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import Mathlib.Tactic.NormNum
import OAI.Computability.UniqueGames.Machines.MachineBinaryNameSearch
import OAI.Computability.UniqueGames.Machines.MachineDrain
import OAI.Computability.UniqueGames.Machines.MachineSubroutineLemmas
namespace OAI
namespace UniqueGamesTheorem.BinaryLiteralMachine
open Turing
open UniqueGamesTheorem.Foundations.Complexity
open MachineComposition
open UniqueGamesTheorem.Reduction.MachineTransfer
variable {K Λ A : Type} [DecidableEq K]
abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := BinaryNameCompare.State (A × Bool)
abbrev clean (ambient : A) (sign : Bool := false) : State A :=
BinaryNameCompare.clean (ambient, sign)
inductive Label
| readSign | scanKey | copyOut | copyBack
| search (label : BinaryNameSearch.Label)
| drainSearch | drainKey | emitIndex | emitSign
deriving DecidableEq, Fintype
/-- Cursor, permanent stream, key, search stream, candidate, key-copy,
copy scratch, index counter, reversed output. -/
def searchTapes (tape : Fin 9 → K) : Fin 6 → K
| 0 => tape 3
| 1 => tape 2
| 2 => tape 4
| 3 => tape 5
| 4 => tape 6
| 5 => tape 7
omit [DecidableEq K] in
theorem searchTapes_injective (tape : Fin 9 → K) (distinct : Function.Injective tape) :
Function.Injective (searchTapes tape) := by
intro i j h
fin_cases i <;> fin_cases j <;> simp only [searchTapes] at h
all_goals first | rfl | have heq := congrArg Fin.val (distinct h); norm_num at heq
def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
.load (fun state => clean state.1.1.1)
(match exit with
| none => .halt
| some label => .goto fun _ => label)
def readSign (tape : Fin 9 → K) (next : Λ) (malformed : Option Λ) :
TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
.push (tape 7) (fun _ => false)
(.pop (tape 0) (fun state head => (state.1, head))
(.branch (fun state => state.2.isSome)
(.load (fun state => clean state.1.1.1 (state.2.getD false)) (.goto fun _ => next))
(finish malformed)))
def emitSign (output : K) (exit : Option Λ) :
TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
.branch (fun state => state.1.1.2)
(.push output (fun _ => true) (.push output (fun _ => false) (finish exit)))
(.push output (fun _ => false) (finish exit))
def instruction (tape : Fin 9 → K) (labels : Label → Λ)
(exit malformed : Option Λ) : Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
| .readSign => readSign tape (labels .scanKey) malformed
| .scanKey => MachineStateEquiv.statement (BinaryNameCompare.scanStateEquiv (A × Bool))
(BinaryNameMachine.scan (tape 0) (tape 2) (labels .scanKey)
(some (labels .copyOut)) malformed)
| .copyOut => loopAt (tape 1) (tape 6) id false
(labels .copyOut) (some (labels .copyBack))
| .copyBack => MachineCopy.forkLoop (tape 6) (tape 1) (tape 3) false
(labels .copyBack) (some (labels (.search .sign)))
| .search l => BinaryNameSearch.instruction (searchTapes tape)
(fun l => labels (.search l)) (some (labels .drainSearch)) malformed malformed l
| .drainSearch => MachineDrain.drain (tape 3) (labels .drainSearch)
(some (labels .drainKey))
| .drainKey => MachineDrain.drain (tape 2) (labels .drainKey)
(some (labels .emitIndex))
| .emitIndex => loopAt (tape 7) (tape 8) id false
(labels .emitIndex) (some (labels .emitSign))
| .emitSign => emitSign (tape 8) exit
