File size: 15,627 Bytes
8df23f4 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 | import OAI.Computability.BinPacking.Computation.MachineRegularTable
namespace OAI
namespace BinPackingGames.Foundations.Complexity.MachineRegularTable.Top
section
open Turing MachineComposition
open BinPackingGames.Foundations.PCP
private theorem joinTrace {A : Type*} {f : A → A} {m n : Nat} {a b c : A}
(first : f^[m] a = b) (second : f^[n] b = c) : f^[m + n] a = c := by
rw [Nat.add_comm m n, Function.iterate_add_apply, first, second]
private theorem normalizeEmptyAppend {A : Type*} (output : List Bool → A)
(bits : List Bool) : output ([] ++ bits) = output bits :=
congrArg output (List.nil_append bits)
theorem headerStartTrace (H : BaseTable) (t : GraphTables.Table) :
(advance (TM2.step (program H)))^[Header.totalTime PreprocessingRegularTables.internalDegree t []]
(some (cfg H (some (.header .init)) (inputData t []) (fun _ => [])
(counterStacks [] [] [] []))) =
some (cfg H (some .oldSeed) (headerData t (headerOutput t)) (headerStacks t)
(counterStacks [] [] [] [])) := by
have run := headerTrace H t [] (counterStacks [] [] [] [])
exact run.trans (normalizeEmptyAppend
(fun output : List Bool => some (cfg H (some Label.oldSeed)
(headerData t output) (headerStacks t) (counterStacks [] [] [] [])))
(Header.headerBits PreprocessingRegularTables.internalDegree t))
theorem headerOldTrace (H : BaseTable) (t : GraphTables.Table) :
(advance (TM2.step (program H)))^[
Header.totalTime PreprocessingRegularTables.internalDegree t [] +
(1+2*(t.darts+2)+(oldElapsed H t (headerOutput t) t.darts+1))]
(some (cfg H (some (.header .init)) (inputData t []) (fun _ => [])
(counterStacks [] [] [] []))) =
some (cfg H (some .ownerSeed) (oldData t t.darts (oldOutput H t)) (headerStacks t)
(counterStacks [] [] [] [])) := by
have first := headerStartTrace H t
have second := oldStageTrace H t (headerOutput t) (headerStacks t)
have joined := joinTrace (f := advance (TM2.step (program H))) first second
have output_eq : headerOutput t ++ oldPrefix H t t.darts = oldOutput H t := rfl
exact joined.trans (congrArg
(fun output : List Bool => some (cfg H (some Label.ownerSeed)
(oldData t t.darts output) (headerStacks t) (counterStacks [] [] [] []))) output_eq)
theorem beforeCleanupTrace (H : BaseTable) (t : GraphTables.Table) :
(advance (TM2.step (program H)))^[
Header.totalTime PreprocessingRegularTables.internalDegree t [] +
(1+2*(t.darts+2)+(oldElapsed H t (headerOutput t) t.darts+1)) +
(1+2*(t.vertices+2)+(ownerElapsed H t (oldOutput H t) t.vertices+1))]
(some (cfg H (some (.header .init)) (inputData t []) (fun _ => [])
(counterStacks [] [] [] []))) =
some ⟨clearEntry 0, readyState H, finalStacks t (finalOutput H t)⟩ := by
have first := headerOldTrace H t
have second := ownerStageTrace H t (oldOutput H t)
have joined := joinTrace (f := advance (TM2.step (program H))) first second
have output_eq : oldOutput H t ++ ownerPrefix H t t.vertices = finalOutput H t := rfl
exact joined.trans (congrArg
(fun output : List Bool => some (cfg H (clearEntry 0)
(ownerData t t.vertices output) (headerStacks t) (counterStacks [] [] [] []))) output_eq)
theorem cleanupFinalTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool) :
(advance (TM2.step (program H)))^[cleanupTime (finalStacks t output) 7]
(some ⟨clearEntry 0, readyState H, finalStacks t output⟩) =
some (cfg H none (inputData t output) (fun _ => []) (counterStacks [] [] [] [])) :=
(cleanupTrace H (finalStacks t output)).trans
(congrArg (fun terminalStacks => some (⟨none, readyState H, terminalStacks⟩ : TM2.Cfg Alphabet Label State))
(cleaned_finalStacks t output))
