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import OAI.Computability.UniqueGames.Machines.MachineCompositionLemmas
import OAI.Computability.UniqueGames.Machines.MachineCopy
namespace OAI
namespace PerfectCompleteness.BinaryNameCompare
open Turing
open UniqueGamesTheorem.Foundations.Complexity
open MachineComposition
variable {K Λ A : Type} [DecidableEq K]
abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := (A × Bool × Option Bool) × Option Bool
def clean (ambient : A) : State A := ((ambient, false, none), none)
def scanStateEquiv (A : Type) : BinaryNameMachine.State (A × Bool) ≃ State A where
toFun s := ((s.1.1, s.1.2, s.2.1), s.2.2)
invFun s := ((s.1.1, s.1.2.1), s.1.2.2, s.2)
left_inv := by rintro ⟨⟨a, b⟩, c, d⟩; rfl
right_inv := by rintro ⟨⟨a, b, c⟩, d⟩; rfl
def compareStateEquiv (A : Type) : MachineCompare.State A ≃ State A where
toFun s := ((s.1, s.2.1, s.2.2.1), s.2.2.2)
invFun s := (s.1.1, s.1.2.1, s.1.2.2, s.2)
left_inv := by rintro ⟨a, b, c, d⟩; rfl
right_inv := by rintro ⟨⟨a, b, c⟩, d⟩; rfl
inductive Label
| scan | copyOut | copyBack | compare
deriving DecidableEq
protected abbrev Label.enumList : List Label := [.scan, .copyOut, .copyBack, .compare]
protected theorem Label.enumList_getElem?_ctorIdx_eq (x : Label) :
Label.enumList[x.ctorIdx]? = some x := by
cases x <;> rfl
protected theorem Label.enumList_nodup : Label.enumList.Nodup := by decide
instance : Fintype Label where
elems := ⟨Label.enumList, Label.enumList_nodup⟩
complete x := by cases x <;> decide
def instruction (tape : Fin 5 → K) (labels : Label → Λ)
(equalExit differentExit malformedExit : Option Λ) :
Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
| .scan => MachineStateEquiv.statement (scanStateEquiv A)
(BinaryNameMachine.scan (tape 0) (tape 2) (labels .scan)
(some (labels .copyOut)) malformedExit)
| .copyOut => UniqueGamesTheorem.Reduction.MachineTransfer.loopAt (tape 1) (tape 4) id false
(labels .copyOut) (some (labels .copyBack))
| .copyBack => MachineCopy.forkLoop (tape 4) (tape 1) (tape 3) false
(labels .copyBack) (some (labels .compare))
| .compare => MachineStateEquiv.statement (compareStateEquiv A)
(MachineCompare.loop (tape 2) (tape 3) (labels .compare) equalExit differentExit)
def steps (payloadLength keyLength : Nat) : Nat :=
(payloadLength + 1) + 2 * (keyLength + 1) + (max payloadLength keyLength + 1)
theorem steps_le (payloadLength keyLength : Nat) :
steps payloadLength keyLength ≤ 2 * payloadLength + 3 * keyLength + 4 := by
unfold steps
omega
private theorem joinTrace {X : Type*} {f : X → X} {a b c : X} {n m : Nat}
(first : f^[n] a = b) (second : f^[m] b = c) : f^[n + m] a = c := by
rw [Nat.add_comm, Function.iterate_add_apply, first, second]
theorem compareTrace (tape : Fin 5 → K) (distinct : Function.Injective tape)
(labels : Label → Λ) (equalExit differentExit malformedExit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
(atLabels : ∀ l, program (labels l) =
instruction tape labels equalExit differentExit malformedExit l)
(base : K → List Bool) (bits suffix key : List Bool)
(canonical : BinaryNameMachine.canonical bits = true)
(sourceWord : base (tape 0) = BinaryNameMachine.frame bits ++ suffix)
(keyWord : base (tape 1) = key)
(candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
(scratchEmpty : base (tape 4) = []) (ambient : A) :
(advance (TM2.step program))^[steps bits.length key.length]
(some ⟨some (labels .scan), clean ambient, base⟩) =
some ⟨if bits.reverse = key then equalExit else differentExit,
