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Lax52.MSOAutomataEquivalence

Büchi-Elgot-Trakhtenbrot theorem for finite words

concepts/Lax52/MSOAutomataEquivalence.lean · lax-52

proven

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    Theorem

    Over a finite alphabet, a language of finite words is recognizable by a nondeterministic finite automaton if and only if it is definable by a monadic second-order sentence in the ordered word structure with one unary predicate for each letter.

    Lean source view on GitHub

    1import Lax52.NFAToMSO
    2import Lax52.MSOToNFA
    3
    4/-!
    5---
    6title: Büchi-Elgot-Trakhtenbrot theorem for finite words
    7type: theorem
    8---
    9
    10Over a finite alphabet, a language of finite words is recognizable by a
    11nondeterministic finite automaton if and only if it is definable by a monadic
    12second-order sentence in the ordered word structure with one unary predicate
    13for each letter.
    14-/
    15
    16namespace Lax52.MSOAutomataEquivalence
    17
    18open Lax52.MSOSyntax
    19open Lax52.WordStructure
    20open Lax52.NFARecognizable
    21
    22universe u
    23
    24axiom nfaRecognizable_iff_msoDefinable {Sigma : Type u} [Fintype Sigma]
    25 (L : _root_.Language Sigma) :
    26 NFARecognizable L ↔
    27 ∃ phi : MSOSyntax.Sentence (wordLanguage Sigma), L = sentenceLanguage phi
    28
    29end Lax52.MSOAutomataEquivalence
    30
    Show Proof

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