : Prop :=
∀ u ∈ S, ∀ v ∈ S, u ≠ v → G.adj u v = false
instance (G : Graph) (S : Finset (Fin G.vertices)) : Decidable (G.Independent S) :=
inferInstanceAs (Decidable (∀ u ∈ S, ∀ v ∈ S, u ≠ v → G.adj u v = false))
def Graph.independenceNumber (G : Graph) : ℕ :=
((Finset.univ : Finset (Fin G.vertices)).powerset.filter G.Independent).sup Finset.card
def Graph.ThreeColorable (G : Graph) : Prop :=
∃ color : Fin G.vertices → Fin 3,
∀ u v, G.adj u v = true → color u ≠ color v
/-- Full explicit adjacency matrix, row-major, with binary vertex count. -/
def graphBits (G : Graph) : List Bool :=
nameBits G.vertices ++ (List.finRange G.vertices).flatMap
(fun u => (List.finRange G.vertices).map (G.adj u))
open scoped Classical
noncomputable section
/-- Polynomial-time graph reduction with a real independence-ratio threshold. -/
structure RealGraphReduction (δ : ℝ) where
reduce : List Bool → Graph
computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) graphBits reduce
finiteAlphabets : ∀ k : computation.tm.K, Finite (computation.tm.Γ k)
outputBound : Polynomial ℕ
outputSize : ∀ b, (graphBits (reduce b)).length ≤ outputBound.eval b.length
completeness : ∀ φ : Formula, φ.Satisfiable → (reduce (formulaBits φ)).ThreeColorable
soundness : ∀ φ : Formula, ¬φ.Satisfiable →
((reduce (formulaBits φ)).independenceNumber : ℝ) <
δ * (reduce (formulaBits φ)).vertices
end
end LargeIndependentSets
end OAI
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