Lax Lean Archive

Undecidability of positive first-order definability

Lax503819.Undecidability · concepts/Lax503819/Undecidability.lean · lax-503819

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    Natural Language Statement

    Theorem

    There is no algorithm that, given a finite automaton over a finite powerset alphabet, decides whether its language is definable by a positive first-order sentence. This is Theorem 6.1 of Kuperberg, Positive First-order Logic on Words and Graphs, LMCS 19(3:7), 2023.

    The input includes the number of predicates: the theorem is uniform over finite powerset alphabets, and does not assert undecidability for one fixed alphabet. Definability uses the concrete syntax and semantics in PositiveLogic; algorithmic decidability uses mathlib's partial-recursive computability model. ComputablePredComputablePred must not be replaced with Lean's DecidableDecidable, which would express a different, classically trivial notion here.

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

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    Lean source view on GitHub

    1import Lax503819.Automata
    2import Mathlib.Computability.Reduce
    3
    4/-!
    5---
    6title: Undecidability of positive first-order definability
    7type: theorem
    8---
    9There is no algorithm that, given a finite automaton over a finite powerset
    10alphabet, decides whether its language is definable by a positive first-order
    11sentence. This is Theorem 6.1 of Kuperberg, *Positive First-order Logic on
    12Words and Graphs*, LMCS 19(3:7), 2023.
    13
    14The input includes the number of predicates: the theorem is uniform over
    15finite powerset alphabets, and does not assert undecidability for one fixed
    16alphabet. Definability uses the concrete syntax and semantics in PositiveLogic;
    17algorithmic decidability uses mathlib's partial-recursive computability model.
    18`ComputablePred` must not be replaced with Lean's `Decidable`, which would
    19express a different, classically trivial notion here.
    20-/
    21
    22namespace Lax503819.Undecidability
    23
    24axiom positive_definability_undecidable :
    25 ¬ ComputablePred Lax503819.Automata.PositiveDefinable
    26
    27end Lax503819.Undecidability
    28
    Show Proof
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