clean ambient, Function.update base (tape 0) suffix⟩ := by
have hd (i j : Fin 5) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
let finalBase := Function.update base (tape 0) suffix
let afterScan := Function.update finalBase (tape 2) bits.reverse
let afterCopy := Function.update afterScan (tape 3) key
have scanRun : (advance (TM2.step program))^[bits.length + 1]
(some ⟨some (labels .scan), clean ambient, base⟩) =
some ⟨some (labels .copyOut), clean ambient, afterScan⟩ := by
let back := MachineStateEquiv.program (scanStateEquiv A).symm program
have atScan : back (labels .scan) =
BinaryNameMachine.scan (tape 0) (tape 2) (labels .scan)
(some (labels .copyOut)) malformedExit := by
change MachineStateEquiv.statement (scanStateEquiv A).symm
(program (labels .scan)) = _
rw [atLabels .scan]
exact MachineStateEquiv.statement_symm_statement (scanStateEquiv A) _
have backForward : MachineStateEquiv.program (scanStateEquiv A) back = program := by
funext l
change MachineStateEquiv.statement (scanStateEquiv A)
(MachineStateEquiv.statement (scanStateEquiv A).symm (program l)) = _
simpa only [Equiv.symm_symm] using
MachineStateEquiv.statement_symm_statement (scanStateEquiv A).symm (program l)
have initialTapes : UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt (tape 0) (tape 2)
base (BinaryNameMachine.frame bits ++ suffix) [] = base := by
rw [← sourceWord, ← candidateEmpty]
exact UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt_self _ _ _
have native := BinaryNameMachine.framedTrace (tape 0) (tape 2) (hd 0 2 (by decide))
(labels .scan) (some (labels .copyOut)) malformedExit back atScan base (ambient, false)
bits suffix [] none
simp only [canonical, ↓reduceIte, initialTapes, List.append_nil] at native
have transported := MachineStateEquiv.trace (scanStateEquiv A) back _ _ _ native
rw [backForward] at transported
simpa only [MachineStateEquiv.configuration, scanStateEquiv, Equiv.coe_fn_mk, BinaryNameMachine.clean,
clean, afterScan, finalBase, UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt] using transported
have afterScanKey : afterScan (tape 1) = key := by
simpa [afterScan, finalBase, hd 1 2 (by decide), hd 1 0 (by decide)] using keyWord
have afterScanCopy : afterScan (tape 3) = [] := by
simpa [afterScan, finalBase, hd 3 2 (by decide), hd 3 0 (by decide)] using copyEmpty
have afterScanScratch : afterScan (tape 4) = [] := by
simpa [afterScan, finalBase, hd 4 2 (by decide), hd 4 0 (by decide)] using scratchEmpty
have copyRun : (advance (TM2.step program))^[2 * (key.length + 1)]
(some ⟨some (labels .copyOut), clean ambient, afterScan⟩) =
some ⟨some (labels .compare), clean ambient, afterCopy⟩ := by
have h := MachineCopy.copyTrace (tape 1) (tape 3) (tape 4)
(hd 1 3 (by decide)) (hd 1 4 (by decide)) (hd 3 4 (by decide)) false
(labels .copyOut) (labels .copyBack) (some (labels .compare)) program
(atLabels .copyOut) (atLabels .copyBack) afterScan afterScanScratch
(ambient, false, none) none
simpa only [afterScanKey, afterScanCopy, List.append_nil, clean, afterCopy] using h
have candidateWord : afterCopy (tape 2) = bits.reverse := by
simp [afterCopy, afterScan, hd 2 3 (by decide)]
have copiedWord : afterCopy (tape 3) = key := by simp [afterCopy]
have restored : UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt (tape 2) (tape 3)
afterCopy [] [] = finalBase := by
funext k
by_cases hc : k = tape 2
· subst k
simp [UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt, afterCopy, afterScan, finalBase,
