File size: 6,200 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePhysical
import OAI.Combinatorics.ContingencyTables.Sampling.ExactPublicCost
namespace OAI
/-! # The cheap gate prefix on public binary input -/
namespace ContingencyTables.ExactGatePublic
open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost Turing
open DensePrograms ProfilePrograms SmallGraphProfiles
open ExactTablePrefix (code decode code_decode decode_eq_iff input)
open scoped BigOperators Classical
noncomputable section
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
attribute [local irreducible] exactOrdinary
lemma feasible (htotal : (∑ i,r i)=∑ j,c j) :
(ExactGatePrefix.experiment (List.ofFn r,List.ofFn c)).All (FeasibleOutcome r c) := by
rw [ExactGatePrefix.experiment,Experiment.all_bind]
apply (Experiment.all_true _).mono
intro rare _
cases rare with
| false =>
rw [ite_eq_right Bool.false_ne_true,Experiment.all_map]
exact exactOrdinary_feasible r c htotal
| true => trivial
lemma print_bound (htotal : (∑ i,r i)=∑ j,c j) (B : ℕ)
(hB : ∀ X : Table r c,GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)≤B) :
(ExactGatePrefix.experiment (List.ofFn r,List.ofFn c)).All
(fun o => o.elim 0 GreedyFeasibleTable.printMatrixRealizer.cost≤B) := by
exact (feasible r c htotal).mono (by
intro o ho
cases o with
| none => exact Nat.zero_le _
| some X => obtain ⟨T,rfl⟩ := ho; exact hB T)
def experiment : Experiment (Option (Table r c)) :=
(ExactGatePrefix.experiment (List.ofFn r,List.ofFn c)).map (decode r c)
lemma failure (htotal : (∑ i,r i)=∑ j,c j) :
(experiment r c).mean (fun o => if o=none then 1 else 0)=1/(2:ℚ)^(correctionPrecision r c) := by
rw [experiment,Experiment.mean_map]
have he := Experiment.mean_congr_all _
(fun o => if decode r c o=none then (1:ℚ) else 0)
(fun o => if o=none then (1:ℚ) else 0)
((feasible r c htotal).mono (by
intro o ho
simp only [decode_eq_iff r c ho none,code,Option.map_none]))
rw [he,ExactGatePrefix.failure,exactPrecision_ofFn]
noncomputable def output (o : Option (Table r c)) :
Cfg Symbol ExactPhysicalProgram.program.State := ExactGatePhysical.output (input r c) (code r c o)
lemma output_injective : Function.Injective (output r c) := by
intro a b h
cases a with
| none =>
cases b with
| none => rfl
| some Y =>
have he := congrArg Cfg.state h
change (ExactGatePhysical.output (input r c) none).state=
(ExactGatePhysical.output (input r c) (some (matrixCode Y.val))).state at he
rw [ExactGatePhysical.output_some] at he
exact False.elim (ExactGatePhysical.output_none_live _ he)
| some X =>
cases b with
| none =>
have he := congrArg Cfg.state h
change (ExactGatePhysical.output (input r c) (some (matrixCode X.val))).state=
(ExactGatePhysical.output (input r c) none).state at he
rw [ExactGatePhysical.output_some] at he
exact False.elim (ExactGatePhysical.output_none_live _ he.symm)
| some Y =>
apply congrArg some
apply PackedMatrixInjective.table_injective X Y
change ExactGatePhysical.output _ (some (matrixCode X.val))=
ExactGatePhysical.output _ (some (matrixCode Y.val)) at h
rw [ExactGatePhysical.output_some,ExactGatePhysical.output_some] at h
simpa only [Program.final,Cfg.mk.injEq,true_and] using h
theorem table_prefix (htotal : (∑ i,r i)=∑ j,c j) (B : ℕ)
(hB : ∀ X : Table r c,GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)≤B) :
∃ p,Implements ExactPhysicalProgram.program.table (output r c)
(ExactPhysicalProgram.program.initial (Tape.mk₁ (TreePost.packed (input r c)))) p ∧
(∀ g : Option (Table r c) → ℚ,p.mean g=(experiment r c).mean g) ∧
p.cost≤ExactGatePhysical.budget (input r c) B := by
have hb : (ExactGatePrefix.experiment (exactParsedMargins (input r c))).All
(fun o => o.elim 0 GreedyFeasibleTable.printMatrixRealizer.cost≤B) := by
rw [show exactParsedMargins (input r c)=(List.ofFn r,List.ofFn c) from exactParsedMargins_spec r c]
exact print_bound r c htotal B hB
obtain ⟨p,hp,hpm,hpc⟩ := ExactGatePhysical.prefix_correct_cost (input r c) B hb
have hm (g : Option MatrixCode → ℚ) :
p.mean g=(ExactGatePrefix.experiment (List.ofFn r,List.ofFn c)).mean g := by
simpa only [input,exactParsedMargins_spec] using hpm g
have hall : p.All (FeasibleOutcome r c) := Experiment.all_of_mean_eq hm _ (feasible r c htotal)
refine ⟨p.map (decode r c),hp.map_all (decode r c) (output r c)
(hall.mono (by intro o ho; unfold output; rw [code_decode r c ho])),?_,?_⟩
· intro g
rw [Experiment.mean_map,hm,experiment,Experiment.mean_map]
· simpa only [Experiment.cost_map] using hpc
noncomputable def budget (B L : ℕ) : ℕ :=
LiteralProgram.packTime (input r c).length+ExactGatePhysical.budget (input r c) B+LiteralProgram.postTime L
noncomputable def publicOutput := ExactPublicCost.output r c (output r c)
lemma publicOutput_halted (X : Table r c) : (publicOutput r c (some X)).state=none := rfl
theorem prefix_correct_cost (htotal : (∑ i,r i)=∑ j,c j) (B L : ℕ)
(hB : ∀ X : Table r c,GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)≤B)
(hL : ∀ X : Table r c,(ExactPublicPrefix.word r c X).length≤L) :
∃ p,Implements ExactPublicPrefix.program.table (publicOutput r c)
(ExactPublicPrefix.program.initial (Tape.mk₁ (input r c))) p ∧
(∀ g : Option (Table r c) → ℚ,p.mean g=(experiment r c).mean g) ∧ p.cost≤budget r c B L := by
obtain ⟨p,hp,hm,hc⟩ := table_prefix r c htotal B hB
have hout (X : Table r c) : output r c (some X)=ExactPhysicalProgram.program.final
(Tape.mk₁ (TreePost.packed (ExactPublicPrefix.word r c X))) := ExactGatePhysical.output_some _ _
obtain ⟨q,hq,hqm,hqc⟩ := ExactPublicCost.wrap_prefix r c (output r c)
(output_injective r c) hout L hL p hp
refine ⟨q,hq,(fun g => (hqm g).trans (hm g)),?_⟩
exact hqc.trans (by unfold budget; omega)
end
end ContingencyTables.ExactGatePublic
end OAI
|