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import OAI.Computability.MaxCut.PCP.LocalEquation
import OAI.Computability.MaxCut.Machines.MachineRegisterEmit
namespace OAI
/-! Emit one fixed descriptor equation from three preserved unary base registers.
The output order is first address, second address, third address, RHS. The actual
stack program emits those fields in reverse order. -/
namespace MaxCutGames.Foundations.Hastad.SourceEquationEmit
open Turing Complexity
open SourceAddressDescriptors
open MaxCutGames.Reduction.CloneGap
open MaxCutGames.Reduction.MachineSubstitution
inductive Label
| seed | thirdDrain | thirdFork | thirdEmit
| secondDrain | secondFork | secondEmit | firstDrain | firstFork | firstEmit
deriving DecidableEq
protected abbrev Label.enumList : List Label := [.seed, .thirdDrain, .thirdFork, .thirdEmit,
.secondDrain, .secondFork, .secondEmit, .firstDrain, .firstFork, .firstEmit]
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
abbrev Alphabet {K : Type} (_ : K) := Bool
abbrev State (σ : Type) := σ × Option Bool
def rhsBit (e : Equation Descriptor) : Nat := if e.rhs then 1 else 0
def steps (values : Fin 3 → Nat) (e : Equation Descriptor) : Nat :=
2 * (values e.first.1 + values e.second.1 + values e.third.1) + 16
variable {K Λ σ : Type} [DecidableEq K]
def statement (sources : Fin 3 → K) (destination scratch : K)
(e : Equation Descriptor) (labels : Label → Λ) (exit : Option Λ) :
Label → TM2.Stmt (Alphabet (K := K)) Λ (State σ)
| .seed => pushWord destination (encodeWord (rhsBit e)).reverse
(.goto fun _ => labels .thirdDrain)
| .thirdDrain => Reduction.MachineTransfer.loopAt (sources e.third.1) scratch id false
(labels .thirdDrain) (some (labels .thirdFork))
| .thirdFork => MachineCopy.forkLoop scratch (sources e.third.1) destination false
(labels .thirdFork) (some (labels .thirdEmit))
| .thirdEmit => MachineRegisterEmit.emitStatement destination e.third.2
(some (labels .secondDrain))
| .secondDrain => Reduction.MachineTransfer.loopAt (sources e.second.1) scratch id false
(labels .secondDrain) (some (labels .secondFork))
| .secondFork => MachineCopy.forkLoop scratch (sources e.second.1) destination false
(labels .secondFork) (some (labels .secondEmit))
| .secondEmit => MachineRegisterEmit.emitStatement destination e.second.2
(some (labels .firstDrain))
| .firstDrain => Reduction.MachineTransfer.loopAt (sources e.first.1) scratch id false
(labels .firstDrain) (some (labels .firstFork))
| .firstFork => MachineCopy.forkLoop scratch (sources e.first.1) destination false
(labels .firstFork) (some (labels .firstEmit))
| .firstEmit => MachineRegisterEmit.emitStatement destination e.first.2 exit
def program (sources : Fin 3 → K) (destination scratch : K) (e : Equation Descriptor) :
Label → TM2.Stmt (Alphabet (K := K)) Label (State σ) :=
statement sources destination scratch e id none
def prefixTapes (destination : K) (base : K → List Bool) (written : List Nat) :
K → List Bool := Function.update base destination (encodeWords written ++ base destination)
theorem prefixTapes_other (destination : K) (base : K → List Bool) (written : List Nat)
(k : K) (hk : k ≠ destination) : prefixTapes destination base written k = base k := by
simp [prefixTapes, hk]
theorem equationTrace (sources : Fin 3 → K) (destination scratch : K)
(sourceDestination : ∀ side, sources side ≠ destination)
(sourceScratch : ∀ side, sources side ≠ scratch)
(destinationScratch : destination ≠ scratch) (e : Equation Descriptor)
(labels : Label → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
