| import OAI.Probability.SATComputability.HierarchyTable |
| import OAI.Probability.SATComputability.FiniteTrialEvaluation |
|
|
| namespace OAI |
|
|
| namespace FixedClauseThreshold.Computability |
|
|
| open DilutedSpinGlass Nat.Partrec FiniteArithmetic |
| attribute [local irreducible] normalizedBranches branchWeights |
|
|
| noncomputable def leafFields (L : ℕ) (cs : List Code) : Fin L → ℝ := |
| fun i => decodedRational (cs[i.val]?.getD .zero) |
|
|
| theorem leafFields_ofFn {L : ℕ} (cs : Fin L → Code) : |
| leafFields L (List.ofFn cs) = fun i => (decodedRational (cs i) : ℝ) := by |
| funext i |
| simp [leafFields, i.isLt] |
|
|
| theorem listLogMean_eq_finite {L : ℕ} (ms : List ℚ) (g : (Fin L → ℝ) → ℝ) |
| (cs : Fin L → Code) : |
| listLogMean ms (fun ds => g (leafFields L ds)) (List.ofFn cs) = |
| finiteLogMean ms.length g (fun i => (ms.get i : ℝ)) |
| (fun i => decodeTree ms.length (cs i)) := by |
| induction ms generalizing cs with |
| | nil => |
| change g (leafFields L (List.ofFn cs)) = g (fun i => (decodedRational (cs i) : ℝ)) |
| rw [leafFields_ofFn] |
| | cons m ms ih => |
| rw [listLogMean, chooseChildren_expect_ofFn] |
| simp_rw [ih] |
| simp only [List.length_cons, finiteLogMean, decodeTree, List.get_eq_getElem, |
| List.getElem_cons_zero, List.getElem_cons_succ, Fin.val_zero, Fin.val_succ] |
| rw [show (instDecidableEqFin L : DecidableEq (Fin L)) = |
| (fun a b => Classical.propDecidable (a = b)) from Subsingleton.elim _ _] |
|
|
| theorem listTrialLog_eq_finite {L : ℕ} (ms : List ℚ) (g : (Fin L → ℝ) → ℝ) |
| (root : Code) : |
| branchSum (chooseChildren (List.replicate L root)) |
| (listLogMean ms (fun ds => g (leafFields L ds))) = |
| finiteTrialLog ms.length (decodeTree (ms.length+1) root) |
| (fun i => (ms.get i : ℝ)) g := by |
| rw [← List.ofFn_const L root, chooseChildren_expect_ofFn] |
| simp_rw [listLogMean_eq_finite] |
| simp only [finiteTrialLog, decodeTree] |
| rw [show (instDecidableEqFin L : DecidableEq (Fin L)) = |
| (fun a b => Classical.propDecidable (a = b)) from Subsingleton.elim _ _] |
| rfl |
|
|
| end FixedClauseThreshold.Computability |
|
|
| end OAI |
|
|
