| 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 |
|
|
