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2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Computability.PerfectCompleteness.Construction.AscendingBranchingLemmas
import OAI.Computability.PerfectCompleteness.Foundations.SourceOddListsLemmas
import OAI.Computability.PerfectCompleteness.Machines.PreprocessingMachineLemmas
import OAI.Computability.PerfectCompleteness.Reduction.CompletionLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedRows
open InitialParameters UpperParameterScalars
open UniqueGamesTheorem.Appendix UniqueGamesTheorem.Fourier
noncomputable section
structure Plan (δ : ℚ) where
depth : Nat
depth_ge : 2 ≤ depth
depth_mass : 4 ≤ (depth : ℝ) * epsilon δ
order : Nat
order_pos : 1 ≤ order
density : ℝ
density_pos : 0 < density
density_lt : density < 1
spectral : MatrixLevelBridge.levelCutoffConstant order density +
RankLevelFilter.node (order + 1) < inverse δ / 2
rootFirstRows : Nat → Nat
rows_order : ∀ k, order ≤ rootFirstRows k
rows_pos : ∀ k, 1 ≤ rootFirstRows k
rows_correction : ∀ k, 1 / (2 : ℝ) ^ (rootFirstRows k - order) <
rhoPred (useful δ) density / 8
repeatCount : Nat → Nat
repeats_min : ∀ ℓ, 6 ≤ repeatCount ℓ
repeats_error : ∀ ℓ, (7 / 8 : ℝ) ^ repeatCount ℓ ≤
min (h order ℓ / 2) (tau (useful δ) density order ℓ ^ 2 / 2)
lowerRequirements : ∀ (hδ : 0 < δ) k (i : Fin k),
(DescendingRows.lowerChoice (useful_pos hδ) density_pos (inverse_pos hδ)
order (rootFirstRows i.val)).minimumRows ≤ rootFirstRows k
theorem exists_plan {δ : ℚ} (hδ : 0 < δ) (hδ' : δ < 1) : Nonempty (Plan δ) := by
obtain ⟨depth, r, hd, hmass, hr, α, hα, hα', hspectral⟩ :=
exists_initial hδ hδ'
obtain ⟨rows, hrows⟩ := DescendingRows.exists_rows
(useful_pos hδ) hα (inverse_pos hδ) r 1
have hreps : ∀ ℓ : Nat, ∃ R : Nat, 6 ≤ R ∧ (7 / 8 : ℝ) ^ R ≤
min (h r ℓ / 2) (tau (useful δ) α r ℓ ^ 2 / 2) :=
fun ℓ => exists_upper_repeats (useful_pos hδ) hα r ℓ
choose repeats hrepeats using hreps
exact ⟨{
depth := depth
depth_ge := hd
depth_mass := hmass
order := r
order_pos := hr
density := α
density_pos := hα
density_lt := hα'
spectral := hspectral
rootFirstRows := rows
rows_order := fun k => (hrows k).2.1
rows_pos := fun k => (hrows k).2.2.1
rows_correction := fun k => (hrows k).2.2.2.1
repeatCount := repeats
repeats_min := fun ℓ => (hrepeats ℓ).1
repeats_error := fun ℓ => (hrepeats ℓ).2
lowerRequirements := fun _ k => (hrows k).2.2.2.2 }⟩
variable {δ : ℚ} (plan : Plan δ)
def rows : Nat → Nat := DescendingRows.treeRows plan.depth plan.rootFirstRows
def repeats : Nat → Nat := fun height => plan.repeatCount (rows plan height)
def cutoff (hδ : 0 < δ) : Nat :=
DescendingRows.commonCutoff (useful_pos hδ) plan.density_pos (inverse_pos hδ)
plan.order plan.depth plan.rootFirstRows
theorem cutoff_pos (hδ : 0 < δ) : 0 < cutoff plan hδ :=
DescendingRows.commonCutoff_pos (useful_pos hδ) plan.density_pos (inverse_pos hδ)
plan.order plan.depth plan.rootFirstRows
instance cutoff_neZero (hδ : 0 < δ) : NeZero (cutoff plan hδ) :=
⟨Nat.ne_of_gt (cutoff_pos plan hδ)⟩
def sourceLength (hδ : 0 < δ) : Nat :=
SourceOddLists.repetitionLength PCPSource.sourceGap (cutoff plan hδ)
PCPSource.sourceGap_pos PCPSource.sourceGap_le_one
theorem sourceLength_pos (hδ : 0 < δ) : 0 < sourceLength plan hδ :=
SourceOddLists.repetitionLength_positive PCPSource.sourceGap (cutoff plan hδ)
PCPSource.sourceGap_pos PCPSource.sourceGap_le_one
theorem source_oddList_value_le_half (hδ : 0 < δ)
(input : List Bool) (unsat : ¬BinaryLanguage.language input) :
(SourceOddLists.game (cutoff plan hδ)
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(sourceLength plan hδ)).value ≤ 1 / 2 :=
SourceOddLists.value_le_half
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
PCPSource.sourceGap PCPSource.sourceGap_pos PCPSource.sourceGap_le_one
(PCPSourceMachine.rawSource_clauseGap input unsat)
theorem lower_row_requirement (hδ : 0 < δ) {i j : Nat}
(hij : i < j) (hj : j ≤ plan.depth) :
(DescendingRows.lowerChoice (useful_pos hδ) plan.density_pos (inverse_pos hδ)
plan.order (rows plan j)).minimumRows ≤ rows plan i :=
DescendingRows.tree_pair_requirement (useful_pos hδ) plan.density_pos (inverse_pos hδ)
plan.order plan.depth plan.rootFirstRows (plan.lowerRequirements hδ) hij hj
end
end PerfectCompleteness.FixedRows
end
end OAI
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