File size: 4,918 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactTablePrefix
namespace OAI
/-! # The fixed exact-sampling machine on the literal public binary tape -/
namespace ContingencyTables.ExactPublicPrefix
open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost Turing
open DensePrograms
open scoped Classical
noncomputable section
/-- These two finite programs are independent of dimensions and margins. -/
def middle : Program := ExactPhysicalProgram.program.seq LiteralProgram.post
def program : Program := LiteralProgram.pack.seq middle
def machine : RandomMachine := program.machine
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
def word (X : Table r c) : List Symbol := GreedyFeasibleTable.printMatrix (matrixCode X.val)
lemma word_eq (X : Table r c) : word r c X=Algorithms.encodeMatrix X.val :=
GreedyFeasibleTable.printMatrix_ofFn X.val
def middleOutput : Option (Table r c) → Cfg Symbol middle.State
| none => leftCfg LiteralProgram.post.entry (ExactTablePrefix.output r c none)
| some X => rightCfg (LiteralProgram.post.final (Tape.mk₁ (word r c X)))
def output (o : Option (Table r c)) : Cfg Symbol program.State := rightCfg (middleOutput r c o)
lemma middle_prefix (htotal : (∑ i,r i)=∑ j,c j) (N : ℕ) :
∃ p,Implements middle.table (middleOutput r c)
(middle.initial (Tape.mk₁ (TreePost.packed (ExactTablePrefix.input r c)))) p ∧
∀ g : Option (Table r c) → ℚ,p.mean g=(ExactTablePrefix.experiment r c N).mean g := by
obtain ⟨p,hp,hpm⟩ := ExactTablePrefix.physical_prefix r c htotal N
obtain ⟨q,hq,hqm,_⟩ := PhysicalPartialSequence.left_prefix_injective
ExactPhysicalProgram.program.table LiteralProgram.post.table LiteralProgram.post.entry
(ExactTablePrefix.output r c) (ExactTablePrefix.output_injective r c) _ p hp
have next (o : Option (Table r c)) : ∃ u,
Implements middle.table (middleOutput r c)
(leftCfg LiteralProgram.post.entry (ExactTablePrefix.output r c o)) u ∧
∀ g : Option (Table r c) → ℚ,u.mean g=g o := by
cases o with
| none => exact ⟨.done none,.done none,fun _ => rfl⟩
| some X =>
obtain ⟨u,hu,hum,_⟩ := LiteralProgram.post_realizes (word r c X)
(ProfilePrograms.printMatrix_alphabet _)
have hu' := hu.right (M := ExactPhysicalProgram.program.table) LiteralProgram.post.entry
have hall : u.All (fun T => T=Tape.mk₁ (word r c X)) :=
Experiment.all_of_mean_eq hum _ rfl
refine ⟨u.map (fun _ => some X),?_,?_⟩
· have hv := hu'.map_all (fun _ => some X) (middleOutput r c)
(hall.mono (by intro T hT; rw [hT]; rfl))
simpa only [middle,Program.seq,ExactTablePrefix.output,ExactTablePrefix.code,Option.map_some,
ExactPhysicalProgram.output_some,word,leftCfg,Program.final,Program.initial,
Option.elim_none,rightCfg,Option.map_some] using hv
· intro g
rw [Experiment.mean_map,PhysicalPartialSequence.mean_const]
choose u hu hum using next
refine ⟨q.bind u,hq.bind u _ hu,?_⟩
intro g
rw [Experiment.mean_bind]
simp_rw [hum]
exact (hqm g).trans (hpm g)
theorem prefix_correct (htotal : (∑ i,r i)=∑ j,c j) (N : ℕ) :
∃ p,Implements program.table (output r c)
(program.initial (Tape.mk₁ (ExactTablePrefix.input r c))) p ∧
∀ g : Option (Table r c) → ℚ,p.mean g=(ExactTablePrefix.experiment r c N).mean g := by
obtain ⟨p,hp,hpm⟩ := middle_prefix r c htotal N
have hi : ∀ a∈ExactTablePrefix.input r c,a≠0 := by
intro a ha
exact GreedyFeasibleTable.encodeMargins_nonzero r c a ha
obtain ⟨u,hu,hum,_⟩ := LiteralProgram.pack_realizes (ExactTablePrefix.input r c) hi
obtain ⟨v,hv,hvm,_⟩ := PhysicalPartialSequence.left_prefix LiteralProgram.pack.table middle.table
middle.entry LiteralProgram.pack.final _ u hu
have hall : v.All (fun x => x=LiteralProgram.pack.final
(Tape.mk₁ (TreePost.packed (ExactTablePrefix.input r c)))) := by
rw [Experiment.all_indicator,hvm,hum]
simp only [Experiment.mean,ite_true]
have hp' := hp.right (M := LiteralProgram.pack.table) middle.entry
refine ⟨v.bind (fun _ => p),?_,?_⟩
· apply hv.bind_all (fun _ => p) (output r c)
exact hall.mono (by intro x hx; rw [hx]; exact hp')
· intro g
rw [Experiment.mean_bind,PhysicalPartialSequence.mean_const]
exact hpm g
lemma output_none_live : (output r c none).state≠none := by
change (some _ : Option program.State)≠none
exact Option.some_ne_none _
lemma output_halted (X : Table r c) : (output r c (some X)).state=none := rfl
lemma output_tape (X : Table r c) :
(output r c (some X)).tape.right₀=ListBlank.mk (Algorithms.encodeMatrix X.val) := by
change (Tape.mk₁ (word r c X)).right₀=_
rw [word_eq]
simp only [Tape.mk₁,Tape.mk₂,Tape.mk'_right₀]
end
end ContingencyTables.ExactPublicPrefix
end OAI
|