Two-variable Fragment Of First-order Logic Research Articles

The regular languages of wire linear AC$$^0$$

In this paper, the regular languages of wire linear \(\hbox {AC}^0\)are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, \(\mathrm{FO}^2[\mathrm{reg}]\). Additionally, they are characterized as the languages recognized by the algebraic class \(\mathbf {QLDA}\). The class is shown to be decidable and examples of languages in and outside of it are presented.

Weighted First-Order Model Counting in the Two-Variable Fragment With Counting Quantifiers

It is known due to the work of Van den Broeck, Meert and Darwiche that weighted first-order model counting (WFOMC) in the two-variable fragment of first-order logic can be solved in time polynomial in the number of domain elements. In this paper we extend this result to the two-variable fragment with counting quantifiers.

TWO-VARIABLE LOGIC HAS WEAK, BUT NOT STRONG, BETH DEFINABILITY

AbstractWe prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

Managing Change in Graph-Structured Data Using Description Logics

In this article, we consider the setting of graph-structured data that evolves as a result of operations carried out by users or applications. We study different reasoning problems, which range from deciding whether a given sequence of actions preserves the satisfaction of a given set of integrity constraints, for every possible initial data instance, to deciding the (non)existence of a sequence of actions that would take the data to an (un)desirable state, starting either from a specific data instance or from an incomplete description of it. For describing states of the data instances and expressing integrity constraints on them, we use description logics (DLs) closely related to the two-variable fragment of first-order logic with counting quantifiers. The updates are defined as actions in a simple yet flexible language, as finite sequences of conditional insertions and deletions, which allow one to use complex DL formulas to select the (pairs of) nodes for which (node or arc) labels are added or deleted. We formalize the preceding data management problems as a static verification problem and several planning problems and show that, due to the adequate choice of formalisms for describing actions and states of the data, most of these data management problems can be effectively reduced to the (un)satisfiability of suitable formulas in decidable logical formalisms. Leveraging this, we provide algorithms and tight complexity bounds for the formalized problems, both for expressive DLs and for a variant of the popular DL-Lite, advocated for data management in recent years.

Two-variable Logic with Counting and a Linear Order

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata.

Decidable Verification of Golog Programs over Non-Local Effect Actions

The Golog action programming language is a powerful means to express high-level behaviours in terms of programs over actions defined in a Situation Calculus theory. In particular for physical systems, verifying that the program satisfies certain desired temporal properties is often crucial, but undecidable in general, the latter being due to the language's high expressiveness in terms of first-order quantification, range of action effects, and program constructs. So far, approaches to achieve decidability involved restrictions where action effects either had to be context-free (i.e. not depend on the current state), local (i.e. only affect objects mentioned in the action's parameters), or at least bounded (i.e. only affect a finite number of objects). In this paper, we introduce two new, more general classes of action theories that allow for context-sensitive, non-local, unbounded effects, i.e. actions that may affect an unbounded number of possibly unnamed objects in a state-dependent fashion. We contribute to the further exploration of the boundary between decidability and undecidability for Golog, showing that for our new classes of action theories in the two-variable fragment of first-order logic, verification of CTL* properties of programs over ground actions is decidable.

Metric propositional neighborhood logic with an equivalence relation

The propositional interval logic of temporal neighborhood (PNL for short) features two modalities that make it possible to access intervals adjacent to the right (modality $$\langle A \rangle $$źAź) and to the left (modality $$\langle \overline{A}\rangle $$źA¯ź) of the current interval. PNL stands at a central position in the realm of interval temporal logics, as it is expressive enough to encode meaningful temporal conditions and decidable (undecidability rules over interval temporal logics, while PNL is NEXPTIME-complete). Moreover, it is expressively complete with respect to the two-variable fragment of first-order logic extended with a linear order $$\text {FO} ^2[<]$$FO2[<]. Various extensions of PNL have been studied in the literature, including metric, hybrid, and first-order ones. Here, we study the effects of the addition of an equivalence relation $$\sim $$~ to Metric PNL (MPNL$${\sim }$$~). We first show that the finite satisfiability problem for PNL extended with $$\sim $$~ is still NEXPTIME-complete. Then, we prove that the same problem for MPNL$${\sim }$$~ can be reduced to the decidable 0---0 reachability problem for vector addition systems and vice versa (EXPSPACE-hardness immediately follows).