hd 2 3 (by decide), hd 2 0 (by decide), candidateEmpty]
· by_cases hk : k = tape 3
· subst k
simp [UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt, afterCopy, afterScan, finalBase,
hd 3 0 (by decide), copyEmpty]
· simp [UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt, afterCopy, afterScan, hc, hk]
have comparisonRun : (advance (TM2.step program))^[max bits.length key.length + 1]
(some ⟨some (labels .compare), clean ambient, afterCopy⟩) =
some ⟨if bits.reverse = key then equalExit else differentExit,
clean ambient, finalBase⟩ := by
let back := MachineStateEquiv.program (compareStateEquiv A).symm program
have atCompare : back (labels .compare) =
MachineCompare.loop (tape 2) (tape 3) (labels .compare) equalExit differentExit := by
change MachineStateEquiv.statement (compareStateEquiv A).symm
(program (labels .compare)) = _
rw [atLabels .compare]
exact MachineStateEquiv.statement_symm_statement (compareStateEquiv A) _
have backForward : MachineStateEquiv.program (compareStateEquiv A) back = program := by
funext l
change MachineStateEquiv.statement (compareStateEquiv A)
(MachineStateEquiv.statement (compareStateEquiv A).symm (program l)) = _
simpa only [Equiv.symm_symm] using
MachineStateEquiv.statement_symm_statement (compareStateEquiv A).symm (program l)
have native := MachineCompare.compareTrace_fromTapes (tape 2) (tape 3)
(hd 2 3 (by decide)) (labels .compare) equalExit differentExit back atCompare afterCopy
ambient none none
rw [candidateWord, copiedWord, restored, List.length_reverse] at native
have transported := MachineStateEquiv.trace (compareStateEquiv A) back _ _ _ native
rw [backForward] at transported
simpa only [MachineStateEquiv.configuration, compareStateEquiv, Equiv.coe_fn_mk, clean] using transported
exact joinTrace (joinTrace scanRun copyRun) comparisonRun
def compareInTime (tape : Fin 5 → K) (distinct : Function.Injective tape)
(labels : Label → Λ) (equalExit differentExit malformedExit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
(atLabels : ∀ l, program (labels l) =
instruction tape labels equalExit differentExit malformedExit l)
(base : K → List Bool) (bits suffix key : List Bool)
(canonical : BinaryNameMachine.canonical bits = true)
(sourceWord : base (tape 0) = BinaryNameMachine.frame bits ++ suffix)
(keyWord : base (tape 1) = key)
(candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
(scratchEmpty : base (tape 4) = []) (ambient : A) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some (labels .scan), clean ambient, base⟩
(some ⟨if bits.reverse = key then equalExit else differentExit,
clean ambient, Function.update base (tape 0) suffix⟩)
(2 * bits.length + 3 * key.length + 4) where
steps := steps bits.length key.length
evals_in_steps := compareTrace tape distinct labels equalExit differentExit malformedExit
program atLabels base bits suffix key canonical sourceWord keyWord
candidateEmpty copyEmpty scratchEmpty ambient
steps_le_m := steps_le bits.length key.length
def machine : FinTM2 where
K := Fin 5
k₀ := 0
k₁ := 1
Γ _ := Bool
Λ := Label ⊕ Fin 3
main := .inl .scan
σ := State Unit
initialState := clean ()
m label := match label with
| .inl l => instruction id Sum.inl (some (.inr 0)) (some (.inr 1)) (some (.inr 2)) l
| .inr _ => .halt
theorem machine_finiteAlphabet (k : machine.K) : Finite (machine.Γ k) := by
change Finite Bool
infer_instance
end PerfectCompleteness.BinaryNameCompare
end OAI
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