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import OAI.Computability.UniqueGames.Machines.MachineFiniteTable
namespace OAI
namespace PerfectCompleteness.DescriptorProjectionMachine
open Turing
open UniqueGamesTheorem.Foundations.Complexity
open SignedCompletionSchedule
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n t q : Nat} {rows repeats : Nat → Nat}
(hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth (TreeCanonical.locationCount branch n t) ≤ q)
def projection (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
ProjectionTable q :=
LocalCompletionSerialization.projection
(FiniteConstraintMachine.result_correct hq large ⟨descriptor.1, descriptor.2.1⟩)
descriptor.2.2
theorem projection_apply
(descriptor : Descriptor (rows := rows) (repeats := repeats) hq large)
(a : Fin (2 * q)) :
(projection hq large descriptor).images[a] = descriptorTable hq large descriptor a :=
(descriptorTable_eq_parsed hq large descriptor.1 descriptor.2 a).symm
def outputWords (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
List Nat := Encoding.tableWords (projection hq large descriptor)
theorem outputWords_eq
(descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
outputWords hq large descriptor =
List.ofFn (fun a : Fin (2 * q) => (descriptorTable hq large descriptor a).val) :=
(LocalCompletionSerialization.rawTableWords_eq
(FiniteConstraintMachine.result_correct hq large ⟨descriptor.1, descriptor.2.1⟩)
descriptor.2.2).symm
@[simp] theorem outputWords_length
(descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
(outputWords hq large descriptor).length = 2 * q :=
Encoding.tableWords_length _
def outputBits (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
List Bool := encodeWords (outputWords hq large descriptor)
theorem outputBits_length_le
(descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
(outputBits hq large descriptor).length ≤ (2 * q) * (q + 1) := by
have hword : ∀ value ∈ outputWords hq large descriptor, value ≤ q := by
intro value hvalue
rw [outputWords_eq] at hvalue
obtain ⟨a, ha⟩ := List.mem_ofFn.mp hvalue
have hrange := (descriptorTable hq large descriptor a).isLt
omega
simpa only [outputBits, outputWords_length] using
encodeWords_length_le (outputWords hq large descriptor) q hword
theorem outputBits_parse
(descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) :
decodeWords (outputBits hq large descriptor) >>= Encoding.parseProjection q =
some (projection hq large descriptor) := by
rw [outputBits, decodeWords_encodeWords]
exact Encoding.parseProjection_encoded _
variable {σ : Type} [Fintype σ] [DecidableEq σ]
(select : σ → Descriptor (rows := rows) (repeats := repeats) hq large)
def keys (σ : Type) [Fintype σ] : List σ := (Finset.univ : Finset σ).toList
omit [DecidableEq σ] in
theorem keys_complete (state : σ) : state ∈ keys σ := by
simp [keys]
def pushBound : Nat :=
((keys σ).map (fun state => (outputBits hq large (select state)).length)).sum
def statement {K Λ : Type} (dst : K) (exit : Λ) :
TM2.Stmt (fun _ : K => Bool) Λ σ :=
MachineFiniteTable.emit dst (fun state => outputBits hq large (select state)) (keys σ)
(.goto (fun _ => exit))
theorem statement_trace {K Λ : Type} [DecidableEq K]
(dst : K) (exit : Λ) (state : σ) (tapes : K → List Bool) :
TM2.stepAux (statement hq large select dst exit) state tapes = {
l := some exit
var := state
stk := Function.update tapes dst (outputBits hq large (select state) ++ tapes dst)
} := by
rw [statement, MachineFiniteTable.stepAux_emit dst _ (keys σ) (keys_complete)]
rfl
def emitInTime {K Λ : Type} [DecidableEq K]
(dst : K) (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ)
