Modal Logic Is More Succinct Iff Bi-Implication Is Available in Some Form

Authors Christoph Berkholz , Dietrich Kuske, Christian Schwarz

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Christoph Berkholz
  • Technische Universität Ilmenau, Germany
Dietrich Kuske
  • Technische Universität Ilmenau, Germany
Christian Schwarz
  • Technische Universität Ilmenau, Germany

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Christoph Berkholz, Dietrich Kuske, and Christian Schwarz. Modal Logic Is More Succinct Iff Bi-Implication Is Available in Some Form. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Is it possible to write significantly smaller formulae, when using more Boolean operators in addition to the De Morgan basis (and, or, not)? For propositional logic a negative answer was given by Pratt: every formula with additional operators can be translated to the De Morgan basis with only polynomial increase in size. Surprisingly, for modal logic the picture is different: we show that adding bi-implication allows to write exponentially smaller formulae. Moreover, we provide a complete classification of finite sets of Boolean operators showing they are either of no help (allow polynomial translations to the De Morgan basis) or can express properties as succinct as modal logic with additional bi-implication. More precisely, these results are shown for the modal logic T (and therefore for K). We complement this result showing that the modal logic S5 behaves as propositional logic: no additional Boolean operators make it possible to write significantly smaller formulae.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • succinctness
  • modal logic


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