File size: 13,839 Bytes
760f90e | 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 | import OAI.Computability.UniqueGames.Machines.MachineCompositionLemmas
import OAI.Computability.UniqueGames.Reduction.MachineTransfer
namespace OAI
section
namespace UniqueGamesTheorem.Foundations.Complexity.MachineCopy
open Turing
variable {K Λ σ β : Type} [DecidableEq K]
abbrev Alphabet (β : Type) (_ : K) := β
/-- Drain one tape into two reverse-order accumulators. -/
def forkLoop (source left right : K) (fallback : β) (loopLabel : Λ) (exit : Option Λ) :
TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β) :=
.pop source (fun state head => (state.1, head))
(.branch (fun state => state.2.isSome)
(.push left (fun state => state.2.getD fallback)
(.push right (fun state => state.2.getD fallback) (.goto fun _ => loopLabel)))
(Reduction.MachineTransfer.exitAt right exit))
def forkTapes (source left right : K) (base : K → List β)
(input leftOutput rightOutput : List β) : K → List β :=
Function.update (Function.update (Function.update base source input) left leftOutput)
right rightOutput
@[simp] theorem forkTapes_source (source left right : K)
(sourceLeft : source ≠ left) (sourceRight : source ≠ right)
(base : K → List β) (input leftOutput rightOutput : List β) :
forkTapes source left right base input leftOutput rightOutput source = input := by
simp [forkTapes, sourceLeft, sourceRight]
@[simp] theorem forkTapes_left (source left right : K) (leftRight : left ≠ right)
(base : K → List β) (input leftOutput rightOutput : List β) :
forkTapes source left right base input leftOutput rightOutput left = leftOutput := by
simp [forkTapes, leftRight]
@[simp] theorem forkTapes_right (source left right : K)
(base : K → List β) (input leftOutput rightOutput : List β) :
forkTapes source left right base input leftOutput rightOutput right = rightOutput := by
simp [forkTapes]
@[simp] theorem forkTapes_self (source left right : K) (base : K → List β) :
forkTapes source left right base (base source) (base left) (base right) = base := by
simp [forkTapes]
private theorem update_fork_source (source left right : K)
(sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
(base : K → List β) (input leftOutput rightOutput replacement : List β) :
Function.update (forkTapes source left right base input leftOutput rightOutput)
source replacement = forkTapes source left right base replacement leftOutput rightOutput := by
funext k
by_cases hs : k = source
· subst k; simp [forkTapes, sourceLeft, sourceRight]
· by_cases hl : k = left
· subst k; simp [forkTapes, Ne.symm sourceLeft, leftRight]
· by_cases hr : k = right
· subst k; simp [forkTapes, Ne.symm sourceRight]
· simp [forkTapes, hs, hl, hr]
private theorem update_fork_left (source left right : K) (leftRight : left ≠ right)
(base : K → List β) (input leftOutput rightOutput replacement : List β) :
Function.update (forkTapes source left right base input leftOutput rightOutput)
left replacement = forkTapes source left right base input replacement rightOutput := by
funext k
by_cases hl : k = left
· subst k; simp [forkTapes, leftRight]
· by_cases hr : k = right
· subst k; simp [forkTapes, Ne.symm leftRight]
· simp [forkTapes, hl, hr]
private theorem update_fork_right (source left right : K)
(base : K → List β) (input leftOutput rightOutput replacement : List β) :
Function.update (forkTapes source left right base input leftOutput rightOutput)
right replacement = forkTapes source left right base input leftOutput replacement := by
simp [forkTapes]
theorem forkStep_empty (source left right : K)
(sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
(fallback : β) (loopLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
(base : K → List β) (leftOutput rightOutput : List β) (ambient : σ) (register : Option β) :
TM2.step program
⟨some loopLabel, (ambient, register), forkTapes source left right base [] leftOutput rightOutput⟩ =
some ⟨exit, (ambient, none), forkTapes source left right base [] leftOutput rightOutput⟩ := by
change some (TM2.stepAux (program loopLabel) (ambient, register)