theorem fullTrace (H : BaseTable) (t : GraphTables.Table) :
(advance (TM2.step (program H)))^[totalTime H t]
(some (cfg H (some (.header .init)) (inputData t []) (fun _ => [])
(counterStacks [] [] [] []))) =
some (cfg H none (inputData t (finalOutput H t)) (fun _ => [])
(counterStacks [] [] [] [])) :=
joinTrace (f := advance (TM2.step (program H))) (beforeCleanupTrace H t) (cleanupFinalTrace H t (finalOutput H t))
@[simp] theorem update_frame_output (data : Data) (header : Header.Tape → List Bool)
(counters : Fin 4 → List Bool) (replacement : List Bool) :
Function.update (frame data header counters) (coreTape 8) replacement =
frame { data with output := replacement } header counters := by
funext j
cases j with
| inl j => cases j with
| inl i => fin_cases i <;> rfl
| inr e => cases e with
| inl e => cases e <;> rfl
| inr e => rfl
| inr j => cases j <;> rfl
theorem input_stacks_eq (H : BaseTable) (t : GraphTables.Table) :
frame (inputData t []) (fun _ => []) (counterStacks [] [] [] []) =
(Turing.initList (machine H) (GraphTables.tableBits t)).stk := by
funext j
cases j with
| inl j => cases j with
| inl i => (fin_cases i <;>
simp [Turing.initList, machine, coreTape, frame, inputData,
MachineRegularOriginalBody.frame, MachineRegularMetadata.frame]); rfl
| inr e => cases e with
| inl e => cases e <;>
simp [Turing.initList, machine, coreTape, frame, inputData,
MachineRegularOriginalBody.frame, MachineRegularMetadata.frame]
| inr e =>
simp [Turing.initList, machine, coreTape, frame,
MachineRegularOriginalBody.frame]
| inr j => cases j with
| inl j => simp [Turing.initList, machine, coreTape, frame]
| inr j => fin_cases j <;> simp [Turing.initList, machine, coreTape, frame, counterStacks]
theorem initial_cfg_eq (H : BaseTable) (t : GraphTables.Table) :
cfg H (some (.header .init)) (inputData t []) (fun _ => []) (counterStacks [] [] [] []) =
Turing.initList (machine H) (GraphTables.tableBits t) := by
change (⟨some (.header .init), readyState H,
frame (inputData t []) (fun _ => []) (counterStacks [] [] [] [])⟩ : TM2.Cfg Alphabet Label State) =
⟨some (.header .init), readyState H, (Turing.initList (machine H) (GraphTables.tableBits t)).stk⟩
rw [input_stacks_eq]
theorem trace (H : BaseTable) (t : GraphTables.Table) :
(advance (TM2.step (program H)))^[totalTime H t]
(some (Turing.initList (machine H) (GraphTables.tableBits t))) =
some ⟨none, readyState H,
Function.update (Turing.initList (machine H) (GraphTables.tableBits t)).stk
(coreTape 8) (PortTables.tableBits (PreprocessingRegularTables.regularize H t))⟩ := by
have full := fullTrace H t
rw [initial_cfg_eq, finalOutput_eq] at full
have final_stacks :
frame (inputData t (PortTables.tableBits (PreprocessingRegularTables.regularize H t)))
(fun _ => []) (counterStacks [] [] [] []) =
Function.update (Turing.initList (machine H) (GraphTables.tableBits t)).stk (coreTape 8)
(PortTables.tableBits (PreprocessingRegularTables.regularize H t)) := by
rw [← input_stacks_eq]
exact (update_frame_output (inputData t []) (fun _ => []) (counterStacks [] [] [] [])
(PortTables.tableBits (PreprocessingRegularTables.regularize H t))).symm
change (advance (TM2.step (program H)))^[totalTime H t]
(some (Turing.initList (machine H) (GraphTables.tableBits t))) =
some ⟨none, readyState H,
frame (inputData t (PortTables.tableBits (PreprocessingRegularTables.regularize H t)))
(fun _ => []) (counterStacks [] [] [] [])⟩ at full
rw [final_stacks] at full
exact full
end
open Turing MachineComposition PCP PreprocessingRegularTables
open PreprocessingRegularBounds PreprocessingMachineBounds
noncomputable def oldLoopPolynomial : Polynomial Nat :=
