| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! |
| Proper clauses choose distinct variables with independent fair signs. Formulas are |
| independent clause draws with replacement, and their satisfiability probability is |
| the literal finite cardinality ratio. The empty formula is satisfiable. When |
| there are fewer variables than literals, nonempty formula lengths have probability |
| zero by the convention that division by zero is zero. |
| The threshold assertion concerns fixed clause size and fixed density only, with |
| no assertion at the critical density. |
| -/ |
|
|
| namespace FixedClauseThreshold |
|
|
| abbrev Assignment (n : ℕ) := Fin n → Bool |
|
|
| abbrev ProperClause (n k : ℕ) := |
| { C : Fin n → Option Bool // (Finset.univ.filter fun i => (C i).isSome).card = k } |
|
|
| abbrev Formula (n k m : ℕ) := Fin m → ProperClause n k |
|
|
| def SatisfiesClause {n k : ℕ} (σ : Assignment n) (C : ProperClause n k) : Prop := |
| ∃ i : Fin n, C.val i = some (σ i) |
|
|
| def Satisfies {n k m : ℕ} (σ : Assignment n) (F : Formula n k m) : Prop := |
| ∀ j, SatisfiesClause σ (F j) |
|
|
| def Satisfiable {n k m : ℕ} (F : Formula n k m) : Prop := |
| ∃ σ : Assignment n, Satisfies σ F |
|
|
| noncomputable def properSATProbability (n k m : ℕ) : ℝ := by |
| classical |
| exact (Fintype.card { F : Formula n k m // Satisfiable F } : ℝ) / |
| (Fintype.card (Formula n k m) : ℝ) |
|
|
| def HasLimitingThreshold (k : ℕ) : Prop := |
| ∃ α : ℝ, 0 < α ∧ |
| (∀ c : ℝ, 0 ≤ c → c < α → |
| Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊) |
| Filter.atTop (nhds 1)) ∧ |
| (∀ c : ℝ, α < c → |
| Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊) |
| Filter.atTop (nhds 0)) |
|
|
| end FixedClauseThreshold |
|
|
| end OAI |
|
|