(atLabels : ∀ label, p (labels label) = statement sources destination scratch e labels exit label)
(base : K → List Bool) (values : Fin 3 → Nat)
(sourceWords : ∀ side, base (sources side) = encodeWord (values side))
(scratchEmpty : base scratch = []) (ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step p))^[steps values e]
(some ⟨some (labels .seed), (ambient, register), base⟩) =
some ⟨exit, (ambient, none), prefixTapes destination base (words values e)⟩ := by
have hw (written : List Nat) (side : Fin 3) :
prefixTapes destination base written (sources side) = encodeWord (values side) := by
rw [prefixTapes_other _ _ _ _ (sourceDestination side), sourceWords side]
have hs (written : List Nat) : prefixTapes destination base written scratch = [] := by
rw [prefixTapes_other _ _ _ _ (Ne.symm destinationScratch), scratchEmpty]
have hseed : (MachineComposition.advance (TM2.step p))^[1]
(some ⟨some (labels .seed), (ambient, register), base⟩) =
some ⟨some (labels .thirdDrain), (ambient, register),
prefixTapes destination base [rhsBit e]⟩ := by
change some (TM2.stepAux (p (labels .seed)) (ambient, register) base) = _
rw [atLabels .seed]
simp only [statement, stepAux_pushWord, List.reverse_reverse, TM2.stepAux,
prefixTapes, encodeWords, List.append_nil]
have hthird : (MachineComposition.advance (TM2.step p))^[2 * values e.third.1 + 5]
(some ⟨some (labels .thirdDrain), (ambient, register),
prefixTapes destination base [rhsBit e]⟩) =
some ⟨some (labels .secondDrain), (ambient, none),
prefixTapes destination base [realize values e.third, rhsBit e]⟩ := by
have h := MachineRegisterEmit.registerEmitTrace (sources e.third.1) destination scratch
(sourceDestination _) (sourceScratch _) destinationScratch e.third.2
(labels .thirdDrain) (labels .thirdFork) (labels .thirdEmit) (some (labels .secondDrain))
p (atLabels .thirdDrain) (atLabels .thirdFork) (atLabels .thirdEmit)
(prefixTapes destination base [rhsBit e]) (values e.third.1)
(hw _ _) (hs _) ambient register
simpa only [prefixTapes, Function.update_self, Function.update_idem, realize,
encodeWords, List.append_nil, List.append_assoc] using h
have hsecond : (MachineComposition.advance (TM2.step p))^[2 * values e.second.1 + 5]
(some ⟨some (labels .secondDrain), (ambient, none),
prefixTapes destination base [realize values e.third, rhsBit e]⟩) =
some ⟨some (labels .firstDrain), (ambient, none),
prefixTapes destination base [realize values e.second, realize values e.third, rhsBit e]⟩ := by
have h := MachineRegisterEmit.registerEmitTrace (sources e.second.1) destination scratch
(sourceDestination _) (sourceScratch _) destinationScratch e.second.2
(labels .secondDrain) (labels .secondFork) (labels .secondEmit) (some (labels .firstDrain))
p (atLabels .secondDrain) (atLabels .secondFork) (atLabels .secondEmit)
(prefixTapes destination base [realize values e.third, rhsBit e]) (values e.second.1)
(hw _ _) (hs _) ambient none
simpa only [prefixTapes, Function.update_self, Function.update_idem, realize,
encodeWords, List.append_nil, List.append_assoc] using h
have hfirst : (MachineComposition.advance (TM2.step p))^[2 * values e.first.1 + 5]
(some ⟨some (labels .firstDrain), (ambient, none),
prefixTapes destination base [realize values e.second, realize values e.third, rhsBit e]⟩) =
some ⟨exit, (ambient, none), prefixTapes destination base (words values e)⟩ := by
have h := MachineRegisterEmit.registerEmitTrace (sources e.first.1) destination scratch
(sourceDestination _) (sourceScratch _) destinationScratch e.first.2
(labels .firstDrain) (labels .firstFork) (labels .firstEmit) exit