Polynomial.X *
(MachineRegularOriginalBodyBounds.timePolynomial.comp
(Polynomial.X + regularPolynomial) + 2)
noncomputable def ownerLoopPolynomial : Polynomial Nat :=
Polynomial.X *
(MachineRegularOwnerBodyBounds.timePolynomial.comp
(Polynomial.X + regularPolynomial) + 2)
noncomputable def timePolynomial : Polynomial Nat :=
Header.timePolynomial internalDegree + oldLoopPolynomial + ownerLoopPolynomial +
4 * Polynomial.X + 12 +
Polynomial.C (7 * (ExpanderFamily.growth * (internalDegree + 4) + 4)) *
(Polynomial.X + 1)
theorem oldElapsed_prefix_le (H : BaseTable) (t : GraphTables.Table) (k : Nat) :
oldElapsed H t (Header.headerBits internalDegree t) k ≤
k * (MachineRegularOriginalBodyBounds.timePolynomial.eval
(inputLength t + regularPolynomial.eval (inputLength t)) + 2) := by
induction k with
| zero => simp [oldElapsed]
| succ k ih =>
simp only [oldElapsed]
split_ifs with hk
· have raw := MachineRegularOriginalBodyBounds.totalSteps_le H t ⟨k, hk⟩
(Header.headerBits internalDegree t ++ oldPrefix H t k)
have hprefix :
(Header.headerBits internalDegree t ++ oldPrefix H t k).length ≤
regularPolynomial.eval (inputLength t) := by
exact regular_originalPrefix_le t H k
have hcall := raw.trans (natPolynomial_eval_mono
MachineRegularOriginalBodyBounds.timePolynomial (Nat.add_le_add_left hprefix _))
rw [Nat.succ_mul]
omega
· rw [Nat.succ_mul]
omega
theorem oldElapsed_le (H : BaseTable) (t : GraphTables.Table) :
oldElapsed H t (Header.headerBits internalDegree t) t.darts ≤
oldLoopPolynomial.eval (inputLength t) := by
have bound := (oldElapsed_prefix_le H t t.darts).trans
(Nat.mul_le_mul_right _ (GraphTables.darts_le_tableBits_length t))
simpa only [oldLoopPolynomial, inputLength, Polynomial.eval_mul, Polynomial.eval_X,
Polynomial.eval_add, Polynomial.eval_comp, Polynomial.eval_ofNat] using bound
theorem ownerElapsed_prefix_le (H : BaseTable) (t : GraphTables.Table) (k : Nat) :
ownerElapsed H t
(Header.headerBits internalDegree t ++ oldPrefix H t t.darts) k ≤
k * (MachineRegularOwnerBodyBounds.timePolynomial.eval
(inputLength t + regularPolynomial.eval (inputLength t)) + 2) := by
induction k with
| zero => simp [ownerElapsed]
| succ k ih =>
simp only [ownerElapsed]
split_ifs with hk
· have raw := MachineRegularOwnerBodyBounds.totalSteps_le H t ⟨k, hk⟩
((Header.headerBits internalDegree t ++ oldPrefix H t t.darts) ++ ownerPrefix H t k)
have hprefix :
((Header.headerBits internalDegree t ++ oldPrefix H t t.darts) ++
ownerPrefix H t k).length ≤ regularPolynomial.eval (inputLength t) := by
rw [owner_output_eq_ownerPrefix]
exact regular_ownerPrefix_le t H k
have hcall := raw.trans (natPolynomial_eval_mono
MachineRegularOwnerBodyBounds.timePolynomial (Nat.add_le_add_left hprefix _))
rw [Nat.succ_mul]
omega
· rw [Nat.succ_mul]
omega
theorem ownerElapsed_le (H : BaseTable) (t : GraphTables.Table) :
ownerElapsed H t
(Header.headerBits internalDegree t ++ oldPrefix H t t.darts) t.vertices ≤
ownerLoopPolynomial.eval (inputLength t) := by
have bound := (ownerElapsed_prefix_le H t t.vertices).trans
(Nat.mul_le_mul_right _ (GraphTables.vertices_le_tableBits_length t))
simpa only [ownerLoopPolynomial, inputLength, Polynomial.eval_mul, Polynomial.eval_X,
Polynomial.eval_add, Polynomial.eval_comp, Polynomial.eval_ofNat] using bound
theorem totalTime_le (H : BaseTable) (t : GraphTables.Table) :
totalTime H t ≤ timePolynomial.eval (inputLength t) := by
have header := Header.totalTime_le internalDegree t []
simp only [List.length_nil, Nat.add_zero] at header
change Header.totalTime internalDegree t [] ≤
(Header.timePolynomial internalDegree).eval (inputLength t) at header
