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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactPublicPrefix
import OAI.Combinatorics.ContingencyTables.Sampling.ExactPhysicalCost
import OAI.Combinatorics.ContingencyTables.Sampling.ExactPrintCost
namespace OAI
/-! # Cost-preserving literal wrapping of exact-sampling prefixes -/
namespace ContingencyTables.ExactPublicCost
open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost Turing
open DensePrograms
open scoped Classical
noncomputable section
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
def middleOutput (out : Option (Table r c) → Cfg Symbol ExactPhysicalProgram.program.State) :
Option (Table r c) → Cfg Symbol ExactPublicPrefix.middle.State
| none => leftCfg LiteralProgram.post.entry (out none)
| some X => rightCfg (LiteralProgram.post.final (Tape.mk₁ (ExactPublicPrefix.word r c X)))
def output (out : Option (Table r c) → Cfg Symbol ExactPhysicalProgram.program.State)
(o : Option (Table r c)) : Cfg Symbol ExactPublicPrefix.program.State :=
rightCfg (middleOutput r c out o)
theorem wrap_prefix (out : Option (Table r c) → Cfg Symbol ExactPhysicalProgram.program.State)
(hinj : Function.Injective out)
(hout : ∀ X,out (some X)=ExactPhysicalProgram.program.final
(Tape.mk₁ (TreePost.packed (ExactPublicPrefix.word r c X))))
(L : ℕ) (hL : ∀ X,(ExactPublicPrefix.word r c X).length≤L)
(p : Experiment (Option (Table r c)))
(hp : Implements ExactPhysicalProgram.program.table out
(ExactPhysicalProgram.program.initial (Tape.mk₁ (TreePost.packed (ExactTablePrefix.input r c)))) p) :
∃ q,Implements ExactPublicPrefix.program.table (output r c out)
(ExactPublicPrefix.program.initial (Tape.mk₁ (ExactTablePrefix.input r c))) q ∧
(∀ g : Option (Table r c) → ℚ,q.mean g=p.mean g) ∧
q.cost≤LiteralProgram.packTime (ExactTablePrefix.input r c).length+p.cost+LiteralProgram.postTime L := by
obtain ⟨v,hv,hvm,hvc⟩ := PhysicalPartialSequence.left_prefix_injective
ExactPhysicalProgram.program.table LiteralProgram.post.table LiteralProgram.post.entry out hinj _ p hp
have next (o : Option (Table r c)) : ∃ u,
Implements ExactPublicPrefix.middle.table (middleOutput r c out)
(leftCfg LiteralProgram.post.entry (out o)) u ∧
(∀ g : Option (Table r c) → ℚ,u.mean g=g o) ∧ u.cost≤LiteralProgram.postTime L := by
cases o with
| none => exact ⟨.done none,.done none,(fun _ => rfl),Nat.zero_le _⟩
| some X =>
obtain ⟨u,hu,hum,huc⟩ := LiteralProgram.post_realizes (ExactPublicPrefix.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 (ExactPublicPrefix.word r c X)) :=
Experiment.all_of_mean_eq hum _ rfl
refine ⟨u.map (fun _ => some X),?_,?_,?_⟩
· have h := hu'.map_all (fun _ => some X) (middleOutput r c out)
(hall.mono (by intro T hT; rw [hT]; rfl))
rw [hout]
simpa only [ExactPublicPrefix.middle,Program.seq,leftCfg,Program.final,Program.initial,
Option.elim_none,rightCfg,Option.map_some] using h
· intro g
rw [Experiment.mean_map,PhysicalPartialSequence.mean_const]
· rw [Experiment.cost_map]
exact huc.trans (LiteralProgram.postTime_mono (hL X))
choose u hu hum huc using next
let w := v.bind u
have hw : Implements ExactPublicPrefix.middle.table (middleOutput r c out)
(ExactPublicPrefix.middle.initial (Tape.mk₁ (TreePost.packed (ExactTablePrefix.input r c)))) w :=
hv.bind u _ hu
have hwm (g : Option (Table r c) → ℚ) : w.mean g=p.mean g := by
change (v.bind u).mean g=p.mean g
rw [Experiment.mean_bind]
simp_rw [hum]
exact hvm g
have hwc : w.cost≤p.cost+LiteralProgram.postTime L :=
(Experiment.cost_bind_le v u _ huc).trans (Nat.add_le_add_right hvc _)
have hi : ∀ a∈ExactTablePrefix.input r c,a≠0 :=
GreedyFeasibleTable.encodeMargins_nonzero r c
obtain ⟨a,ha,ham,hac⟩ := LiteralProgram.pack_realizes (ExactTablePrefix.input r c) hi
