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e5a97c6 | 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.SupportedCompiler
import OAI.Combinatorics.MatchingCount.Machines.BoolPostCompiler
namespace OAI
namespace MatchingFPRAS.FixedAlphabetCompiler
open Turing MachineCost
open scoped Classical
variable {Γ Λ σ : Type*} [Inhabited Γ] [Fintype Γ] [Inhabited Λ]
[Inhabited σ] [Fintype σ]
variable {w : ℕ} (enc : Γ → List.Vector Bool w) (dec : List.Vector Bool w → Γ)
(enc0 : enc default = List.Vector.replicate w false)
(encdec : ∀ a,dec (enc a)=a) (M : Λ → TM1.Stmt Γ Λ σ)
(S : Finset Λ) (hS : TM1.Supports M S)
def bitsWord (input : List Γ) : List Bool := input.flatMap (fun a=>(enc a).toList)
def word (input : List Γ) : List Symbol := (bitsWord enc input).map BoolPostCompiler.up
omit [Fintype Γ] [Fintype σ] in
lemma initial_block (input : List Γ) :
TM1to1.trCfg enc enc0 (TM1.init (Λ:=Λ) (σ:=σ) input)=
TM1.init (bitsWord enc input) := by
simp only [TM1to1.trCfg,TM1.init,Option.map_some,TM1to1.trTape,TM1to1.trTape',
Tape.mk₁,Tape.mk₂,Tape.mk'_left,Tape.mk'_right₀,ListBlank.flatMap_mk,
List.flatMap_nil,bitsWord]
rfl
omit [Fintype Γ] [Fintype σ] [Inhabited Λ] [Inhabited σ] in
lemma output_block (c : TM1.Cfg Γ Λ σ) (output : List Γ)
(h : c.Tape.right₀=ListBlank.mk output) :
(TM1to1.trCfg enc enc0 c).Tape.right₀=ListBlank.mk (bitsWord enc output) := by
change (TM1to1.trTape enc0 c.Tape).right₀=_
simp only [TM1to1.trTape,TM1to1.trTape',Tape.mk'_right₀,h,ListBlank.flatMap_mk,bitsWord]
noncomputable def support := TM1to1.trSupp M S
include hS in
omit [Inhabited Γ] [Inhabited σ] [Fintype σ] in
lemma block_support : TM1.Supports (TM1to1.tr enc dec M) (support M S) :=
TM1to1.tr_supports enc dec M hS
noncomputable def compiled : RandomMachine :=
BoolPostCompiler.compiled (TM1to0.tr (TM1to1.tr enc dec M))
(TM1to0.trStmts (TM1to1.tr enc dec M) (support M S))
(TM1to0.tr_supports _ (block_support enc dec M S hS))
noncomputable def cost : ℕ :=
SupportedCompiler.cost (TM1to1.tr enc dec M) (support M S) *
(AlphabetCost.supportedWrites M S+1)
include encdec enc0
theorem compiled_words (input output : List Γ) (n t : ℕ)
(hhalt : (MachineCost.run (TM1.step M) n (TM1.init input)).l=none)
(hout : (MachineCost.run (TM1.step M) n (TM1.init input)).Tape.right₀=
ListBlank.mk output)
(ht : cost enc dec M S*n+2 ≤ t) (bits : Fin t → Bool) :
(MatchingFPRAS.run (compiled enc dec M S hS) (word enc input) bits).state=none ∧
(MatchingFPRAS.run (compiled enc dec M S hS) (word enc input) bits).tape.right₀=
ListBlank.mk (word enc output) := by
have hi : (TM1.init (Λ:=Λ) (σ:=σ) input).l ∈ S.insertNone :=
Finset.some_mem_insertNone.mpr hS.1
obtain ⟨b,hb,he⟩:=AlphabetCost.execution_supported enc dec enc0 encdec M S hS hi n
rw [initial_block] at he
have hS' := block_support enc dec M S hS
have hi' : (TM1.init (Λ:=TM1to1.Λ' Γ Λ σ) (σ:=σ) (bitsWord enc input)).l ∈
(support M S).insertNone := Finset.some_mem_insertNone.mpr hS'.1
obtain ⟨k,hk,hke⟩:=SupportedCompiler.post_time (TM1to1.tr enc dec M) (support M S) hS' hi' b
have initpost : TM1to0.trCfg (TM1to1.tr enc dec M)
(TM1.init (bitsWord enc input))=TM0.init (bitsWord enc input) := rfl
rw [initpost,he] at hke
have hh : TM0.step (TM1to0.tr (TM1to1.tr enc dec M))
(MachineCost.run (TM0.step (TM1to0.tr (TM1to1.tr enc dec M))) k
(TM0.init (bitsWord enc input)))=none := by
rw [hke]
generalize hc:MachineCost.run (TM1.step M) n (TM1.init input)=c at hhalt ⊢
rcases c with ⟨l,v,T⟩
change l=none at hhalt
subst l
rfl
have ho : (MachineCost.run (TM0.step (TM1to0.tr (TM1to1.tr enc dec M))) k
(TM0.init (bitsWord enc input))).Tape.right₀=ListBlank.mk (bitsWord enc output) := by
rw [hke]
exact output_block enc enc0 _ output hout
have hkt : k+2 ≤ t := by
apply (Nat.add_le_add_right (hk.trans (Nat.mul_le_mul_left _ hb)) 2).trans
have hp : SupportedCompiler.cost (TM1to1.tr enc dec M) (support M S) *
(n*(AlphabetCost.supportedWrites M S+1))+2 = cost enc dec M S*n+2 := by
unfold cost
ring
rw [hp]
exact ht
exact BoolPostCompiler.compiled_words _ _ _ (bitsWord enc input) (bitsWord enc output)
k t hh ho hkt bits
end MatchingFPRAS.FixedAlphabetCompiler
end OAI
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