p (atLabels .firstDrain) (atLabels .firstFork) (atLabels .firstEmit)
(prefixTapes destination base [realize values e.second, realize values e.third, rhsBit e])
(values e.first.1) (hw _ _) (hs _) ambient none
simpa only [prefixTapes, Function.update_self, Function.update_idem, words, rhsBit, realize,
encodeWords, List.append_nil, List.append_assoc] using h
rw [show steps values e = (2 * values e.first.1 + 5) +
((2 * values e.second.1 + 5) + ((2 * values e.third.1 + 5) + 1)) by
unfold steps; omega]
rw [Function.iterate_add_apply _ (2 * values e.first.1 + 5),
Function.iterate_add_apply _ (2 * values e.second.1 + 5),
Function.iterate_add_apply _ (2 * values e.third.1 + 5), hseed, hthird, hsecond, hfirst]
def equationInTime (sources : Fin 3 → K) (destination scratch : K)
(sourceDestination : ∀ side, sources side ≠ destination)
(sourceScratch : ∀ side, sources side ≠ scratch)
(destinationScratch : destination ≠ scratch) (e : Equation Descriptor)
(labels : Label → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
(atLabels : ∀ label, p (labels label) = statement sources destination scratch e labels exit label)
(base : K → List Bool) (values : Fin 3 → Nat)
(sourceWords : ∀ side, base (sources side) = encodeWord (values side))
(scratchEmpty : base scratch = []) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step p)
⟨some (labels .seed), (ambient, register), base⟩
(some ⟨exit, (ambient, none), prefixTapes destination base (words values e)⟩)
(steps values e) where
steps := steps values e
evals_in_steps := equationTrace sources destination scratch sourceDestination sourceScratch
destinationScratch e labels exit p atLabels base values sourceWords scratchEmpty ambient register
steps_le_m := Nat.le_refl _
def programInTime (sources : Fin 3 → K) (destination scratch : K)
(sourceDestination : ∀ side, sources side ≠ destination)
(sourceScratch : ∀ side, sources side ≠ scratch)
(destinationScratch : destination ≠ scratch) (e : Equation Descriptor)
(base : K → List Bool) (values : Fin 3 → Nat)
(sourceWords : ∀ side, base (sources side) = encodeWord (values side))
(scratchEmpty : base scratch = []) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step (program (σ := σ) sources destination scratch e))
⟨some .seed, (ambient, register), base⟩
(some ⟨none, (ambient, none), prefixTapes destination base (words values e)⟩)
(steps values e) :=
equationInTime sources destination scratch sourceDestination sourceScratch destinationScratch
e id none (program sources destination scratch e) (fun _ => rfl)
base values sourceWords scratchEmpty ambient register
abbrev Tape := Fin 3 ⊕ Bool
def machine (e : Equation Descriptor) : FinTM2 where
K := Tape
k₀ := .inl 0
k₁ := .inr false
Γ _ := Bool
Λ := Label
main := .seed
σ := State Unit
initialState := ((), none)
m := program Sum.inl (.inr false) (.inr true) e
def machineInTime (e : Equation Descriptor) (base : Tape → List Bool) (values : Fin 3 → Nat)
(sourceWords : ∀ side, base (.inl side) = encodeWord (values side))
(scratchEmpty : base (.inr true) = []) (register : Option Bool) :
StateTransition.EvalsToInTime (machine e).step
⟨some Label.seed, ((), register), base⟩
(some ⟨none, ((), none), prefixTapes (.inr false : Tape) base (words values e)⟩)
(steps values e) :=
programInTime Sum.inl (.inr false : Tape) (.inr true)
(fun _ => Sum.inl_ne_inr) (fun _ => Sum.inl_ne_inr) (by decide)
e base values sourceWords scratchEmpty () register
end MaxCutGames.Foundations.Hastad.SourceEquationEmit
end OAI
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