| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Complexity.GuardMachineBase |
| import OAI.Combinatorics.MatchingCount.Machines.PhysicalPost |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.CoinTM1 |
| open Turing MachineCost PhysicalStream |
|
|
| variable {Γ Λ σ : Type} [Inhabited Γ] |
|
|
| abbrev Node (Γ Λ σ : Type) := Λ ⊕ TM1.Stmt Γ Λ σ |
| abbrev State (Γ Λ σ : Type) := Node Γ Λ σ × σ |
| abbrev Coins (Λ : Type) := Λ → Option (Bool → Λ) |
|
|
| def aux (a : Γ) : TM1.Stmt Γ Λ σ → σ → Option (State Γ Λ σ × TM0.Stmt Γ) |
| | .move d q,v => some ((.inr q,v),.move d) |
| | .write f q,v => some ((.inr q,v),.write (f a v)) |
| | .load f q,v => aux a q (f a v) |
| | .branch f q r,v => if f a v then aux a q v else aux a r v |
| | .goto f,v => some ((.inl (f a v),v),.write a) |
| | .halt,_ => none |
|
|
| def table (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ) : PhysicalPost.Table Γ (State Γ Λ σ) |
| | (.inl l,v),a,b => match coin l with |
| | none => aux a (M l) v |
| | some f => some ((.inl (f b),v),.write a) |
| | (.inr q,v),a,_ => aux a q v |
|
|
| def pending (q : TM1.Stmt Γ Λ σ) (v : σ) (T : Tape Γ) : PhysicalPost.Cfg Γ (State Γ Λ σ) := |
| ⟨some (.inr q,v),T⟩ |
|
|
| def trCfg (c : TM1.Cfg Γ Λ σ) : PhysicalPost.Cfg Γ (State Γ Λ σ) := |
| ⟨c.l.map (fun l=>(Sum.inl l,c.var)),c.Tape⟩ |
|
|
| lemma execute_first_eq (M : PhysicalPost.Table Γ (State Γ Λ σ)) |
| (x y : PhysicalPost.Cfg Γ (State Γ Λ σ)) (h : ∀ b,PhysicalPost.tick M x b=PhysicalPost.tick M y b) |
| {n : ℕ} (hn : 0 < n) (b : Stream) : PhysicalPost.execute M n x b=PhysicalPost.execute M n y b := by |
| cases n with |
| | zero => omega |
| | succ n => simp only [PhysicalPost.execute,h] |
|
|
| theorem statement_ticks (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ) |
| (q : TM1.Stmt Γ Λ σ) (v : σ) (T : Tape Γ) : |
| ∃ k,0 < k ∧ k ≤ statementCost q ∧ ∀ b : Stream, |
| PhysicalPost.execute (table M coin) k (pending q v T) b=trCfg (TM1.stepAux q v T) := by |
| induction q generalizing v T with |
| | move d q ih => |
| obtain ⟨k,hk,hc,he⟩:=ih v (T.move d) |
| refine ⟨k+1,by omega,by simp only [statementCost]; omega,fun b=>?_⟩ |
| simpa only [PhysicalPost.execute,PhysicalPost.tick,pending,table,aux,TM1.stepAux] using he (shift 1 b) |
| | write f q ih => |
| obtain ⟨k,hk,hc,he⟩:=ih v (T.write (f T.head v)) |
| refine ⟨k+1,by omega,by simp only [statementCost]; omega,fun b=>?_⟩ |
| simpa only [PhysicalPost.execute,PhysicalPost.tick,pending,table,aux,TM1.stepAux] using he (shift 1 b) |
| | load f q ih => |
| obtain ⟨k,hk,hc,he⟩:=ih (f T.head v) T |
| refine ⟨k,hk,hc,fun b=>?_⟩ |
| rw [execute_first_eq (table M coin) (pending (.load f q) v T) (pending q (f T.head v) T) (fun _=>rfl) hk] |
| exact he b |
| | branch f q r ihq ihr => |
| cases h:f T.head v with |
| | false => |
| obtain ⟨k,hk,hc,he⟩:=ihr v T |
| refine ⟨k,hk,hc.trans (le_max_right _ _),fun b=>?_⟩ |
| rw [execute_first_eq (table M coin) (pending (.branch f q r) v T) (pending r v T) (fun _=>by |
| simp only [PhysicalPost.tick,pending,table,aux,h,Bool.false_eq_true,ite_false]) hk] |
| simpa only [TM1.stepAux,h,Bool.cond_false] using he b |
| | true => |
| obtain ⟨k,hk,hc,he⟩:=ihq v T |
| refine ⟨k,hk,hc.trans (le_max_left _ _),fun b=>?_⟩ |
| rw [execute_first_eq (table M coin) (pending (.branch f q r) v T) (pending q v T) (fun _=>by |
| simp only [PhysicalPost.tick,pending,table,aux,h,ite_true]) hk] |
| simpa only [TM1.stepAux,h,Bool.cond_true] using he b |
| | goto f => |
| refine ⟨1,by omega,le_rfl,fun b=>?_⟩ |
| simp only [PhysicalPost.execute,PhysicalPost.tick,pending,table,aux,TM1.stepAux,trCfg,Option.map_some,Tape.write_self] |
| | halt => |
| refine ⟨1,by omega,le_rfl,fun b=>?_⟩ |
| rfl |
|
|
| lemma entry_ticks (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ) |
| (l : Λ) (v : σ) (T : Tape Γ) (hl : coin l=none) : |
| ∃ k,0 < k ∧ k ≤ statementCost (M l) ∧ ∀ b : Stream, |
| PhysicalPost.execute (table M coin) k (trCfg ⟨some l,v,T⟩) b= |
| trCfg (TM1.stepAux (M l) v T) := by |
| obtain ⟨k,hk,hc,he⟩:=statement_ticks M coin (M l) v T |
| refine ⟨k,hk,hc,fun b=>?_⟩ |
| rw [execute_first_eq (table M coin) (trCfg ⟨some l,v,T⟩) (pending (M l) v T) (fun _=>by |
| simp only [PhysicalPost.tick,trCfg,Option.map_some,table,hl,pending]) hk] |
| exact he b |
|
|
| lemma coin_tick (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ) |
| (l : Λ) (v : σ) (T : Tape Γ) (f : Bool → Λ) (hl : coin l=some f) (b : Bool) : |
| PhysicalPost.tick (table M coin) (trCfg ⟨some l,v,T⟩) b=trCfg ⟨some (f b),v,T⟩ := by |
| simp only [PhysicalPost.tick,trCfg,Option.map_some,table,hl,Tape.write_self] |
|
|
| end MatchingFPRAS.CoinTM1 |
|
|
| end OAI |
|
|
