File size: 3,839 Bytes
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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactInputBounds
import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePublic
import OAI.Combinatorics.ContingencyTables.Sampling.PhysicalCostPolynomial
namespace OAI
/-! # A uniform polynomial bound for the cheap public prefix -/
namespace ContingencyTables.ExactGatePolynomial
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream
open ProfilePrograms DensePrograms Polynomial
open Algorithms (samplingSize encodeMargins)
open scoped BigOperators Classical
noncomputable section
attribute [local irreducible] ExactTypedProgram.program exactParsedMargins
ExactGatePrefix.budget ExactGateCost.measure
def measure (as : List Symbol) (B : ℕ) : ℕ :=
weight as+ExactGateCost.measure (exactParsedMargins as)+B
theorem budget_polynomial : ∃ p : Polynomial ℕ,∀ as B,
LiteralProgram.packTime as.length+ExactGatePhysical.budget as B+LiteralProgram.postTime B ≤
p.eval (measure as B) := by
obtain ⟨p,hp⟩ := polynomial_exactParsedMargins
obtain ⟨q,hq⟩ := ExactGateCost.budget_polynomial
let t : Polynomial ℕ := p+q+X
obtain ⟨v,hv⟩ := PhysicalCostPolynomial.bound
(OnlineStack.accesses ExactTypedProgram.program.code)
(OnlineStack.overhead ExactTypedProgram.program.code) t
refine ⟨v,?_⟩
intro as B
let M := measure as B
have hw : weight as ≤ M := by unfold M measure; omega
have hg : ExactGateCost.measure (exactParsedMargins as) ≤ M := by unfold M measure; omega
have hB : B ≤ M := by unfold M measure; omega
have ht : ExactGatePhysical.typedBudget as B ≤ t.eval M := by
have hp' := (Nat.le_add_right _ _).trans ((hp as).trans (polynomial_eval_monotone p hw))
have hq' := (hq _).trans (polynomial_eval_monotone q hg)
simpa only [ExactGatePhysical.typedBudget,t,eval_add,eval_X] using
Nat.add_le_add (Nat.add_le_add hp' hq') hB
have hb := hv (weight as) (ExactGatePhysical.typedBudget as B) as.length B M hw ht
((list_length_le_weight as).trans hw) hB
simpa only [ExactGatePhysical.budget,OnlineStack.Executes.factor] using hb
/-- Any fixed polynomial print allowance still gives a uniform public
polynomial bound for the prefix before the rare tabulation starts. -/
theorem public_bound (print : Polynomial ℕ) : ∃ p : Polynomial ℕ,
∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),
(∑ i,r i)=∑ j,c j →
let B := print.eval (ExactPrintCost.matrixSize (n := n) r)
ExactGatePublic.budget r c B B ≤ p.eval (samplingSize n r 0) := by
obtain ⟨p,hp⟩ := budget_polynomial
obtain ⟨q,hq⟩ := SmallGraphProfiles.exact_gate_measure_polynomial
let u : Polynomial ℕ := 91*X^2+q+print.comp (25*X^3)
refine ⟨p.comp u,?_⟩
intro m n r c htotal
dsimp only
have hm : measure (encodeMargins r c) (print.eval (ExactPrintCost.matrixSize (n := n) r)) ≤
u.eval (samplingSize n r 0) := by
simp only [measure,SmallGraphProfiles.exactParsedMargins_spec,u,eval_add,eval_mul,
eval_ofNat,eval_pow,eval_X,eval_comp]
exact Nat.add_le_add (Nat.add_le_add (SmallGraphProfiles.exact_public_weight r c htotal)
(hq m n r c htotal))
(polynomial_eval_monotone print (SmallGraphProfiles.exact_matrix_size (n := n) r))
have hb : LiteralProgram.packTime (encodeMargins r c).length+
ExactGatePhysical.budget (encodeMargins r c) (print.eval (ExactPrintCost.matrixSize (n := n) r))+
LiteralProgram.postTime (print.eval (ExactPrintCost.matrixSize (n := n) r)) ≤
(p.comp u).eval (samplingSize n r 0) :=
(hp (encodeMargins r c) (print.eval (ExactPrintCost.matrixSize (n := n) r))).trans (by
rw [eval_comp]
exact polynomial_eval_monotone p hm)
simpa only [ExactGatePublic.budget,ExactTablePrefix.input] using hb
end
end ContingencyTables.ExactGatePolynomial
end OAI
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