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Lax52.NFAToMSO

Finite automata are MSO-definable

concepts/Lax52/NFAToMSO.lean · lax-52

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Evidence

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    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.

    Lean source view on GitHub

    1import Lax52.NFARecognizable
    2import Lax52.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 Lax52.NFAToMSO
    16
    17open Lax52.MSOSyntax
    18open Lax52.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 Lax52.NFAToMSO
    27
    Show Proof

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