Lax Lean Archive

Finite automata are MSO-definable

Lax146103.NFAToMSO · concepts/Lax146103/NFAToMSO.lean · lax-146103

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    Natural Language Statement

    Theorem

    Every nondeterministic finite automaton over a finite alphabet has a monadic second-order sentence that holds on exactly the words accepted by the automaton.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Lax146103.NFARecognizable
    2import Lax146103.WordStructure
    3
    4/-!
    5---
    6title: Finite automata are MSO-definable
    7type: theorem
    8---
    9
    10Every nondeterministic finite automaton over a finite alphabet has a monadic
    11second-order sentence that holds on exactly the words accepted by the
    12automaton.
    13-/
    14
    15namespace Lax146103.NFAToMSO
    16
    17open Lax146103.MSOSyntax
    18open Lax146103.WordStructure
    19
    20universe u
    21
    22axiom nfa_definable_by_mso {Sigma : Type u} [Fintype Sigma]
    23 {Q : Type} [Fintype Q] (M : NFA Sigma Q) :
    24 ∃ phi : MSOSyntax.Sentence (wordLanguage Sigma), M.accepts = sentenceLanguage phi
    25
    26end Lax146103.NFAToMSO
    27
    Show Proof
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