A Note on C² Interpreted over Finite Data-Words

Authors Bartosz Bednarczyk , Piotr Witkowski

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Author Details

Bartosz Bednarczyk
  • Computational Logic Group, TU Dresden, Germany
  • Institute of Computer Science, University of Wrocław, Poland
Piotr Witkowski
  • Institute of Computer Science, University of Wrocław, Poland

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Bartosz Bednarczyk and Piotr Witkowski. A Note on C² Interpreted over Finite Data-Words. In 27th International Symposium on Temporal Representation and Reasoning (TIME 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 178, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We consider the satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers, interpreted over finite words with data, denoted here with C²[≤ , succ, ∼, π_bin]. In our scenario, we allow for using arbitrary many uninterpreted binary predicates from π_bin, two navigational predicates ≤ and succ over word positions as well as a data-equality predicate ∼. We prove that the obtained logic is undecidable, which contrasts with the decidability of the logic without counting by Montanari, Pazzaglia and Sala [Angelo Montanari et al., 2016]. We supplement our results with decidability for several sub-fragments of C²[≤ , succ, ∼, π_bin], e.g. without binary predicates, without successor succ, or under the assumption that the total number of positions carrying the same data value in a data-word is bounded by an a priori given constant.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Two-variable logic
  • data-words
  • VASS
  • decidability
  • undecidability
  • counting


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