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e5a97c6 | 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 81 82 83 84 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordFinalDecision
namespace OAI
/-! # Time and space guarantees for the bounded randomized decision procedure
`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
private theorem double_time (a b c : ℝ) : 2*((a*2)*b*c) = (4*a)*b*c := by ring
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))
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
/-- One-sided correctness and one-fifth-exponential storage for any fixed
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
end
end OAI
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