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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 | import OAI.Computability.PerfectCompleteness.Foundations.ExactConditionalOutput
import OAI.Computability.PerfectCompleteness.Foundations.RawOccurrenceOrderLemmas
import OAI.Computability.PerfectCompleteness.Machines.FiniteConstraintMachine
namespace OAI
namespace PerfectCompleteness.SignedCompletionSchedule
open MetadataFreeSampler MetadataFreeLaw
open UniqueGamesTheorem.Foundations.Games
open Turing UniqueGamesTheorem.Foundations.Complexity
open scoped BigOperators Classical
noncomputable section
variable {branch : Nat → Nat} {n t q : Nat} {rows repeats : Nat → Nat}
(hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth (TreeCanonical.locationCount branch n t) ≤ q)
def result (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) :
LocalCompletionFamily.Result q := FiniteConstraintMachine.result hq large ⟨signs, e⟩
def active (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) : Nat :=
LocalCompletionFamily.activeCount (result (rows := rows) (repeats := repeats) hq large signs e)
theorem active_positive (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) :
0 < active (rows := rows) (repeats := repeats) hq large signs e := LocalCompletionFamily.activeCount_positive _
theorem active_le (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) :
active (rows := rows) (repeats := repeats) hq large signs e ≤ 2 * q := LocalCompletionFamily.activeCount_le _
abbrev Completed (signs : SignTuple branch n t) :=
Σ e : SignedOutcome signs rows repeats, Fin (active (rows := rows) (repeats := repeats) hq large signs e)
abbrev Descriptor := Σ signs : SignTuple branch n t, Completed (rows := rows) (repeats := repeats) hq large signs
def descriptorTable (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : Fin (2 * q) → Fin q :=
LocalCompletionFamily.table (result (rows := rows) (repeats := repeats) hq large descriptor.1 descriptor.2.1) descriptor.2.2
theorem descriptorTable_eq_parsed (signs : SignTuple branch n t)
(e : Completed (rows := rows) (repeats := repeats) hq large signs) (a : Fin (2 * q)) :
descriptorTable (rows := rows) (repeats := repeats) hq large ⟨signs, e⟩ a =
(LocalCompletionSerialization.projection
(FiniteConstraintMachine.result_correct (rows := rows) (repeats := repeats) hq large ⟨signs, e.1⟩) e.2).images[a] :=
(FiniteConstraintMachine.parsed_projection (rows := rows) (repeats := repeats) hq large ⟨signs, e.1⟩ e.2 a).symm
def seedLaw (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) :
FiniteDistribution (Fin (active (rows := rows) (repeats := repeats) hq large signs e)) := by
letI : Nonempty (Fin (active (rows := rows) (repeats := repeats) hq large signs e)) := ⟨⟨0, active_positive (rows := rows) (repeats := repeats) hq large signs e⟩⟩
exact FiniteDistribution.uniform _
variable (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1))
def law (signs : SignTuple branch n t) : FiniteDistribution (Completed (rows := rows) (repeats := repeats) hq large signs) :=
CompletionSoundness.sigmaLaw (signedLaw signs rows repeats hn hbranch hrows)
(seedLaw (rows := rows) (repeats := repeats) hq large signs)
theorem law_weight (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) :
(law (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).weight e =
(signedLaw signs rows repeats hn hbranch hrows).weight e.1 /
active (rows := rows) (repeats := repeats) hq large signs e.1 := by
change (signedLaw signs rows repeats hn hbranch hrows).weight e.1 *
(1 / (Fintype.card (Fin (active (rows := rows) (repeats := repeats) hq large signs e.1)) : ℝ)) = _
simp only [Fintype.card_fin, div_eq_mul_inv, one_mul]
def count (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) : Nat :=
ExactSeedMultiplicity.perSeed (2 * q) (active (rows := rows) (repeats := repeats) hq large signs e.1)
(SignedMultiplicity.baseCopies branch n t rows repeats hn hbranch hrows signs e.1)
theorem count_positive (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) :
0 < count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e :=
ExactSeedMultiplicity.perSeed_positive (active_positive (rows := rows) (repeats := repeats) hq large signs e.1)
(active_le (rows := rows) (repeats := repeats) hq large signs e.1)
(SignedMultiplicity.baseCopies_positive branch n t rows repeats hn hbranch hrows signs e.1)
theorem count_total (signs : SignTuple branch n t) :
(∑ e, count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e) =
SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q :=
ExactSeedMultiplicity.sum_multiplicity
(SignedMultiplicity.baseCopies branch n t rows repeats hn hbranch hrows signs)
(active (rows := rows) (repeats := repeats) hq large signs) (active_positive (rows := rows) (repeats := repeats) hq large signs) (active_le (rows := rows) (repeats := repeats) hq large signs)
(SignedMultiplicity.baseCopies_total branch n t rows repeats hn hbranch hrows signs)
theorem count_ratio (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) :
(count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e : ℚ) /
SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q =
SignedMultiplicity.weights branch n t rows repeats hn hbranch hrows signs e.1 /
active (rows := rows) (repeats := repeats) hq large signs e.1 := by