def signWord (sign : Bool) : List Bool := encodeWord (if sign then 1 else 0)
def index (tokens : List BinaryNameSearch.Token) (bits : List Bool) : Nat :=
(BinaryNameSearch.payloads tokens).idxOf bits
def outputWord (tokens : List BinaryNameSearch.Token) (bits : List Bool) (sign : Bool) :
List Bool := encodeWord (index tokens bits) ++ signWord sign
def steps (tokens : List BinaryNameSearch.Token) (bits : List Bool) : Nat :=
1 + (bits.length + 1) + 2 * ((BinaryNameSearch.stream tokens).length + 1) +
BinaryNameSearch.steps tokens bits +
((BinaryNameSearch.stream (BinaryNameSearch.afterMatch tokens bits)).length + 1) +
(bits.length + 1) + (index tokens bits + 2) + 1
theorem afterMatch_length_le (tokens : List BinaryNameSearch.Token) (bits : List Bool) :
(BinaryNameSearch.stream (BinaryNameSearch.afterMatch tokens bits)).length ≤
(BinaryNameSearch.stream tokens).length := by
induction tokens with
| nil => simp [BinaryNameSearch.afterMatch]
| cons token tokens ih =>
simp only [BinaryNameSearch.afterMatch, BinaryNameSearch.stream_cons, List.length_append]
split <;> omega
theorem steps_le (tokens : List BinaryNameSearch.Token) (bits : List Bool)
(present : bits ∈ BinaryNameSearch.payloads tokens) :
steps tokens bits ≤
3 * ((BinaryNameSearch.stream tokens).length + bits.length) ^ 2 +
13 * ((BinaryNameSearch.stream tokens).length + bits.length) + 9 := by
have search := BinaryNameSearch.steps_le_stream tokens bits
have suffix := afterMatch_length_le tokens bits
have idx : index tokens bits < tokens.length := by
have h := List.idxOf_lt_length_iff.mpr present
simpa [index, BinaryNameSearch.payloads] using h
have count : tokens.length ≤ (BinaryNameSearch.stream tokens).length := by
rw [BinaryNameSearch.stream_length]
omega
unfold steps
nlinarith [Nat.zero_le ((BinaryNameSearch.stream tokens).length * bits.length)]
@[simp] theorem stepAux_finish (exit : Option Λ) (state : State A) (base : K → List Bool) :
TM2.stepAux (finish exit) state base = ⟨exit, clean state.1.1.1, base⟩ := by
cases exit <;> rfl
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 literalTrace (tape : Fin 9 → K) (distinct : Function.Injective tape)
(labels : Label → Λ) (exit malformed : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
(atLabels : ∀ l, program (labels l) = instruction tape labels exit malformed l)
(base : K → List Bool) (tokens : List BinaryNameSearch.Token)
(sign : Bool) (bits suffix : List Bool)
(canonical : BinaryNameMachine.canonical bits = true)
(tokensCanonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true)
(present : bits ∈ BinaryNameSearch.payloads tokens)
(cursor : base (tape 0) = sign :: (BinaryNameMachine.frame bits ++ suffix))
(permanent : base (tape 1) = BinaryNameSearch.stream tokens)
(empty : ∀ i : Fin 9, 2 ≤ i.val → i.val ≤ 7 → base (tape i) = [])
(ambient : A) :
(advance (TM2.step program))^[steps tokens bits]
(some ⟨some (labels .readSign), clean ambient, base⟩) =
some ⟨exit, clean ambient,
Function.update (Function.update base (tape 0) suffix) (tape 8)
((outputWord tokens bits sign).reverse ++ base (tape 8))⟩ := by
have hd (i j : Fin 9) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
have he (i : Fin 9) (hlo : 2 ≤ i.val := by decide) (hhi : i.val ≤ 7 := by decide) :
base (tape i) = [] := empty i hlo hhi
let s₁ := Function.update (Function.update base (tape 7) [false]) (tape 0)
