File size: 3,211 Bytes
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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordTuple
import OAI.Combinatorics.LowSpaceSubsetSum.Counting.SamplerAttempts
namespace OAI
/-! # A fair-word implementation of the measured bounded rejection loop
Only the polynomial-size candidate tuple is drawn and retained. The same
measured evaluator then supplies output, read count, and test-work count.
-/
namespace LowSpaceSubsetSum.WordRejection
open WordProgram
def run (w k N : ℕ) (evaluate : Fin (2^k) → Bool × ℕ) :
Program w (Option (Fin (2^k)) × ℕ × ℕ) :=
WordProgram.map (fun f => SamplerAttempts.runCostAcc evaluate N
(fun i => ExactWordDraw.low w k (f i)) 0 0) (WordTuple.sample N)
theorem bounded (w k N : ℕ) (evaluate : Fin (2^k) → Bool × ℕ) :
Bounded (run w k N evaluate) N := by
simpa only [run, WordProgram.map, Nat.add_zero] using
WordProgram.bind_bound (WordTuple.sample (w := w) N) N (WordTuple.bounded N)
(fun f => .pure (SamplerAttempts.runCostAcc evaluate N
(fun i => ExactWordDraw.low w k (f i)) 0 0)) 0 (fun _ => Bounded.pure _ 0)
theorem probability (w k : ℕ) (hk : k ≤ w) (N : ℕ)
(A : Fin (2^k) → Prop) [DecidablePred A] [Nonempty {a : Fin (2^k) // A a}]
(evaluate : Fin (2^k) → Bool × ℕ)
(P : (Option (Fin (2^k)) × ℕ × ℕ) → Prop) :
WordProgram.probability (run w k N evaluate) P =
(firstAcceptCoupling A N).law.probability
(fun s => P (SamplerAttempts.runCostAcc evaluate N s.1 0 0)) := by
rw [run, WordProgram.probability_map, WordTuple.probability]
exact (ExactWordDraw.candidates_words w k hk A N
(fun f => P (SamplerAttempts.runCostAcc evaluate N f 0 0))).symm
def sample (w k N : ℕ) (A : Fin (2^k) → Prop) [DecidablePred A] :
Program w (Option {a : Fin (2^k) // A a}) :=
WordProgram.map (fun f => firstAcceptSubtype A N
(fun i => ExactWordDraw.low w k (f i))) (WordTuple.sample N)
theorem sample_bounded (w k N : ℕ) (A : Fin (2^k) → Prop) [DecidablePred A] :
Bounded (sample w k N A) N := by
simpa only [sample, WordProgram.map, Nat.add_zero] using
WordProgram.bind_bound (WordTuple.sample (w := w) N) N (WordTuple.bounded N)
(fun f => .pure (firstAcceptSubtype A N (fun i => ExactWordDraw.low w k (f i)))) 0
(fun _ => Bounded.pure _ 0)
theorem sample_probability (w k : ℕ) (hk : k ≤ w) (N : ℕ)
(A : Fin (2^k) → Prop) [DecidablePred A] [Nonempty {a : Fin (2^k) // A a}]
(P : Option {a : Fin (2^k) // A a} → Prop) :
WordProgram.probability (sample w k N A) P =
(firstAcceptCoupling A N).boundedLaw.probability P := by
rw [sample, WordProgram.probability_map, WordTuple.probability,
SamplerCoupling.boundedLaw, FiniteLaw.map_probability]
exact (ExactWordDraw.firstAccept_words w k hk A N P).symm
theorem sample_expectation (w k : ℕ) (hk : k ≤ w) (N : ℕ)
(A : Fin (2^k) → Prop) [DecidablePred A] [Nonempty {a : Fin (2^k) // A a}]
(f : Option {a : Fin (2^k) // A a} → ℝ) :
WordProgram.expectation (sample w k N A) f =
(firstAcceptCoupling A N).boundedLaw.expectation f :=
WordProgram.expectation_of_probability _ _ (sample_probability w k hk N A) f
end LowSpaceSubsetSum.WordRejection
end OAI
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