| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordFinalDecision |
|
|
| namespace OAI |
|
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| /-! # Time and space guarantees for the bounded randomized decision procedure |
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| `main_resources` bounds the work and storage counters with fixed threshold |
| `T`. `space_corollary` absorbs the polynomial storage factor for every rate |
| above .199. All resource quantifiers range over every random-word outcome, |
| while the procedure and threshold are independent of the polynomial-bit exponent. |
| -/ |
|
|
| noncomputable section |
| namespace LowSpaceSubsetSum.WordMain |
|
|
| open Filter |
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|
| private theorem double_time (a b c : ℝ) : 2*((a*2)*b*c) = (4*a)*b*c := by ring |
|
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| theorem main_resources (T : DecisionThreshold) (c : ℕ) (hc : 0 < c) : |
| ∀ᶠ n : ℕ in atTop, ∀ I : Instance n, inputBits I ≤ n^c → |
| ∀ words : WordFinalDecision.Words I, |
| ((WordFinalDecision.run T I words).1.2 : ℝ) ≤ |
| (4*(3 : ℝ)^624)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) ∧ |
| ((WordFinalDecision.run T I words).2 : ℝ) ≤ |
| 2*(inputScale I : ℝ)^15*(2 : ℝ)^((199/1000 : ℝ)*n) := by |
| filter_upwards [eventually_mainGuard T.value c hc,eventually_ge_atTop 2] with n hg hn |
| intro I hI words |
| have hm : mainGuard T.value n (inputScale I) = true := hg (inputBits I) hI |
| have hb := WordFinalDecision.main_bounds T I hm words |
| constructor |
| · have hnat : ((WordFinalDecision.run T I words).1.2 : ℝ) ≤ |
| ((2*(inputScale I^624*2^((n+1)/2)) : ℕ) : ℝ) := Nat.cast_le.mpr hb.1 |
| rw [Nat.cast_mul,Nat.cast_ofNat] at hnat |
| apply hnat.trans |
| have h := mul_le_mul_of_nonneg_left (WordExecutionAsymptotics.time c hc n hn I hI) |
| (by norm_num : (0 : ℝ) ≤ 2) |
| simpa only [double_time] using h |
| · have hnat : ((WordFinalDecision.run T I words).2 : ℝ) ≤ |
| ((2*writableWordLimit I : ℕ) : ℝ) := Nat.cast_le.mpr hb.2 |
| simp only [writableWordLimit,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hnat |
| rw [←mul_assoc] at hnat |
| exact hnat.trans (mul_le_mul_of_nonneg_left (PreparedAsymptotics.capacity_upper n) |
| (by positivity)) |
|
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| theorem space_corollary (T : DecisionThreshold) (rho : ℝ) |
| (hrho : (199/1000 : ℝ) < rho) (c : ℕ) (hc : 0 < c) : |
| ∀ᶠ n : ℕ in atTop, ∀ I : Instance n, inputBits I ≤ n^c → |
| ∀ words : WordFinalDecision.Words I, |
| ((WordFinalDecision.run T I words).1.2 : ℝ) ≤ |
| (4*(3 : ℝ)^624)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) ∧ |
| ((WordFinalDecision.run T I words).2 : ℝ) ≤ |
| (2*(3 : ℝ)^15)*(2 : ℝ)^(rho*n) := by |
| filter_upwards [main_resources T c hc,absorb_space_polynomial hrho c hc] with n hm hs |
| intro I hI words |
| refine ⟨(hm I hI words).1,(hm I hI words).2.trans ?_⟩ |
| have h := mul_le_mul_of_nonneg_left (hs (inputBits I) hI) (by norm_num : (0 : ℝ) ≤ 2) |
| simpa only [inputScale,Nat.cast_add,Nat.cast_ofNat,mul_assoc] using h |
|
|
| / |
| threshold satisfying the large-input conditions. -/ |
| theorem conditional_main (T : DecisionThreshold) : |
| (∀ (n : ℕ) (I : Instance n) (words : WordFinalDecision.Words I), |
| (WordFinalDecision.run T I words).1.1 = true → Solvable I) ∧ |
| (∀ (n : ℕ) (I : Instance n), Solvable I → |
| (2/3 : ℝ) ≤ (FiniteLaw.uniform : FiniteLaw (WordFinalDecision.Words I)).probability |
| (fun words => (WordFinalDecision.run T I words).1.1 = true)) ∧ |
| (∀ c : ℕ, 0 < c → ∀ᶠ n : ℕ in atTop, |
| ∀ I : Instance n, inputBits I ≤ n^c → ∀ words : WordFinalDecision.Words I, |
| ((WordFinalDecision.run T I words).1.2 : ℝ) ≤ |
| (4*(3 : ℝ)^624)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) ∧ |
| ((WordFinalDecision.run T I words).2 : ℝ) ≤ |
| (2*(3 : ℝ)^15)*(2 : ℝ)^((1/5 : ℝ)*n)) := by |
| refine ⟨fun _ I words => WordFinalDecision.sound T I words, |
| fun _ I => WordFinalDecision.probability T I,?_⟩ |
| intro c hc |
| exact space_corollary T (1/5) (by norm_num) c hc |
|
|
| end LowSpaceSubsetSum.WordMain |
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| end |
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| end OAI |
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