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5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordMain
namespace OAI
/-! # Worst-case Subset Sum in one-fifth-exponential space
The threshold is chosen once from its proved existence theorem. No solution,
deficiency statistic, or bit-length exponent is an input. The finite random-word
experiment has one-sided correctness, success probability at least two thirds,
and bounds on its operation and writable-storage counts on every outcome.
-/
noncomputable section
namespace LowSpaceSubsetSum
def algorithm {n : ℕ} (I : Instance n) (words : WordFinalDecision.Words I) :
(Bool × ℕ) × ℕ := WordFinalDecision.run decisionThreshold I words
/-- One-sided randomized correctness and all-outcome resource-count bounds. -/
theorem worst_case_subset_sum :
(∀ (n : ℕ) (I : Instance n) (words : WordFinalDecision.Words I),
(algorithm 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 => (algorithm I words).1.1 = true)) ∧
(∀ c : ℕ, 0 < c → ∀ᶠ n : ℕ in Filter.atTop,
∀ I : Instance n, inputBits I ≤ n^c → ∀ words : WordFinalDecision.Words I,
((algorithm I words).1.2 : ℝ) ≤
(4*(3 : ℝ)^624)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) ∧
((algorithm I words).2 : ℝ) ≤
2*(inputScale I : ℝ)^15*(2 : ℝ)^((199/1000 : ℝ)*n)) :=
⟨fun _ I words => WordFinalDecision.sound decisionThreshold I words,
fun _ I => WordFinalDecision.probability decisionThreshold I,
WordMain.main_resources decisionThreshold⟩
/-- The writable-storage count has every fixed exponential rate above .199. -/
theorem exponential_space (rho : ℝ) (hrho : (199/1000 : ℝ) < rho) (c : ℕ) (hc : 0 < c) :
∀ᶠ n : ℕ in Filter.atTop, ∀ I : Instance n, inputBits I ≤ n^c →
∀ words : WordFinalDecision.Words I,
((algorithm I words).1.2 : ℝ) ≤
(4*(3 : ℝ)^624)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) ∧
((algorithm I words).2 : ℝ) ≤ (2*(3 : ℝ)^15)*(2 : ℝ)^(rho*n) :=
WordMain.space_corollary decisionThreshold rho hrho c hc
/-- In particular, ordinary one-fifth-exponential writable space. -/
theorem one_fifth_space (c : ℕ) (hc : 0 < c) :
∀ᶠ n : ℕ in Filter.atTop, ∀ I : Instance n, inputBits I ≤ n^c →
∀ words : WordFinalDecision.Words I,
((algorithm I words).1.2 : ℝ) ≤
(4*(3 : ℝ)^624)*(n : ℝ)^(624*c)*(2 : ℝ)^((1/2 : ℝ)*n) ∧
((algorithm I words).2 : ℝ) ≤ (2*(3 : ℝ)^15)*(2 : ℝ)^((1/5 : ℝ)*n) :=
exponential_space (1/5) (by norm_num) c hc
theorem algorithm_word_width {n : ℕ} (I : Instance n) :
wordWidth I = 4*(n+inputBits I+2) := rfl
end LowSpaceSubsetSum
end
end OAI
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