One-variable first-order linear temporal logics with counting

Authors Christopher Hampson, Agi Kurucz

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Christopher Hampson
Agi Kurucz

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Christopher Hampson and Agi Kurucz. One-variable first-order linear temporal logics with counting. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 348-362, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


First-order temporal logics are notorious for their bad computational behaviour. It is known that even the two-variable monadic fragment is highly undecidable over various timelines. However, following the introduction of the monodic formulas (where temporal operators can be applied only to subformulas with at most one free variable), there has been a renewed interest in understanding extensions of the one-variable fragment and identifying those that are decidable. Here we analyse the one-variable fragment of temporal logic extended with counting (to two), interpreted in models with constant, decreasing, and expanding first-order domains. We show that over most classes of linear orders these logics are (sometimes highly) undecidable, even without constant and function symbols, and with the sole temporal operator 'eventually'. A more general result says that the bimodal logic of commuting linear and pseudo-equivalence relations is undecidable. The proofs are by reductions of various counter machine problems.
  • modal and temporal logic
  • counting
  • decision procedures


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