A Note on C² Interpreted over Finite Data-Words

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,∼, πbin ]. In our scenario, we allow for using arbitrary many uninterpreted binary predicates from πbin, 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 [27]. We supplement our results with decidability for several sub-fragments of C2[≤, succ,∼, πbin], 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. 2012 ACM Subject Classification Theory of computation → Logic and verification