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Lax307052.ConfigurationGraph

The bounded configuration graph

concepts/Lax307052/ConfigurationGraph.lean · lax-307052

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    Definition

    Vertices are configurations whose work tape has a fixed finite bound. Edges are machine transitions. Acceptance is reachability from the initial configuration to a terminal accepting configuration. The search predicate replaces reachability by the recursive finite graph test.

    Lean source view on GitHub

    1import Lax307052.BoundedConfigurations
    2import Lax307052.Reachability
    3import Mathlib.Data.Fintype.EquivFin
    4import Mathlib.Data.Nat.Log
    5
    6/-!
    7---
    8title: The bounded configuration graph
    9type: definition
    10---
    11Vertices are configurations whose work tape has a fixed finite bound.
    12Edges are machine transitions. Acceptance is reachability from the initial
    13configuration to a terminal accepting configuration. The search predicate
    14replaces reachability by the recursive finite graph test.
    15-/
    16
    17namespace Lax307052.ConfigurationGraph
    18
    19open Lax434930.PolynomialTime Lax434930.SpaceMachines BoundedConfigurations Reachability
    20
    21noncomputable def numbering (M : Machine) (n s : ℕ) :
    22 Config M n s ≃ Fin (Fintype.card (Config M n s)) := Fintype.equivFin _
    23
    24noncomputable def graph (M : Machine) (w : Word) (s : ℕ) :
    25 Graph (Fintype.card (Config M w.length s)) := by
    26 classical
    27 exact fun a b => decide (M.Step w (expand ((numbering M w.length s).symm a))
    28 (expand ((numbering M w.length s).symm b)))
    29
    30def AcceptsBounded (M : Machine) (w : Word) (s : ℕ) : Prop :=
    31 ∃ a b : Config M w.length s,
    32 expand a = M.initial ∧ M.Terminal w (expand b) ∧ M.accept b.state = true ∧
    33 Reachable (graph M w s) (numbering M w.length s a) (numbering M w.length s b)
    34
    35def SearchAccepts (M : Machine) (w : Word) (s : ℕ) : Prop :=
    36 ∃ a b : Config M w.length s,
    37 expand a = M.initial ∧ M.Terminal w (expand b) ∧ M.accept b.state = true ∧
    38 search (graph M w s) (Nat.clog 2 (Fintype.card (Config M w.length s)))
    39 (numbering M w.length s a) (numbering M w.length s b) = true
    40
    41end Lax307052.ConfigurationGraph
    42

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