Two-variable logic on data words

Abstract

In a data word each position carries a label from a finite alphabet and a data value from some infinite domain. This model has been already considered in the realm of semistructured data, timed automata, and extended temporal logics. This article shows that satisfiability for the two-variable fragment FO 2 (∼,<,+1) of first-order logic with data equality test ∼ is decidable over finite and infinite data words. Here +1 and < are the usual successor and order predicates, respectively. The satisfiability problem is shown to be at least as hard as reachability in Petri nets. Several extensions of the logic are considered; some remain decidable while some are undecidable.

On notions of regularity for data languages

“…Several specification mechanisms for data languages have been suggested, such as register automata [12], pebble automata [15], quasi-regular expressions [13], data automata [3], and LTL with a freeze quantifier [5]. For an overview, see [18].…”

“…Due to the limited access to data values, they can be represented by an equivalence relation. First-order logic on data strings is undecidable, but the two-variable fragment has a decidable satisfiability problem [3]. In its decidability proof, the latter paper introduced a new automaton model for data strings, data automata (DAs).…”

“…As they have the expressive power described above and a decidable emptiness problem, they fulfill, at least to some extent, the requirements of (1) expressiveness and (2) decidability. The other three requirements were not studied in depth in [3]. Nevertheless, the paper gave a characterization of the class R of data languages accepted by DAs in terms of an existential monadic second-order logic and thus established a certain robustness of the class.…”

On pebble automata for data languages with decidable emptiness problem

“…Recently, there has been a significant amount of work in this direction. See, for example, [3,4,6,9,13,15].…”

“…Later in [4] first-order logic for data languages was considered, and, in particular, the so-called data automata was introduced. It was shown that data automata define the fragment of existential monadic second-order logic for data languages in which the first-order part is restricted to two variables only.…”