change (ExactSeedMultiplicity.perSeed (2 * q) (active (rows := rows) (repeats := repeats) hq large signs e.1)
(SignedMultiplicity.baseCopies branch n t rows repeats hn hbranch hrows signs e.1) : ℚ) /
ExactSeedMultiplicity.denominator
(SignedMultiplicity.baseDenominator branch n t rows repeats hn hbranch hrows) (2 * q) = _
rw [ExactSeedMultiplicity.perSeed_ratio
(SignedMultiplicity.baseDenominator_positive branch n t rows repeats hn hbranch hrows)
(active_positive (rows := rows) (repeats := repeats) hq large signs e.1) (active_le (rows := rows) (repeats := repeats) hq large signs e.1),
SignedMultiplicity.baseCopies_ratio]
theorem count_mass (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) :
(count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e : ℝ) /
SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q =
(law (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).weight e := by
have h := congrArg (fun z : ℚ => (z : ℝ)) (count_ratio (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e)
simp only [Rat.cast_div, Rat.cast_natCast] at h
rw [law_weight, (signedRationalLaw signs rows repeats hn hbranch hrows).cast_weight]
exact h
def schedule (signs : SignTuple branch n t) : List (Completed (rows := rows) (repeats := repeats) hq large signs) :=
ExactConditionalOutput.copies (count (rows := rows) (repeats := repeats) hq large hn hbranch hrows) signs
theorem schedule_length (signs : SignTuple branch n t) :
(schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).length =
SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q := by
rw [schedule, ExactConditionalOutput.copies_length, count_total]
theorem schedule_nonempty (signs : SignTuple branch n t) :
schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs ≠ [] := by
intro hempty
have h := schedule_length (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs
rw [hempty, List.length_nil] at h
exact (Nat.ne_of_gt
(SignedMultiplicity.denominator_positive branch n t rows repeats hn hbranch hrows q)) h.symm
theorem schedule_acceptance (signs : SignTuple branch n t)
(accepts : Completed (rows := rows) (repeats := repeats) hq large signs → Bool) :
(((schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).filter accepts).length : ℝ) /
(schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).length =
(law (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).probability accepts := by
rw [schedule_length]
change (((ExactConditionalOutput.copies (count (rows := rows) (repeats := repeats) hq large hn hbranch hrows) signs).filter
accepts).length : ℝ) / _ = _
rw [ExactConditionalOutput.copies_filter_length, Nat.cast_sum, Finset.sum_div,
FiniteDistribution.probability]
apply Finset.sum_congr rfl
intro e _
cases accepts e <;> simp [count_mass]
def descriptorEnum : Descriptor (rows := rows) (repeats := repeats) hq large ≃ Fin (Fintype.card (Descriptor (rows := rows) (repeats := repeats) hq large)) :=
Fintype.equivFin _
def descriptorBits (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : List Bool :=
FiniteBlockMachine.word (descriptorEnum (rows := rows) (repeats := repeats) hq large) descriptor
@[simp] theorem descriptorBits_length (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
(descriptorBits (rows := rows) (repeats := repeats) hq large descriptor).length = Fintype.card (Descriptor (rows := rows) (repeats := repeats) hq large) :=
FiniteBlockMachine.word_length _ _
def outputBits (signs : SignTuple branch n t) : List Bool :=
(schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).flatMap
(fun e => descriptorBits (rows := rows) (repeats := repeats) hq large ⟨signs, e⟩)
def signsEnum (branch : Nat → Nat) (n t : Nat) :
SignTuple branch n t ≃ Fin (Fintype.card (SignTuple branch n t)) := Fintype.equivFin _
def inputBits (signs : SignTuple branch n t) : List Bool :=
FiniteBlockMachine.word (signsEnum branch n t) signs
def computation :
TM2ComputableInPolyTime (inputBits (branch := branch) (n := n) (t := t))
id (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows) :=
FiniteWordMachine.computation (signsEnum branch n t) (fun _ _ => (false, false, false))
(outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows)
def machine_output (signs : SignTuple branch n t) :
TM2OutputsInTime (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm (inputBits signs)
(some (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs)) 1 := by
have h := (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).outputsFun signs
change TM2OutputsInTime (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm
((inputBits signs).map id) (some ((outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).map id))
((1 : Polynomial Nat).eval _) at h
rw [Polynomial.eval_one] at h
exact Eq.mp (congrArg₂
(fun (i o : List Bool) =>
TM2OutputsInTime (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm i (some o) 1)
(List.map_id (inputBits signs))
(List.map_id (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs))) h
theorem workAlphabet_finite (tape : (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm.K) :
Finite ((computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm.Γ tape) := by
change Finite Bool
infer_instance
end
end PerfectCompleteness.SignedCompletionSchedule
end OAI
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