File size: 2,729 Bytes
97c3405 | 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.ContingencyTables.Machines.BooleanChunkProgram
import OAI.Combinatorics.ContingencyTables.Dense.DenseWalkSemantics
namespace OAI
/-! # A polynomial program for the complete flat-word dense walk -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS.TreeTyped GridBoundary ProfilePrograms
def denseWidthRealizer : Realizer denseStepBits := by
let index := composition (composition (pair (constant 8) identity) multiply) ceilLogTwoRealizer
let coins := composition (pair (constant 8) fifthPower) multiply
exact composition (pair index coins) add
theorem polynomial_denseWidth : PolynomialTime denseWidthRealizer := by
unfold denseWidthRealizer
exact polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_composition
(polynomial_pair (polynomial_constant 8) polynomial_identity) polynomial_multiply)
polynomial_ceilLogTwo)
(polynomial_composition (polynomial_pair (polynomial_constant 8) polynomial_fifthPower)
polynomial_multiply)) polynomial_add
def flatWordWalk (z : FlatExponentData×List Bool) : FlatExponentData :=
walkWords (z.1,wordChunks (denseStepBits z.1.2,z.2))
def flatWordWalkRealizer : Realizer flatWordWalk :=
composition (pair first (composition (pair
(composition (composition first second) denseWidthRealizer) second) wordChunksRealizer))
walkWordsRealizer
theorem polynomial_flatWordWalk : PolynomialTime flatWordWalkRealizer :=
polynomial_composition (polynomial_pair polynomial_first (polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_denseWidth)
polynomial_second) polynomial_wordChunks)) polynomial_walkWords
lemma denseStepBits_pos {d : ℕ} (hd : 0<d) : 0<denseStepBits d := by
have h := Nat.mul_pos (by decide : 0<8) (pow_pos hd 5)
exact h.trans_le (Nat.le_add_left _ _)
noncomputable section
open scoped Classical
/-- The literal flat-bit program computes exactly the finite Boolean walk.
The polynomial runtime theorem above applies to every input word. -/
theorem flatWordWalk_code {m n d : ℕ} (hd : 0<d)
(R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(a : Grid (I := Fin (m*n)) (4*d^8)) (steps : ℕ)
(bits : Fin (steps*denseStepBits d) → Bool) :
flatWordWalk ((((marginCode R,marginCode P),gridNatCode a),d),List.ofFn bits)=
(((marginCode R,marginCode P),
gridNatCode (denseBooleanWalk R P d finProdFinEquiv a steps bits)),d) := by
dsimp only [flatWordWalk]
rw [wordChunks_ofFn (denseStepBits_pos hd) bits]
exact walkWords_ofFn R P a steps (gridWordEquiv steps (denseStepBits d) bits)
end
end ContingencyTables.DensePrograms
end OAI
|