File size: 6,573 Bytes
2f887db | 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 | import OAI.Computability.PerfectCompleteness.Machines.CanonicalKeyMachine
namespace OAI
namespace PerfectCompleteness.DescriptorKeyMachine
open Turing UniqueGamesTheorem.Foundations.Complexity MachineComposition
open CanonicalKeys CanonicalKeyShape ClauseSupport
open scoped Classical
noncomputable section
variable {n : Nat} {α K Λ σ : Type} [DecidableEq K]
inductive Label (α : Type) (n : Nat)
| dispatch
| key (descriptor : α) (localLabel : CanonicalKeyMachine.Label n)
deriving DecidableEq, Fintype
abbrev Alphabet (_ : K) := Bool
abbrev State (σ : Type) := KeyMetadataMachine.State σ
def start (n : Nat) : CanonicalKeyMachine.Label n := .inl ()
@[simp] theorem entry_eq (placed : CanonicalKeyMachine.Label n → Λ) (exit : Option Λ) :
CanonicalKeyMachine.entry placed exit = some (placed (start n)) := rfl
def instruction (select : σ → α) (shapes : α → Fin n → Shape)
(modes : α → Fin n → SupportMode) (side : α → Side)
(partition : α → Set (Set (Fin n → ReducedValue)))
(tape : Fin n → Fin 6 → K) (output : K)
(labels : Label α n → Λ) (exit : Option Λ) :
Label α n → TM2.Stmt (Alphabet (K := K)) Λ (State σ)
| .dispatch => .goto (fun state => labels (.key (select state.1) (start n)))
| .key descriptor localLabel =>
CanonicalKeyMachine.instruction (shapes descriptor) (modes descriptor)
(side descriptor) (partition descriptor) tape output
(fun l => labels (.key descriptor l)) exit localLabel
def payload (select : σ → α) (shapes : α → Fin n → Shape)
(modes : α → Fin n → SupportMode) (side : α → Side)
(partition : α → Set (Set (Fin n → ReducedValue)))
(ambient : σ) (occurrences : Fin n → Nat) (variableIDs : Fin n → Fin 3 → Nat) : List Bool :=
CanonicalKeyMachine.payload (shapes (select ambient)) (modes (select ambient))
(side (select ambient)) (partition (select ambient)) occurrences variableIDs
theorem dispatch_step (select : σ → α) (shapes : α → Fin n → Shape)
(modes : α → Fin n → SupportMode) (side : α → Side)
(partition : α → Set (Set (Fin n → ReducedValue)))
(tape : Fin n → Fin 6 → K) (output : K)
(labels : Label α n → Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
(atDispatch : program (labels .dispatch) =
instruction select shapes modes side partition tape output labels exit .dispatch)
(base : K → List Bool) (ambient : σ) :
TM2.step program ⟨some (labels .dispatch), KeyMetadataMachine.clean ambient, base⟩ =
some ⟨CanonicalKeyMachine.entry (fun l => labels (.key (select ambient) l)) exit,
KeyMetadataMachine.clean ambient, base⟩ := by
change some (TM2.stepAux (program (labels .dispatch))
(KeyMetadataMachine.clean ambient) base) = _
rw [atDispatch]
rfl
private def prefixOne {X : Type} (step : X → Option X)
{initial middle : X} {final : Option X} {budget : Nat}
(first : step initial = some middle)
(body : StateTransition.EvalsToInTime step middle final budget) :
StateTransition.EvalsToInTime step initial final (1 + budget) := by
refine
{ steps := 1 + body.steps
evals_in_steps := ?_
steps_le_m := Nat.add_le_add_left body.steps_le_m 1 }
change (advance step)^[1 + body.steps] (some initial) = final
rw [Nat.add_comm 1 body.steps, Function.iterate_add_apply,
Function.iterate_one, advance_some, first]
exact body.evals_in_steps
def keyInTime (select : σ → α) (shapes : α → Fin n → Shape)
(modes : α → Fin n → SupportMode) (side : α → Side)
(partition : α → Set (Set (Fin n → ReducedValue)))
(occurrences : Fin n → Nat) (variableIDs : Fin n → Fin 3 → Nat)
(tape : Fin n → Fin 6 → K) (distinct : ∀ i, Function.Injective (tape i))
(scratch output : K) (scratchOutput : scratch ≠ output)
(sharedScratch : ∀ i, tape i 4 = scratch) (sharedOutput : ∀ i, tape i 5 = output)
(labels : Label α n → Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
(atLabels : ∀ label, program (labels label) =
instruction select shapes modes side partition tape output labels exit label)
(base : K → List Bool)
(sourceWords : ∀ i field, base (KeyMetadataMachine.idTape (tape i) field) =
encodeWord (KeyMetadataMachine.fieldValue (occurrences i) (variableIDs i) field))
(scratchEmpty : base scratch = []) (ambient : σ) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some (labels .dispatch), KeyMetadataMachine.clean ambient, base⟩
(some ⟨exit, KeyMetadataMachine.clean ambient, Function.update base output
(payload select shapes modes side partition ambient occurrences variableIDs ++ base output)⟩)
(1 + (2 * CanonicalKeyMachine.sourceLength tape base + 4 * n + 2)) := by
let body := CanonicalKeyMachine.keyInTime (shapes (select ambient)) (modes (select ambient))
(side (select ambient)) (partition (select ambient)) occurrences variableIDs
tape distinct scratch output scratchOutput sharedScratch sharedOutput
(fun l => labels (.key (select ambient) l)) exit program
(fun l => atLabels (.key (select ambient) l)) base sourceWords scratchEmpty ambient
exact prefixOne (TM2.step program)
(dispatch_step select shapes modes side partition tape output labels exit program
(atLabels .dispatch) base ambient) body
theorem output_sources (tape : Fin n → Fin 6 → K)
(distinct : ∀ i, Function.Injective (tape i)) (output : K)
(sharedOutput : ∀ i, tape i 5 = output) (base : K → List Bool) (bits : List Bool)
(i : Fin n) (field : Fin 4) :
Function.update base output (bits ++ base output) (KeyMetadataMachine.idTape (tape i) field) =
base (KeyMetadataMachine.idTape (tape i) field) :=
CanonicalKeyMachine.output_sources tape distinct output sharedOutput base bits i field
theorem output_scratch (scratch output : K) (hne : scratch ≠ output)
(base : K → List Bool) (bits : List Bool) (hempty : base scratch = []) :
Function.update base output (bits ++ base output) scratch = [] :=
CanonicalKeyMachine.output_scratch scratch output hne base bits hempty
theorem label_finite [Fintype α] : Finite (Label α n) := inferInstance
theorem state_finite [Finite σ] : Finite (State σ) := inferInstance
omit [DecidableEq K] in
theorem alphabet_finite (k : K) : Finite (Alphabet k) := inferInstance
end
end PerfectCompleteness.DescriptorKeyMachine
end OAI
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