One-variable logic meets Presburger arithmetic

Abstract

We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove NP-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.

Two Variable Logic with Ultimately Periodic Counting

“…An atomic formula is either an atom R( u), where R is a predicate, and u is a tuple of variables of appropriate size, or an equality u = u , with u and u variables, or one of the formulas and ⊥ denoting the True and False values. The logic FO 2 Pres is a class of first-order formulas using only variables x and y, built up from atomic formulas and equalities using the usual boolean connectives and also ultimately periodic counting quantification, which is of the form ∃ S x φ where S is a u.p.s. One special case is where S is a singleton {a} with a ∈ N ∞ , which we write ∃ a x φ; in case of a ∈ N, these are counting quantifiers.…”

“…In the search for expressive logics with decidable satisfiability problem, two-variable logic, denoted here as FO 2 , is one yardstick. This logic is expressive enough to subsume basic modal logic and many description logics, while satisfiability and finite satisfiability coincide, and both are decidable [23,15,9].…”

“…Related one-variable fragments in which we have only a unary relational vocabulary and the main quantification is ∃ S x φ(x) are known to be decidable (see, e.g. [2]), and their decidability is the basis for a number of software tools focusing on integration of relational languages with Presburger arithmetic [14]. The decidability of full FO 2 Pres is, to the best of our knowledge, still open [4].…”

“…[2]), and their decidability is the basis for a number of software tools focusing on integration of relational languages with Presburger arithmetic [14]. The decidability of full FO 2 Pres is, to the best of our knowledge, still open [4]. There are a number of other extensions of C 2 that have been shown decidable; for example it has been shown that one can allow a distinguished equivalence relation [22] or a forest-structured relation [6,5].…”

On two-variable guarded fragment logic with expressive local Presburger constraints

“…The extension of one-variable logic with quantifiers of the form ∃ S x φ(x), where S is a ultimately periodic set, is NP-complete [Bed20]. Semantically ∃ S x φ(x) means the number of x where φ(x) holds is in the set S. The extension of two-variable logic with such quantifiers is later shown to be 2-NEXP [BKT20] whose proof makes heavy use of the biregular graph method introduced in [KT15] to analyze the spectrum of two-variable logic with counting quantifiers.…”