(BinaryNameMachine.frame bits ++ suffix)
let s₂ := Function.update (Function.update s₁ (tape 0) suffix) (tape 2) bits.reverse
let s₃ := Function.update s₂ (tape 3) (BinaryNameSearch.stream tokens)
let tail := BinaryNameSearch.stream (BinaryNameSearch.afterMatch tokens bits)
let s₄ := tapesAt (tape 3) (tape 7) s₃ tail (encodeWord (index tokens bits))
let s₅ := Function.update s₄ (tape 3) []
let s₆ := Function.update s₅ (tape 2) []
let s₇ := tapesAt (tape 7) (tape 8) s₆ []
((encodeWord (index tokens bits)).reverse ++ s₆ (tape 8))
let s₈ := Function.update s₇ (tape 8) ((signWord sign).reverse ++ s₇ (tape 8))
have signRun : (advance (TM2.step program))^[1]
(some ⟨some (labels .readSign), clean ambient, base⟩) =
some ⟨some (labels .scanKey), clean ambient sign, s₁⟩ := by
change some (TM2.stepAux (program (labels .readSign)) _ _) = _
rw [atLabels .readSign]
simp [instruction, readSign, TM2.stepAux, clean, BinaryNameCompare.clean,
cursor, he 7, hd 0 7 (by decide), s₁]
have s₁cursor : s₁ (tape 0) = BinaryNameMachine.frame bits ++ suffix := by simp [s₁]
have s₁key : s₁ (tape 2) = [] := by
simp [s₁, hd 2 0 (by decide), hd 2 7 (by decide), he 2]
have keyRun : (advance (TM2.step program))^[bits.length + 1]
(some ⟨some (labels .scanKey), clean ambient sign, s₁⟩) =
some ⟨some (labels .copyOut), clean ambient sign, s₂⟩ := by
let e := BinaryNameCompare.scanStateEquiv (A × Bool)
let back := MachineStateEquiv.program e.symm program
have atScan : back (labels .scanKey) =
BinaryNameMachine.scan (tape 0) (tape 2) (labels .scanKey)
(some (labels .copyOut)) malformed := by
change MachineStateEquiv.statement e.symm (program (labels .scanKey)) = _
rw [atLabels .scanKey]
exact MachineStateEquiv.statement_symm_statement e _
have backForward : MachineStateEquiv.program e back = program := by
funext l
change MachineStateEquiv.statement e (MachineStateEquiv.statement e.symm (program l)) = _
exact MachineStateEquiv.statement_symm_statement e.symm _
have initial : tapesAt (tape 0) (tape 2) s₁
(BinaryNameMachine.frame bits ++ suffix) [] = s₁ := by
rw [← s₁cursor, ← s₁key]
exact tapesAt_self _ _ _
have native := BinaryNameMachine.framedTrace (tape 0) (tape 2) (hd 0 2 (by decide))
(labels .scanKey) (some (labels .copyOut)) malformed back atScan s₁
((ambient, sign), false) bits suffix [] none
simp only [canonical, ↓reduceIte, initial, List.append_nil] at native
have transported := MachineStateEquiv.trace e back _ _ _ native
rw [backForward] at transported
simpa only [MachineStateEquiv.configuration, e, BinaryNameCompare.scanStateEquiv, Equiv.coe_fn_mk,
BinaryNameMachine.clean, clean, BinaryNameCompare.clean, s₂, tapesAt] using transported
have s₂permanent : s₂ (tape 1) = BinaryNameSearch.stream tokens := by
simpa [s₂, s₁, hd 1 0 (by decide), hd 1 2 (by decide), hd 1 7 (by decide)] using permanent
have s₂empty (i : Fin 9) (hi : i = 3 ∨ i = 4 ∨ i = 5 ∨ i = 6) : s₂ (tape i) = [] := by
rcases hi with rfl | rfl | rfl | rfl <;>
simp [s₂, s₁, hd, he]
have copyRun : (advance (TM2.step program))^[2 * ((BinaryNameSearch.stream tokens).length + 1)]
(some ⟨some (labels .copyOut), clean ambient sign, s₂⟩) =
some ⟨some (labels (.search .sign)), clean ambient sign, s₃⟩ := by
have h := MachineCopy.copyTrace (tape 1) (tape 3) (tape 6)
(hd 1 3 (by decide)) (hd 1 6 (by decide)) (hd 3 6 (by decide)) false