have old := oldElapsed_le H t
have owners := ownerElapsed_le H t
have cleanup := cleanupTime_final_le t (finalOutput H t)
have hm : t.darts ≤ inputLength t := GraphTables.darts_le_tableBits_length t
have hn : t.vertices ≤ inputLength t := GraphTables.vertices_le_tableBits_length t
simp only [totalTime, headerOutput, oldOutput, timePolynomial,
Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
Polynomial.eval_ofNat, Polynomial.eval_one]
omega
noncomputable def machineInTime (H : BaseTable) (t : GraphTables.Table) :
StateTransition.EvalsToInTime (TM2.step (program H))
(initList (machine H) (GraphTables.tableBits t))
(some ⟨none, readyState H,
Function.update (initList (machine H) (GraphTables.tableBits t)).stk
(coreTape 8) (PortTables.tableBits (regularize H t))⟩)
(timePolynomial.eval (inputLength t)) where
steps := totalTime H t
evals_in_steps := trace H t
steps_le_m := totalTime_le H t
end BinPackingGames.Foundations.Complexity.MachineRegularTable.Top
namespace BinPackingGames.Foundations.Complexity.MachineRegularTableRuntime
open Turing
open PCP
open MachineRegularTable.Top (Tape Label State coreTape readyState)
def rawProgram (H : PreprocessingRegularTables.BaseTable) :
MachineCanonicalOutput.Program Tape Label State where
input := coreTape 0
output := coreTape 8
main := .header .init
initial := readyState H
code := MachineRegularTable.Top.program H
theorem sourceMachine_eq (H : PreprocessingRegularTables.BaseTable) :
MachineCanonicalOutput.sourceMachine (rawProgram H) =
MachineRegularTable.Top.machine H := rfl
noncomputable def cleanupTapes : List Tape :=
(Finset.univ : Finset Tape).toList.filter (fun k => decide (k ≠ coreTape 8))
theorem cleanup_complete (H : PreprocessingRegularTables.BaseTable) (k : Tape) :
k ∈ cleanupTapes ↔ k ≠ (rawProgram H).output := by
simp [cleanupTapes, rawProgram]
noncomputable def terminalRun (H : PreprocessingRegularTables.BaseTable)
(t : GraphTables.Table) :
MachineCanonicalOutput.TerminalRun (rawProgram H) (GraphTables.tableBits t)
(PortTables.inputBits (PreprocessingStageMaps.regular H t))
(MachineRegularTable.Top.timePolynomial.eval (GraphTables.tableBits t).length) where
state := readyState H
tapes := Function.update
(initList (MachineRegularTable.Top.machine H) (GraphTables.tableBits t)).stk
(coreTape 8) (PortTables.inputBits (PreprocessingStageMaps.regular H t))
execution := by
simpa only [sourceMachine_eq, FinTM2.Cfg, FinTM2.step, MachineRegularTable.Top.machine,
PreprocessingMachineBounds.inputLength, PreprocessingStageMaps.regular,
PortTables.inputBits] using!
MachineRegularTable.Top.machineInTime H t
output_eq := by
change Function.update _ (coreTape 8) _ (coreTape 8) = _
exact Function.update_self _ _ _
noncomputable def computableInPolyTime (H : PreprocessingRegularTables.BaseTable) :
TM2ComputableInPolyTime GraphTables.tableBits
(PortTables.inputBits (ports := PreprocessingRegularTables.internalDegree + 1))
(PreprocessingStageMaps.regular H) :=
MachineCanonicalOutput.computableInPolyTime (rawProgram H) cleanupTapes
(cleanup_complete H) GraphTables.tableBits PortTables.inputBits
(PreprocessingStageMaps.regular H) MachineRegularTable.Top.timePolynomial (terminalRun H)
theorem finite_alphabet (H : PreprocessingRegularTables.BaseTable) :
∀ k, Finite ((computableInPolyTime H).tm.Γ k) :=
MachineCanonicalOutput.computableInPolyTime_finite_alphabet (rawProgram H) cleanupTapes
(cleanup_complete H) GraphTables.tableBits PortTables.inputBits
(PreprocessingStageMaps.regular H) MachineRegularTable.Top.timePolynomial (terminalRun H)
end BinPackingGames.Foundations.Complexity.MachineRegularTableRuntime
end OAI
|