| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
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| namespace RandomKSAT |
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|
| abbrev Assignment (n : ℕ) := Fin n → Bool |
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| def Clause (n k : ℕ) := |
| (s : {s : Finset (Fin n) // s.card = k}) × (s.1 → Bool) |
| |
| instance clauseFintype (n k : ℕ) : Fintype (Clause n k) := by |
| classical |
| unfold Clause |
| infer_instance |
|
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| instance clauseMeasurableSpace (n k : ℕ) : MeasurableSpace (Clause n k) := ⊤ |
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| instance clauseMeasurableSingletonClass (n k : ℕ) : |
| MeasurableSingletonClass (Clause n k) := ⟨fun _ => trivial⟩ |
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| def Satisfies {n k : ℕ} (c : Clause n k) (a : Assignment n) : Prop := |
| ∃ v : c.1.1, a v = c.2 v |
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| abbrev Stream (n k : ℕ) := ℕ → Clause n k |
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| def clauseLaw (n k : ℕ) : Measure (Clause n k) := uniformOn Set.univ |
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| def streamLaw (n k : ℕ) : Measure (Stream n k) := |
| Measure.infinitePi (fun _ : ℕ => clauseLaw n k) |
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| def PrefixSAT {n k : ℕ} (ω : Stream n k) (m : ℕ) : Prop := |
| ∃ a : Assignment n, ∀ i < m, Satisfies (ω i) a |
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| def firstFailure {n k : ℕ} (ω : Stream n k) : ℕ∞ := |
| sInf ((fun m : ℕ => (m : ℕ∞)) '' {m | ¬ PrefixSAT ω m}) |
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| def H {n k : ℕ} (ω : Stream n k) : ℝ := (firstFailure ω).toNat |
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| def cap (n : ℕ) (B : ℝ) : ℕ := ⌊B * n⌋₊ |
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| def T {n k : ℕ} (B : ℝ) (ω : Stream n k) : ℝ := |
| (min (firstFailure ω) (cap n B : ℕ∞)).toNat |
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| def ell (k n : ℕ) : ℝ := |
| if k = 3 then Real.log (Real.exp 1 * n) else 1 |
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| def U (k : ℕ) : ℝ := Real.log 2 / (-Real.log (1 - ((2 : ℝ) ^ k)⁻¹)) |
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| end RandomKSAT |
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| end |
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| end OAI |
| |
