| import Mathlib |
| import OAI.Computability.MaxCut.Machines.MachineCopy |
|
|
| namespace OAI |
|
|
| namespace MaxCutGames.Foundations.Complexity.MachineFiniteTable |
|
|
| open Turing |
| open MaxCutGames.Reduction.MachineSubstitution |
|
|
| variable {K Λ σ : Type} [DecidableEq K] [DecidableEq σ] |
| variable {Γ : K → Type} |
|
|
| def dispatch (dst : K) (table : σ → List (Γ dst)) |
| (next : TM2.Stmt Γ Λ σ) : List σ → TM2.Stmt Γ Λ σ |
| | [] => next |
| | key :: rest => |
| .branch (fun state => decide (state = key)) |
| (pushWord dst (table key).reverse next) (dispatch dst table next rest) |
|
|
| theorem stepAux_dispatch (dst : K) (table : σ → List (Γ dst)) |
| (next : TM2.Stmt Γ Λ σ) (keys : List σ) (state : σ) |
| (hstate : state ∈ keys) (tapes : ∀ k, List (Γ k)) : |
| TM2.stepAux (dispatch dst table next keys) state tapes = |
| TM2.stepAux next state (Function.update tapes dst (table state ++ tapes dst)) := by |
| induction keys with |
| | nil => simp at hstate |
| | cons key rest ih => |
| by_cases h : state = key |
| · subst state |
| simp only [dispatch, TM2.stepAux, decide_true, Bool.cond_true] |
| simpa only [List.reverse_reverse] using |
| stepAux_pushWord dst (table key).reverse next key tapes |
| · have hm : state ∈ rest := (List.mem_cons.mp hstate).resolve_left h |
| simpa only [dispatch, TM2.stepAux, h, decide_false, Bool.cond_false] using ih hm |
|
|
| omit [DecidableEq K] in |
| /-- A finite upper bound on all literal push chains in the decision tree. -/ |
| theorem dispatch_push_bound (dst : K) (table : σ → List (Γ dst)) |
| (next : TM2.Stmt Γ Λ σ) (keys : List σ) : |
| Runtime.statementPushBound (dispatch dst table next keys) ≤ |
| (keys.map (fun state => (table state).length)).sum + |
| Runtime.statementPushBound next := by |
| induction keys with |
| | nil => simp [dispatch] |
| | cons key rest ih => |
| simp only [dispatch, Runtime.statementPushBound, statementPushBound_pushWord, |
| List.length_reverse, List.map_cons, List.sum_cons] |
| apply max_le <;> omega |
|
|
| def emit (dst : K) (table : σ → List (Γ dst)) (keys : List σ) |
| (next : TM2.Stmt Γ Λ σ) : TM2.Stmt Γ Λ σ := |
| dispatch dst table next keys |
|
|
| def tableBound (dst : K) (table : σ → List (Γ dst)) (keys : List σ) : ℕ := |
| (keys.map (fun state => (table state).length)).sum |
|
|
| theorem stepAux_emit (dst : K) (table : σ → List (Γ dst)) |
| (keys : List σ) (hkeys : ∀ state, state ∈ keys) (next : TM2.Stmt Γ Λ σ) (state : σ) (tapes : ∀ k, List (Γ k)) : |
| TM2.stepAux (emit dst table keys next) state tapes = |
| TM2.stepAux next state (Function.update tapes dst (table state ++ tapes dst)) := |
| stepAux_dispatch dst table next keys state (hkeys state) tapes |
|
|
| omit [DecidableEq K] in |
| theorem emit_push_bound (dst : K) (table : σ → List (Γ dst)) |
| (keys : List σ) (next : TM2.Stmt Γ Λ σ) : |
| Runtime.statementPushBound (emit dst table keys next) ≤ |
| tableBound dst table keys + Runtime.statementPushBound next := |
| dispatch_push_bound dst table next _ |
|
|
| /-- One actual transition emits the table entry and enters the continuation. |
| The caller supplies its program equation, not an execution assumption. -/ |
| def emitInTime (dst : K) (table : σ → List (Γ dst)) |
| (keys : List σ) (hkeys : ∀ state, state ∈ keys) |
| (program : Λ → TM2.Stmt Γ Λ σ) (label exitLabel : Λ) |
| (atLabel : program label = emit dst table keys (.goto (fun _ => exitLabel))) |
| (state : σ) (tapes : ∀ k, List (Γ k)) : |
| StateTransition.EvalsToInTime (TM2.step program) |
| ⟨some label, state, tapes⟩ |
| (some ⟨some exitLabel, state, |
| Function.update tapes dst (table state ++ tapes dst)⟩) 1 where |
| steps := 1 |
| evals_in_steps := by |
| change some (TM2.stepAux (program label) state tapes) = _ |
| rw [atLabel, stepAux_emit dst table keys hkeys] |
| rfl |
| steps_le_m := Nat.le_refl _ |
|
|
| omit [DecidableEq σ] in |
| theorem output_frame (dst : K) (table : σ → List (Γ dst)) |
| (state : σ) (tapes : ∀ k, List (Γ k)) (k : K) (h : k ≠ dst) : |
| (Function.update tapes dst (table state ++ tapes dst)) k = tapes k := by |
| simp [Function.update, h] |
|
|
| end MaxCutGames.Foundations.Complexity.MachineFiniteTable |
|
|
| end OAI |
|
|
