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Büchi-Elgot-Trakhtenbrot theorem for finite words

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

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    Natural Language Statement

    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.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax146103.NFAToMSO
    2import Lax146103.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 Lax146103.MSOAutomataEquivalence
    17
    18open Lax146103.MSOSyntax
    19open Lax146103.WordStructure
    20open Lax146103.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 Lax146103.MSOAutomataEquivalence
    30
    Show Proof
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