| import OAI.Combinatorics.ContingencyTables.Sampling.ExactPublicPrefix |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactPhysicalCost |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactPrintCost |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| 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 |
|
|
