Decidability of weak logics with deterministic transitive closure

Abstract

The deterministic transitive closure operator, added to languages containing even only two variables, allows to express many natural properties of a binary relation, including being a linear order, a tree, a forest or a partial function. This makes it a potentially attractive ingredient of computer science formalisms. In this paper we consider the extension of the two-variable fragment of first-order logic by the deterministic transitive closure of a single binary relation, and prove that the satisfiability and finite satisfiability problems for the obtained logic are decidable and EXPSPACE-complete. This contrasts with the undecidability of two-variable logic with the deterministic transitive closures of several binary relations, known before. We also consider the class of universal first-order formulas in prenex form. Its various extensions by deterministic closure operations were earlier considered by other authors, leading to both decidability and undecidability results. We examine this scenario in more details.

Two-variable Logic with Counting and a Linear Order

“…Among possible directions for future work one can choose combination of C 2 with other interpreted binary relations like preorders [22] or transitive relations [33]. Another possibility is to consider C 2 with closure operations on some relations, like equivalence closure [15] or deterministic transitive closure [3].…”

“…Many extensions of FO 2 , in particular with transitive closure or least fixed-point operators, quickly lead to undecidability [8,12]. Extensions that go beyond first order logic, but their (finite) satisfiability problem remains decidable, include FO 2 over restricted classes of structures where one [16] or two relation symbols [17] are interpreted as equivalence relations (but there are no other binary symbols); where one [26] or two relations are interpreted as linear orders [31]; where two relations are interpreted as successors of two linear orders [21,7,4]; where one relation is interpreted as linear order, one as its successor and another one as equivalence [1]; where one relation is transitive [33]; where an equivalence closure can be applied to two binary predicates [15]; where deterministic transitive closure can be applied to one binary relation [3]. It is known that the finite satisfiability problem is undecidable for FO 2 with two transitive relations [14], with three equivalence relations [16], with one transitive and one equivalence relation [17], with three linear orders [13], with two linear orders and their two corresponding successors [21].…”

“…When studying extensions of FO 2 it is enough to consider relational signatures with symbols of arity at most 2 [9]. Some of the above mentioned decidability results, e. g., [1,3,7,21,31], hold under the assumption that there are no other binary predicates in the signature; some, like [4,15,16,17,26,33] are valid in the general setting. Similarly, for undecidability results additional binary symbols are required e. g., in [8,12,17], but not required in [13,14,16,21].…”

Two-Variable Separation Logic and Its Inner Circle

“…By contrast, decidability results of FO2 interpreted over linear structures can be found in Otto [2001]; FO2 on data words is decidable too and somewhat equivalent to the reachability problem for Petri nets [Bojańczyk et al 2011]. There are also other decidable extensions of FO2 (see Pacholski et al [1997], Grädel et al [1997b], Kieroński et al [2012], Szwast and Tendera [2013], and Charatonik et al [2014]). At the propositional level, bounding the number of propositional variables also makes sense to restrict syntactic resources and to study its impact on the complexity of decision problems; see examples for modal and temporal logics in Halpern [1995] and Demri and Schnoebelen [2002].…”

Separation logics and modalities: a survey

“…Recently, FO2 augmented with the deterministic transitive closure of a single binary relation is shown to have a decidable and EXPSPACE-complete satisfiability problem [CKM14]. The works [GOR99] and [CKM14] contain numerous undecidability results related to the deterministic transitive closure operator but this involves more than one binary relation, whereas the models for 1SL have a unique deterministic binary relation. However, several results presented in [CKM14] are quite optimal with respect to the syntactic resources.…”