A Hypersequent Calculus with Clusters for Tense Logic over Ordinals

Authors David Baelde, Anthony Lick, Sylvain Schmitz

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David Baelde
  • LSV, ENS Paris-Saclay & CNRS & Inria, Université Paris-Saclay, France
Anthony Lick
  • LSV, CNRS & ENS Paris-Saclay, Université Paris-Saclay, France
Sylvain Schmitz
  • LSV, ENS Paris-Saclay & CNRS, Université Paris-Saclay, France

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David Baelde, Anthony Lick, and Sylvain Schmitz. A Hypersequent Calculus with Clusters for Tense Logic over Ordinals. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Prior's tense logic forms the core of linear temporal logic, with both past- and future-looking modalities. We present a sound and complete proof system for tense logic over ordinals. Technically, this is a hypersequent system, enriched with an ordering, clusters, and annotations. The system is designed with proof search algorithms in mind, and yields an optimal coNP complexity for the validity problem. It entails a small model property for tense logic over ordinals: every satisfiable formula has a model of order type at most omega^2. It also allows to answer the validity problem for ordinals below or exactly equal to a given one.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Modal and temporal logics
  • modal logic
  • proof system
  • hypersequent


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