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Positive and Monotone Fragments of FO and LTL

Authors Simon Iosti, Denis Kuperberg , Quentin Moreau

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ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Regular languages
  • Theory of computation → Logic and verification
Keywords
  • Positive logic
  • LTL
  • separation
  • first-order
  • monotone

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Abstract

We study the positive logic FO^+ on finite words, and its fragments, pursuing and refining the work initiated in [Denis Kuperberg, 2023]. First, we transpose notorious logic equivalences into positive first-order logic: FO^+ is equivalent to LTL^+, and its two-variable fragment FO^{2+} with (resp. without) successor available is equivalent to UTL^+ with (resp. without) the "next" operator X available. This shows that despite previous negative results, the class of FO^+-definable languages exhibits some form of robustness. We then exhibit an example of an FO-definable monotone language on one predicate, that is not FO^+-definable, refining the example from [Denis Kuperberg, 2023] with 3 predicates. Moreover, we show that such a counter-example cannot be FO²-definable. Finally, we provide a new example distinguishing the positive and monotone versions of FO² without quantifier alternation. This does not rely on a variant of the previously known counter-example, and witnesses a new phenomenon.

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Simon Iosti, Denis Kuperberg, and Quentin Moreau. Positive and Monotone Fragments of FO and LTL. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 162:1-162:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.162

Author Details

Simon Iosti
  • LIP, ENS Lyon, France
Denis Kuperberg
  • CNRS, LIP, ENS Lyon, France
Quentin Moreau
  • ENS Lyon, France

Funding

  • Kuperberg, Denis: ANR ReCiProg.

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