(forkTapes source left right base [] leftOutput rightOutput)) = _
rw [atLoop]
cases exit <;>
simp [forkLoop, Reduction.MachineTransfer.exitAt, TM2.stepAux, sourceLeft, sourceRight,
leftRight, update_fork_source]
theorem forkStep_cons (source left right : K)
(sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
(fallback : β) (loopLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
(base : K → List β) (head : β) (input leftOutput rightOutput : List β)
(ambient : σ) (register : Option β) :
TM2.step program
⟨some loopLabel, (ambient, register),
forkTapes source left right base (head :: input) leftOutput rightOutput⟩ =
some ⟨some loopLabel, (ambient, some head),
forkTapes source left right base input (head :: leftOutput) (head :: rightOutput)⟩ := by
change some (TM2.stepAux (program loopLabel) (ambient, register)
(forkTapes source left right base (head :: input) leftOutput rightOutput)) = _
rw [atLoop]
simp [forkLoop, TM2.stepAux, sourceLeft, sourceRight, leftRight,
update_fork_source, update_fork_left, update_fork_right]
/-- Each symbol and the final empty test take one actual transition. -/
theorem forkTrace (source left right : K)
(sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
(fallback : β) (loopLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
(base : K → List β) (input leftOutput rightOutput : List β)
(ambient : σ) (register : Option β) :
(MachineComposition.advance (TM2.step program))^[input.length + 1]
(some ⟨some loopLabel, (ambient, register),
forkTapes source left right base input leftOutput rightOutput⟩) =
some ⟨exit, (ambient, none), forkTapes source left right base []
(input.reverse ++ leftOutput) (input.reverse ++ rightOutput)⟩ := by
induction input generalizing leftOutput rightOutput register with
| nil =>
simpa only [List.length_nil, Nat.zero_add, Function.iterate_one,
MachineComposition.advance_some, List.reverse_nil, List.nil_append] using
forkStep_empty source left right sourceLeft sourceRight leftRight fallback loopLabel exit
program atLoop base leftOutput rightOutput ambient register
| cons head input ih =>
rw [List.length_cons, Function.iterate_succ_apply]
change (MachineComposition.advance (TM2.step program))^[input.length + 1]
(TM2.step program ⟨some loopLabel, (ambient, register),
forkTapes source left right base (head :: input) leftOutput rightOutput⟩) = _
rw [forkStep_cons source left right sourceLeft sourceRight leftRight fallback loopLabel exit
program atLoop base head input leftOutput rightOutput ambient register, ih]
simp only [List.reverse_cons, List.append_assoc, List.singleton_append]
def forkInTime (source left right : K)
(sourceLeft : source ≠ left) (sourceRight : source ≠ right) (leftRight : left ≠ right)
(fallback : β) (loopLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atLoop : program loopLabel = forkLoop source left right fallback loopLabel exit)
(base : K → List β) (ambient : σ) (register : Option β) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some loopLabel, (ambient, register), base⟩
(some ⟨exit, (ambient, none), forkTapes source left right base []
((base source).reverse ++ base left) ((base source).reverse ++ base right)⟩)
((base source).length + 1) where
steps := (base source).length + 1
evals_in_steps := by
change (MachineComposition.advance (TM2.step program))^[(base source).length + 1]
(some ⟨some loopLabel, (ambient, register), base⟩) = _
simpa only [forkTapes_self] using
forkTrace source left right sourceLeft sourceRight leftRight fallback loopLabel exit
program atLoop base (base source) (base left) (base right) ambient register
steps_le_m := Nat.le_refl _
def drainedTapes (source scratch : K) (base : K → List β) : K → List β :=
Reduction.MachineTransfer.tapesAt source scratch base [] (base source).reverse
theorem restoredTapes (source destination scratch : K)
(sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
(destinationScratch : destination ≠ scratch) (base : K → List β)
(scratchEmpty : base scratch = []) :
forkTapes scratch source destination (drainedTapes source scratch base) []
