File size: 3,371 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 | import OAI.Computability.PerfectCompleteness.Machines.NormalizedTargetMachine
import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryOutputLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedTarget
open Turing UniqueGamesTheorem.Foundations.Complexity
open FixedParameters FixedRows FixedPreliminaryGame
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
def alphabet : Nat :=
FinitePreliminaryCompletion.alphabet (branch p) p.plan.depth
(sourceLength p.plan hδ) δ
theorem alphabet_ge_two : 2 ≤ alphabet p :=
CanonicalLocalCompletion.alphabet_ge_two
(TreeCanonical.locationCount (branch p) p.plan.depth (sourceLength p.plan hδ)) δ
theorem alphabet_pos : 0 < alphabet p :=
lt_of_lt_of_le (by decide : 0 < 2) (alphabet_ge_two p)
def construct (input : List Bool) : Instance (alphabet p) :=
NormalizedTarget.construct (branch := branch p) (n := p.plan.depth)
(t := sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ
(NormalizedSourceInput.raw input)
def computation :
TM2ComputableInPolyTime (id : List Bool → List Bool) Encoding.gameBits
(construct p) := by
change TM2ComputableInPolyTime (id : List Bool → List Bool) Encoding.gameBits
(fun input => NormalizedTarget.construct (branch := branch p) (n := p.plan.depth)
(t := sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ
(NormalizedSourceInput.raw input))
exact MachineSequential.composeBits
NormalizedSourceInput.rawComputation
(NormalizedTargetMachine.targetComputation (branch := branch p) (n := p.plan.depth)
(t := sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ)
theorem finiteAlphabet :
MachineFiniteAlphabet.FiniteAlphabet (computation p).tm :=
MachineFiniteAlphabet.composeBits
NormalizedSourceInput.rawComputation
(NormalizedTargetMachine.targetComputation (branch := branch p) (n := p.plan.depth)
(t := sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ)
NormalizedSourceInput.raw_finiteAlphabet
(NormalizedTargetMachine.finiteAlphabet (branch := branch p) (n := p.plan.depth)
(t := sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ)
theorem perfectlyComplete (input : List Bool) (accepted : BinaryLanguage.language input) :
PerfectlyComplete (construct p input) :=
NormalizedTarget.perfectlyComplete (branch := branch p) (n := p.plan.depth)
(t := sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ
(NormalizedSourceInput.raw input)
((NormalizedSourceInput.raw_satisfiable_iff input).mpr accepted)
theorem value_eq_one (input : List Bool) (accepted : BinaryLanguage.language input) :
(construct p input).value = 1 :=
((construct p input).perfectlyComplete_iff_value_eq_one (alphabet_pos p)).mp
(perfectlyComplete p input accepted)
end
end PerfectCompleteness.FixedTarget
end
end OAI
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