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d5b7741 | 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.ContingencyTables.Sampling.ExactExpectedTime
import OAI.Combinatorics.ContingencyTables.Sampling.ExactRarePublicBound
namespace OAI
/-! # The exact public sampler in uniform expected polynomial bit time -/
namespace ContingencyTables.SmallGraphProfiles
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream Turing
open ProfilePrograms DensePrograms Polynomial Algorithms
open scoped BigOperators Classical
noncomputable section
attribute [local irreducible] ExactPublicPrefix.program
theorem exact_expected_polynomial : ∃ p : Polynomial ℕ,
∀ (_hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q)
(_hCheeger : ∀ q k,PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*k^8))),
∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), (∑ i,r i)=∑ j,c j →
(∑' (t : ℕ), (1 - haltMass ExactPublicPrefix.machine (encodeMargins r c) t)) ≤
((p.eval (samplingSize n r 0) : ℕ) : ℝ) := by
obtain ⟨print,hprint⟩ := ExactPrintCost.polynomial_print_bound
obtain ⟨p,hp⟩ := ExactGatePolynomial.public_bound print
obtain ⟨q,hq⟩ := exact_public_correction_bound print
refine ⟨p+11*q,?_⟩
intro hPL hCheeger m n r c htotal
let B := print.eval (ExactPrintCost.matrixSize (n := n) r)
let S := ExactGatePublic.budget r c B B
let T := ExactPublicCost.budget r c 1 B B+1
let den : ℝ := (2:ℝ)^correctionPrecision r c
have hB (X : Table r c) : GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val) ≤ B :=
(Nat.le_add_right _ _).trans (hprint m n r c X)
have hL (X : Table r c) : (ExactPublicPrefix.word r c X).length ≤ B :=
(list_length_le_weight _).trans ((Nat.le_add_left _ _).trans (hprint m n r c X))
have htail := (ExactExpectedTime.bound r c hPL hCheeger htotal B B hB hL).2
have hgate : (S:ℝ) ≤ ((p.eval (samplingSize n r 0):ℕ):ℝ) := by exact_mod_cast hp m n r c htotal
have hT : (T:ℝ) ≤ den*((q.eval (samplingSize n r 0):ℕ):ℝ) := by
have hn : T ≤ 2^(correctionPrecision r c)*q.eval (samplingSize n r 0) := hq m n r c htotal
have ht : (T:ℝ) ≤ ((2^(correctionPrecision r c)*q.eval (samplingSize n r 0):ℕ):ℝ) := by
exact_mod_cast hn
simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,den] using ht
have hrare : (1/den)*T ≤ ((q.eval (samplingSize n r 0):ℕ):ℝ) := by
have hden : 0 < den := by unfold den; positivity
have ht : (T:ℝ)/den ≤ ((q.eval (samplingSize n r 0):ℕ):ℝ) :=
(div_le_iff₀ hden).mpr (by simpa only [mul_comm] using hT)
simpa only [one_div,div_eq_mul_inv,mul_comm,one_mul] using ht
have he : (fun t => 1-haltMass ExactPublicPrefix.machine (encodeMargins r c) t)=
PhysicalRetryTail.tail ExactPublicPrefix.program.table
(ExactPublicPrefix.program.initial (Tape.mk₁ (encodeMargins r c))) := by
funext t
exact PhysicalPublicLaw.tail_eq _ _ _
rw [←he] at htail
change _ ≤ (S:ℝ)+11*(1/den)*T at htail
simp only [eval_add,eval_mul,eval_ofNat,Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat]
nlinarith
/-- Paper 200-01, main theorem (ii), conditional on the cited finite-box
Prékopa--Leindler and finite-grid Cheeger inputs. The machine is fixed,
including parsing, printing, and the unbiased-bit implementation. -/
theorem exactSampling_conditional
(hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : ∀ q k,PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*k^8))) :
ExactSamplingStatement := by
obtain ⟨p,hp⟩ := exact_expected_polynomial
refine ⟨ExactPublicPrefix.machine,p.eval 1+1,p.natDegree,by omega,?_⟩
intro m n r c htotal
dsimp only
refine ⟨ExactPublicLaw.halted_feasible r c hPL hCheeger htotal,
ExactPublicLaw.tableMass_limit r c hPL hCheeger htotal,
ExactPublicLaw.haltMass_limit r c hPL hCheeger htotal,
ExactExpectedTime.summable r c hPL hCheeger htotal,?_⟩
have hS : 1 ≤ samplingSize n r 0 := by unfold samplingSize; omega
have hpoly : p.eval (samplingSize n r 0) ≤ p.eval 1*(samplingSize n r 0)^p.natDegree := by
simpa only [one_mul,pow_one] using polynomial_scaled_power p 1 (samplingSize n r 0) 1 hS
have hnat := hpoly.trans (Nat.mul_le_mul_right _ (Nat.le_succ _))
have hreal : ((p.eval (samplingSize n r 0):ℕ):ℝ) ≤
(p.eval 1+1:ℕ)*(samplingSize n r 0:ℝ)^p.natDegree := by exact_mod_cast hnat
exact (hp hPL hCheeger m n r c htotal).trans hreal
end
end ContingencyTables.SmallGraphProfiles
end OAI
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