(labels .copyOut) (labels .copyBack) (some (labels (.search .sign))) program
(atLabels .copyOut) (atLabels .copyBack) s₂
(s₂empty 6 (by simp)) ((ambient, sign), false, none) none
simpa only [s₂permanent, s₂empty 3 (by simp), List.append_nil,
clean, BinaryNameCompare.clean, s₃] using h
have s₃key : s₃ (tape 2) = bits.reverse := by
simp [s₃, s₂, hd 2 3 (by decide)]
have s₃counter : s₃ (tape 7) = [false] := by
simp [s₃, s₂, s₁, hd 7 3 (by decide), hd 7 2 (by decide), hd 7 0 (by decide)]
have s₃stream : s₃ (tape 3) = BinaryNameSearch.stream tokens := by simp [s₃]
have searchRun : (advance (TM2.step program))^[BinaryNameSearch.steps tokens bits]
(some ⟨some (labels (.search .sign)), clean ambient sign, s₃⟩) =
some ⟨some (labels .drainSearch), clean ambient sign, s₄⟩ := by
have h := BinaryNameSearch.searchTrace (searchTapes tape)
(searchTapes_injective tape distinct) (fun l => labels (.search l))
(some (labels .drainSearch)) malformed malformed program
(fun l => atLabels (.search l)) s₃ (ambient, sign) tokens bits [] [false]
tokensCanonical present s₃key
(by simpa [s₃, searchTapes, hd 4 3 (by decide)] using s₂empty 4 (by simp))
(by simpa [s₃, searchTapes, hd 5 3 (by decide)] using s₂empty 5 (by simp))
(by simpa [s₃, searchTapes, hd 6 3 (by decide)] using s₂empty 6 (by simp))
have initial : tapesAt (tape 3) (tape 7) s₃ (BinaryNameSearch.stream tokens) [false] = s₃ := by
rw [← s₃stream, ← s₃counter]
exact tapesAt_self _ _ _
simpa only [searchTapes, List.append_nil, initial, clean, BinaryNameSearch.clean,
s₄, tail, encodeWord, index] using h
have s₄stream : s₄ (tape 3) = tail := by simp [s₄, hd 3 7 (by decide)]
have drainSearchRun : (advance (TM2.step program))^[tail.length + 1]
(some ⟨some (labels .drainSearch), clean ambient sign, s₄⟩) =
some ⟨some (labels .drainKey), clean ambient sign, s₅⟩ := by
have h := MachineDrain.drainTrace (tape 3) (labels .drainSearch)
(some (labels .drainKey)) program (atLabels .drainSearch) s₄ tail
((ambient, sign), false, none) none
have initial : Function.update s₄ (tape 3) tail = s₄ := by
rw [← s₄stream]; exact Function.update_eq_self _ _
simpa only [initial, clean, BinaryNameCompare.clean, s₅] using h
have s₅key : s₅ (tape 2) = bits.reverse := by
simp [s₅, s₄, tapesAt, hd 2 3 (by decide), hd 2 7 (by decide), s₃key]
have drainKeyRun : (advance (TM2.step program))^[bits.length + 1]
(some ⟨some (labels .drainKey), clean ambient sign, s₅⟩) =
some ⟨some (labels .emitIndex), clean ambient sign, s₆⟩ := by
have h := MachineDrain.drainTrace (tape 2) (labels .drainKey)
(some (labels .emitIndex)) program (atLabels .drainKey) s₅ bits.reverse
((ambient, sign), false, none) none
have initial : Function.update s₅ (tape 2) bits.reverse = s₅ := by
rw [← s₅key]; exact Function.update_eq_self _ _
simpa only [initial, List.length_reverse, clean, BinaryNameCompare.clean, s₆] using h
have s₆index : s₆ (tape 7) = encodeWord (index tokens bits) := by
simp [s₆, s₅, s₄, hd 7 2 (by decide), hd 7 3 (by decide)]
have indexRun : (advance (TM2.step program))^[index tokens bits + 2]
(some ⟨some (labels .emitIndex), clean ambient sign, s₆⟩) =
some ⟨some (labels .emitSign), clean ambient sign, s₇⟩ := by
change (nextAt (Γ := Alphabet) (tape 8) program)^[index tokens bits + 2]
(some ⟨some (labels .emitIndex), (((ambient, sign), false, none), none), s₆⟩) =