obtain ⟨b,hb,hbm,hbc⟩ := PhysicalPartialSequence.left_prefix LiteralProgram.pack.table
ExactPublicPrefix.middle.table ExactPublicPrefix.middle.entry LiteralProgram.pack.final _ a ha
have hall : b.All (fun x => x=LiteralProgram.pack.final
(Tape.mk₁ (TreePost.packed (ExactTablePrefix.input r c)))) := by
rw [Experiment.all_indicator,hbm,ham]
simp only [Experiment.mean,ite_true]
have hw' := hw.right (M := LiteralProgram.pack.table) ExactPublicPrefix.middle.entry
refine ⟨b.bind (fun _ => w),?_,?_,?_⟩
· apply hb.bind_all (fun _ => w) (output r c out)
exact hall.mono (by intro x hx; rw [hx]; exact hw')
· intro g
rw [Experiment.mean_bind,PhysicalPartialSequence.mean_const]
exact hwm g
· have h := Experiment.cost_bind_le b (fun _ => w) w.cost (fun _ => le_rfl)
omega
/-- Table decoding preserves the finite-prefix work bound. -/
theorem table_prefix (htotal : (∑ i,r i)=∑ j,c j) (N B : ℕ)
(hB : ∀ X : Table r c,GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)≤B) :
∃ p,Implements ExactPhysicalProgram.program.table (ExactTablePrefix.output r c)
(ExactPhysicalProgram.program.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) ∧
p.cost≤ExactPhysicalCost.budget (ExactTablePrefix.input r c) N B := by
have hprint : (ExactCoreProgram.prefixExperiment
(ProfilePrograms.exactParsedMargins (ExactTablePrefix.input r c)) N).All
(fun o => o.elim 0 GreedyFeasibleTable.printMatrixRealizer.cost≤B) := by
rw [show ProfilePrograms.exactParsedMargins (ExactTablePrefix.input r c)=
(List.ofFn r,List.ofFn c) from SmallGraphProfiles.exactParsedMargins_spec r c]
exact ExactPrintCost.prefix_print_bound r c htotal B hB N
obtain ⟨p,hp,hpm,hpc⟩ := ExactPhysicalCost.prefix_correct_cost (ExactTablePrefix.input r c) N B hprint
have hm (g : Option MatrixCode → ℚ) :
p.mean g=(ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).mean g := by
simpa only [ExactTablePrefix.input,SmallGraphProfiles.exactParsedMargins_spec] using hpm g
have hall : p.All (SmallGraphProfiles.FeasibleOutcome r c) :=
Experiment.all_of_mean_eq hm _ (SmallGraphProfiles.exactPrefix_feasible r c htotal N)
refine ⟨p.map (ExactTablePrefix.decode r c),hp.map_all (ExactTablePrefix.decode r c)
(ExactTablePrefix.output r c) (hall.mono (by
intro o ho
unfold ExactTablePrefix.output
rw [ExactTablePrefix.code_decode r c ho])),?_,?_⟩
· intro g
rw [Experiment.mean_map,hm,ExactTablePrefix.experiment,Experiment.mean_map]
· simpa only [Experiment.cost_map] using hpc
noncomputable def budget (N B L : ℕ) : ℕ :=
LiteralProgram.packTime (ExactTablePrefix.input r c).length+
ExactPhysicalCost.budget (ExactTablePrefix.input r c) N B+LiteralProgram.postTime L
theorem prefix_correct_cost (htotal : (∑ i,r i)=∑ j,c j) (N 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 (ExactPublicPrefix.output r c)
(ExactPublicPrefix.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) ∧
p.cost≤budget r c N B L := by
obtain ⟨p,hp,hm,hc⟩ := table_prefix r c htotal N B hB
have hout (X : Table r c) : ExactTablePrefix.output r c (some X)=
ExactPhysicalProgram.program.final
(Tape.mk₁ (TreePost.packed (ExactPublicPrefix.word r c X))) := by
exact ExactPhysicalProgram.output_some _ _
obtain ⟨q,hq,hqm,hqc⟩ := wrap_prefix r c (ExactTablePrefix.output r c)
(ExactTablePrefix.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)
lemma quadratic_budget (N B L : ℕ) :
budget r c N B L≤budget r c 1 B L*(N+1)^2 := by
have hp := ExactPhysicalCost.quadratic_budget (ExactTablePrefix.input r c) N B
have hsq : 1≤(N+1)^2 := Nat.one_le_pow _ _ (by omega)
have hc := Nat.mul_le_mul_left (LiteralProgram.packTime (ExactTablePrefix.input r c).length+
LiteralProgram.postTime L) hsq
unfold budget
nlinarith
end
end ContingencyTables.ExactPublicCost
end OAI
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