(base source) (base source ++ base destination) =
Function.update base destination (base source ++ base destination) := by
funext k
by_cases hs : k = source
· subst k
simp [forkTapes, sourceDestination]
· by_cases hd : k = destination
· subst k; simp [forkTapes]
· by_cases ht : k = scratch
· subst k
simp [forkTapes, Ne.symm sourceScratch, Ne.symm destinationScratch, scratchEmpty]
· simp [forkTapes, drainedTapes, Reduction.MachineTransfer.tapesAt, hs, hd, ht]
theorem copyTrace (source destination scratch : K)
(sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
(destinationScratch : destination ≠ scratch)
(fallback : β) (firstLabel secondLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atFirst : program firstLabel =
Reduction.MachineTransfer.loopAt source scratch id fallback firstLabel (some secondLabel))
(atSecond : program secondLabel = forkLoop scratch source destination fallback secondLabel exit)
(base : K → List β) (scratchEmpty : base scratch = [])
(ambient : σ) (register : Option β) :
(MachineComposition.advance (TM2.step program))^[2 * ((base source).length + 1)]
(some ⟨some firstLabel, (ambient, register), base⟩) =
some ⟨exit, (ambient, none),
Function.update base destination (base source ++ base destination)⟩ := by
have first := Reduction.MachineTransfer.transferAt_fromTapes source scratch sourceScratch
id fallback firstLabel (some secondLabel) program atFirst base ambient register
change (MachineComposition.advance (TM2.step program))^[(base source).length + 1]
(some ⟨some firstLabel, (ambient, register), base⟩) = _ at first
simp only [List.map_id, scratchEmpty, List.append_nil] at first
have second := forkTrace scratch source destination (Ne.symm sourceScratch)
(Ne.symm destinationScratch) sourceDestination fallback secondLabel exit program atSecond
(drainedTapes source scratch base)
(drainedTapes source scratch base scratch) (drainedTapes source scratch base source)
(drainedTapes source scratch base destination) ambient none
simp only [forkTapes_self] at second
have scratchWord : drainedTapes source scratch base scratch = (base source).reverse := by
simp [drainedTapes]
have sourceWord : drainedTapes source scratch base source = [] := by
simp [drainedTapes, sourceScratch]
have destinationWord : drainedTapes source scratch base destination = base destination := by
simp [drainedTapes, Reduction.MachineTransfer.tapesAt, Ne.symm sourceDestination,
destinationScratch]
rw [scratchWord, sourceWord, destinationWord] at second
simp only [List.reverse_reverse, List.append_nil, List.length_reverse] at second
rw [restoredTapes source destination scratch sourceDestination sourceScratch destinationScratch
base scratchEmpty] at second
rw [show 2 * ((base source).length + 1) =
((base source).length + 1) + ((base source).length + 1) by omega,
Function.iterate_add_apply, first]
exact second
/-- A timed witness whose stored step count is the exact two-loop trace length. -/
def copyInTime (source destination scratch : K)
(sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
(destinationScratch : destination ≠ scratch)
(fallback : β) (firstLabel secondLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atFirst : program firstLabel =
Reduction.MachineTransfer.loopAt source scratch id fallback firstLabel (some secondLabel))
(atSecond : program secondLabel = forkLoop scratch source destination fallback secondLabel exit)
(base : K → List β) (scratchEmpty : base scratch = [])
(ambient : σ) (register : Option β) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some firstLabel, (ambient, register), base⟩
(some ⟨exit, (ambient, none),
Function.update base destination (base source ++ base destination)⟩)
(2 * ((base source).length + 1)) where
steps := 2 * ((base source).length + 1)
evals_in_steps := copyTrace source destination scratch sourceDestination sourceScratch
destinationScratch fallback firstLabel secondLabel exit program atFirst atSecond base
scratchEmpty ambient register
steps_le_m := Nat.le_refl _
end UniqueGamesTheorem.Foundations.Complexity.MachineCopy
end
end OAI
|