some ⟨some (labels .emitSign), (((ambient, sign), false, none), none), s₇⟩
have h := transferAt_fromTapes (Γ := Alphabet) (tape 7) (tape 8)
(hd 7 8 (by decide)) id false (labels .emitIndex) (some (labels .emitSign)) program
(atLabels .emitIndex) s₆ ((ambient, sign), false, none) none
have time : index tokens bits + 1 + 1 = index tokens bits + 2 := by omega
simpa only [s₆index, encodeWord_length, List.map_id_fun, id_eq, time, s₇] using h
have emitRun : (advance (TM2.step program))^[1]
(some ⟨some (labels .emitSign), clean ambient sign, s₇⟩) =
some ⟨exit, clean ambient, s₈⟩ := by
change some (TM2.stepAux (program (labels .emitSign)) _ _) = _
rw [atLabels .emitSign]
cases sign <;> simp [instruction, emitSign, clean, BinaryNameCompare.clean,
TM2.stepAux, signWord, encodeWord, s₈, Function.update_idem]
have full := joinTrace (joinTrace (joinTrace (joinTrace (joinTrace (joinTrace
(joinTrace signRun keyRun) copyRun) searchRun) drainSearchRun) drainKeyRun) indexRun) emitRun
have finalTapes : s₈ = Function.update (Function.update base (tape 0) suffix) (tape 8)
((outputWord tokens bits sign).reverse ++ base (tape 8)) := by
funext k
by_cases h8 : k = tape 8
· subst k
simp [s₈, s₇, s₆, s₅, s₄, s₃, s₂, s₁, tapesAt, outputWord,
List.reverse_append, List.append_assoc, hd]
· by_cases h0 : k = tape 0
· subst k
simp [s₈, s₇, s₆, s₅, s₄, s₃, s₂, s₁, tapesAt, hd]
· by_cases h2 : k = tape 2
· subst k
simp [s₈, s₇, s₆, s₅, s₄, s₃, s₂, s₁, tapesAt, hd, he 2]
· by_cases h3 : k = tape 3
· subst k
simp [s₈, s₇, s₆, s₅, s₄, s₃, s₂, s₁, tapesAt, hd, he 3]
· by_cases h7 : k = tape 7
· subst k
simp [s₈, s₇, s₆, s₅, s₄, s₃, s₂, s₁, tapesAt, hd, he 7]
· simp [s₈, s₇, s₆, s₅, s₄, s₃, s₂, s₁, tapesAt, h8, h0, h2, h3, h7]
simpa only [steps, tail, finalTapes] using full
/-- Polynomial resource bound attached to the proved literal execution. -/
def literalInTime (tape : Fin 9 → K) (distinct : Function.Injective tape)
(labels : Label → Λ) (exit malformed : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
(atLabels : ∀ l, program (labels l) = instruction tape labels exit malformed l)
(base : K → List Bool) (tokens : List BinaryNameSearch.Token)
(sign : Bool) (bits suffix : List Bool)
(canonical : BinaryNameMachine.canonical bits = true)
(tokensCanonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true)
(present : bits ∈ BinaryNameSearch.payloads tokens)
(cursor : base (tape 0) = sign :: (BinaryNameMachine.frame bits ++ suffix))
(permanent : base (tape 1) = BinaryNameSearch.stream tokens)
(empty : ∀ i : Fin 9, 2 ≤ i.val → i.val ≤ 7 → base (tape i) = [])
(ambient : A) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some (labels .readSign), clean ambient, base⟩
(some ⟨exit, clean ambient,
Function.update (Function.update base (tape 0) suffix) (tape 8)
((outputWord tokens bits sign).reverse ++ base (tape 8))⟩)
(3 * ((BinaryNameSearch.stream tokens).length + bits.length) ^ 2 +
13 * ((BinaryNameSearch.stream tokens).length + bits.length) + 9) where
steps := steps tokens bits
evals_in_steps := literalTrace tape distinct labels exit malformed program atLabels base
tokens sign bits suffix canonical tokensCanonical present cursor permanent empty ambient
steps_le_m := steps_le tokens bits present
end UniqueGamesTheorem.BinaryLiteralMachine
end OAI
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