(label exit : Λ)
(atLabel : program label = statement hq large select dst exit)
(state : σ) (tapes : K → List Bool) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some label, state, tapes⟩
(some ⟨some exit, state,
Function.update tapes dst (outputBits hq large (select state) ++ tapes dst)⟩) 1 :=
MachineFiniteTable.emitInTime (Γ := fun _ : K => Bool)
dst (fun state => outputBits hq large (select state))
(keys σ) keys_complete program label exit
(by simpa only [statement] using atLabel) state tapes
omit [Fintype σ] [DecidableEq σ] in
theorem output_frame {K : Type} [DecidableEq K]
(dst : K) (state : σ) (tapes : K → List Bool) (k : K) (hne : k ≠ dst) :
(Function.update tapes dst (outputBits hq large (select state) ++ tapes dst)) k =
tapes k :=
MachineFiniteTable.output_frame dst (fun state => outputBits hq large (select state))
state tapes k hne
theorem statement_pushBound {K Λ : Type} (dst : K) (exit : Λ) :
Runtime.statementPushBound (statement hq large select dst exit) ≤
pushBound hq large select := by
have h := MachineFiniteTable.emit_push_bound (Γ := fun _ : K => Bool)
dst (fun state => outputBits hq large (select state)) (keys σ)
(.goto (fun _ => exit))
simpa only [statement, pushBound, MachineFiniteTable.tableBound,
Runtime.statementPushBound, Nat.add_zero] using h
def reverseStatement {K Λ : Type} (dst : K) (exit : Λ) :
TM2.Stmt (fun _ : K => Bool) Λ σ :=
MachineFiniteTable.emit dst (fun state => (outputBits hq large (select state)).reverse)
(keys σ) (.goto (fun _ => exit))
theorem reverse_statement_trace {K Λ : Type} [DecidableEq K]
(dst : K) (exit : Λ) (state : σ) (tapes : K → List Bool) :
TM2.stepAux (reverseStatement hq large select dst exit) state tapes = {
l := some exit
var := state
stk := Function.update tapes dst
((outputBits hq large (select state)).reverse ++ tapes dst)
} := by
rw [reverseStatement, MachineFiniteTable.stepAux_emit dst _ (keys σ) keys_complete]
rfl
def reverseEmitInTime {K Λ : Type} [DecidableEq K]
(dst : K) (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ)
(label exit : Λ)
(atLabel : program label = reverseStatement hq large select dst exit)
(state : σ) (tapes : K → List Bool) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some label, state, tapes⟩
(some ⟨some exit, state,
Function.update tapes dst
((outputBits hq large (select state)).reverse ++ tapes dst)⟩) 1 :=
MachineFiniteTable.emitInTime (Γ := fun _ : K => Bool) dst
(fun state => (outputBits hq large (select state)).reverse)
(keys σ) keys_complete program label exit
(by simpa only [reverseStatement] using atLabel) state tapes
omit [Fintype σ] [DecidableEq σ] in
theorem reverse_output_frame {K : Type} [DecidableEq K]
(dst : K) (state : σ) (tapes : K → List Bool) (k : K) (hne : k ≠ dst) :
(Function.update tapes dst
((outputBits hq large (select state)).reverse ++ tapes dst)) k = tapes k :=
MachineFiniteTable.output_frame dst
(fun state => (outputBits hq large (select state)).reverse) state tapes k hne
theorem reverse_statement_pushBound {K Λ : Type} (dst : K) (exit : Λ) :
Runtime.statementPushBound (reverseStatement hq large select dst exit) ≤
pushBound hq large select := by
have h := MachineFiniteTable.emit_push_bound (Γ := fun _ : K => Bool)
dst (fun state => (outputBits hq large (select state)).reverse) (keys σ)
(.goto (fun _ => exit))
simpa only [reverseStatement, pushBound, MachineFiniteTable.tableBound,
List.length_reverse, Runtime.statementPushBound, Nat.add_zero] using h
end
end PerfectCompleteness.DescriptorProjectionMachine
end OAI
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