File size: 6,520 Bytes
e320e82 | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | import OAI.Combinatorics.EditApproximation.Specification.ConcreteFootprintBridges
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI.EditApproximation
open Filter MeasureTheory
attribute [local instance] queryRoundsNeZero
def computedQueryTemporaryCellsFootprint (source target : List ℕ) (N : ℕ)
(occurrence : BinaryMemo PositionCounts) (symbolBits positionBits : ℕ)
(input : ComputedFamilyInput source target N) : ℕ :=
BitQuery.Footprint.rankedBitMemoCells (SourceRanked.rank ChargedPhysicalRequest.rank)
(computedQueryMemoNormalized source target N occurrence symbolBits positionBits input)
(computedQueryMemoFootprint source target N occurrence symbolBits positionBits input)
(16384 * inputHeight N + 5) (computedFamilyMemoCode source target N (inputHeight N ^ 2))
(fun key => (computedFamilyMemoCodeWithWork source target N (inputHeight N ^ 2) key).2)
(.inr (localFinalRootRequest target N (integerParameters N).T (integerParameters N).S
(inputHeight N) (Nat.two_pow_pos _) (finalTargetState target.length))) .empty
noncomputable def integerAcceptedSpaceTermsFootprint (source target : List ℤ) (N : ℕ)
(epsilon : BinaryFraction)
(input : ComputedFamilyInput (source.map integerSymbolCode) (target.map integerSymbolCode) N) : List ℕ :=
let x := source.map integerSymbolCode
let y := target.map integerSymbolCode
[8 * integerPreevaluationWork source target N epsilon,
2 * integerStoredDataSpace source target,
computedQuerySetupAllocation N,
(computedQueryMemoResult x y N (integerOccurrenceIndex source target)
(integerInputSymbolBits source target) (integerInputPositionBits target) input).memory.fractionStorage,
computedQueryTemporaryCellsFootprint x y N (integerOccurrenceIndex source target)
(integerInputSymbolBits source target) (integerInputPositionBits target) input,
computedFamilyCoarseCopyStorage x y N input,
computedFamilyCoarseArrayCapacity x y N input,
computedFamilyCoarseScratchCapacity x y N input,
computedFamilyCoarseControlCapacity x y N input,
4 * (inputHeight N + (integerParameters N).B + 4),
integerFinalAllocation source target N input,
8 * (N + 1)]
noncomputable def integerBinarySpaceFootprint (source target : List ℤ) (N : ℕ) (epsilon : BinaryFraction)
(outcome : ComputedFullOutcome (source.map integerSymbolCode) (target.map integerSymbolCode) N) : ℕ :=
if physicalLargeInput N epsilon then
match outcome with
| .inl _ => 8 * integerPreevaluationWork source target N epsilon +
2 * integerStoredDataSpace source target +
suffixDPAllocation (source.map integerSymbolCode) (target.map integerSymbolCode) + 8 * (N + 1)
| .inr execution => (integerAcceptedSpaceTermsFootprint source target N epsilon
(computedExecutionAccepted _ _ N execution)).sum
else 8 * integerPreevaluationWork source target N epsilon +
2 * integerStoredDataSpace source target +
suffixDPAllocation (source.map integerSymbolCode) (target.map integerSymbolCode) + 8 * (N + 1)
theorem computedQueryTemporaryCellsFootprint_eq (source target : List ℕ) (N : ℕ)
(occurrence : BinaryMemo PositionCounts) (symbolBits positionBits : ℕ)
(input : ComputedFamilyInput source target N) :
computedQueryTemporaryCellsFootprint source target N occurrence symbolBits positionBits input =
computedQueryTemporaryCells source target N occurrence symbolBits positionBits input := by
have h : computedQueryMemoFootprint source target N occurrence symbolBits positionBits input =
(fun key => (computedQueryMemoHeap source target N occurrence symbolBits positionBits input key).footprint) := by
funext key
exact (computedQueryMemoHeap_footprint source target N occurrence symbolBits positionBits input key).symm
unfold computedQueryTemporaryCellsFootprint computedQueryTemporaryCells
rw [h]
exact BitQuery.Heap.footprint_rankedBitMemoCells _ _ _ _ _ _ _ _
theorem integerAcceptedSpaceTermsFootprint_eq (source target : List ℤ) (N : ℕ)
(epsilon : BinaryFraction)
(input : ComputedFamilyInput (source.map integerSymbolCode) (target.map integerSymbolCode) N) :
integerAcceptedSpaceTermsFootprint source target N epsilon input =
integerAcceptedSpaceTerms source target N epsilon input := by
simp only [integerAcceptedSpaceTermsFootprint, integerAcceptedSpaceTerms,
computedQueryTemporaryCellsFootprint_eq]
theorem integerBinarySpaceFootprint_eq (source target : List ℤ) (N : ℕ) (epsilon : BinaryFraction)
(outcome : ComputedFullOutcome (source.map integerSymbolCode) (target.map integerSymbolCode) N) :
integerBinarySpaceFootprint source target N epsilon outcome =
integerBinarySpace source target N epsilon outcome := by
simp only [integerBinarySpaceFootprint, integerBinarySpace, integerAcceptedSpaceTermsFootprint_eq]
rfl
theorem almostLinearEditDistance_main (epsilon : BinaryFraction)
(hepsilon : 0 < epsilon.value) (_hepsilonOne : epsilon.value < 1)
:
(∀ (source target : List ℤ) (N : ℕ), source.length + target.length ≤ N →
(2 / 3 : ℝ) ≤ (computedFullLaw (source.map integerSymbolCode) (target.map integerSymbolCode) N epsilon).real
{outcome | EditDistortion.edit source target ≤ integerBinaryAnswer source target N epsilon outcome ∧
(integerBinaryAnswer source target N epsilon outcome : ℝ) ≤
(1 + (epsilon.value : ℝ)) * EditDistortion.edit source target}) ∧
(∀ (source target : List ℤ) (N : ℕ), source = target → ∀ outcome,
integerBinaryAnswer source target N epsilon outcome = 0) ∧
(∀ (C : ℕ) (eta : ℝ), 0 < eta → ∀ᶠ N : ℕ in atTop,
∀ source target : List ℤ, source.length + target.length ≤ N →
(∀ z ∈ source ++ target, z.natAbs ≤ (N + 2) ^ C) →
(∫ outcome, integerBinaryWork source target N epsilon outcome
∂computedFullLaw (source.map integerSymbolCode) (target.map integerSymbolCode) N epsilon) ≤
(N + 2 : ℝ) ^ (1 + eta)) ∧
(∀ C : ℕ, ∀ᶠ N : ℕ in atTop, ∀ source target : List ℤ,
source.length + target.length ≤ N →
(∀ z ∈ source ++ target, z.natAbs ≤ (N + 2) ^ C) →
∀ outcome, integerBinarySpaceFootprint source target N epsilon outcome ≤ (N + 2) ^ (C + 65564)) := by
simpa only [integerBinarySpaceFootprint_eq] using
almostLinearEditDistance epsilon hepsilon _hepsilonOne
end OAI.EditApproximation
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