On Polynomial-Time Decidability of k-Negations Fragments of FO Theories (Extended Abstract)

Authors Christoph Haase , Alessio Mansutti , Amaury Pouly

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Author Details

Christoph Haase
  • University of Oxford, UK
Alessio Mansutti
  • IMDEA Software Institute, Madrid, Spain
Amaury Pouly
  • University of Oxford, UK
  • Université Paris Cité, CNRS, IRIF, France

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Christoph Haase, Alessio Mansutti, and Amaury Pouly. On Polynomial-Time Decidability of k-Negations Fragments of FO Theories (Extended Abstract). In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1-31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • first-order theories
  • arithmetic theories
  • fixed-parameter tractability


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