Using tableau to decide description logics with full role negation and identity

This article presents a tableau approach for deciding expressive description logics with full role negation and role identity. We consider the description logic ALBO id , which is ALC extended with the Boolean role operators, inverse of roles, the identity role, and includes full support for individuals and singleton concepts. ALBO id is expressively equivalent to the two-variable fragment of first-order logic with equality and subsumes Boolean modal logic. In this article, we define a sound, complete, and terminating tableau calculus for ALBO id that provides the basis for decision procedures for this logic and all its sublogics. An important novelty of our approach is the use of a generic unrestricted blocking mechanism. Unrestricted blocking is based on equality reasoning and a conceptually simple rule, which performs case distinctions over the identity of individuals. The blocking mechanism ties the proof of termination of tableau derivations to the finite model property of ALBO id .

Two-Variable First-Order Logic with Equivalence Closure

We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of first-order logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2-NExpTime, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2-NExpTime-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite satisfiability problems. We further show that the satisfiability (${=}$ finite satisfiability) problem for the two-variable fragment of first-order logic with equivalence closure applied to a single binary predicate is NExpTime-complete.

From XQuery to relational logics

Predicate logic has long been seen as a good foundation for querying relational data. This is embodied in the correspondence between relational calculus and first-order logic, and can also be seen in mappings from fragments of the standard relational query language SQL to extensions of first-order logic (e.g. with counting). A key question is what is the analog to this correspondence for querying tree-structured data, as seen, for example, in XML documents. We formalize this as the question of the appropriate logical query language for defining transformations on tree-structured data. The predominant practitioner paradigm for defining such transformations is top-down tree building . This is embodied by the XQuery query language, which builds the output tree in parallel starting at the root, based on variable bindings and nodeset queries in the XPath language. The goal of this article is to compare the expressiveness of top-down tree-building languages based on a benchmark of predicate logic. We start by giving a formalized XQuery XQ that can serve as a representative of the top-down approach. We show that all queries in XQ with only atomic equality are equivalent to first-order interpretations, an analog to first-order logic (FO) in the setting of transformations of tree-structured data. We then consider fragments of atomic XQ . We identify a fragment that maps efficiently into first-order, a fragment that maps into existential first-order logic, and a fragment that maps into the navigationally two-variable fragment of first-order logic—an analog of two-variable logic in the setting where data values are unbounded. When XQ is considered with deep equality, we find that queries can be translated into FO with counting ( FO (Cnt)). Translations from XQ to logical languages on relations have a number of consequences. We use them to derive complexity bounds for XQ fragments, and to bound the Boolean expressiveness of XQ fragments.

Notes on Logics of Metric Spaces

In [14], we studied the computational behaviour of various first-order and modal languages interpreted in metric or weaker distance spaces. [13] gave an axiomatisation of an expressive and decidable metric logic. The main result of this paper is in showing that the technique of representing metric spaces by means of Kripke frames can be extended to cover the modal (hybrid) language that is expressively complete over metric spaces for the (undecidable) two-variable fragment of first-order logic with binary pred-icates interpreting the metric. The frame conditions needed correspond rather directly with a Boolean modal logic that is, again, of the same expressivity as the two-variable fragment. We use this representation to derive an axiomatisation of the modal hybrid variant of the two-variable fragment, discuss the compactness property in distance logics, and derive some results on (the failure of) interpolation in distance logics of various expressive power.

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.

Complexity of the Two-Variable Fragment with Counting Quantifiers

The satisfiability and finite satisfiability problems for the two-variable fragment of first-order logic with counting quantifiers are both in NEXPTIME, even when counting quantifiers are coded succinctly.

A Two-Variable Fragment of English

Controlled languages are regimented fragments of natural language designed to make the processing of natural language more efficient and reliable. This paper defines a controlled language, E2V, whose principal grammatical resources include determiners, relative clauses, reflexives and pronouns. We provide a formal syntax and semantics for E2V, in which anaphoric ambiguities are resolved in a linguistically natural way. We show that the expressive power of E2V is equal to that of the two-variable fragment of first-order logic. It follows that the problem of determining the satisfiability of a set of E2V sentences is NEXPTIME complete. We show that E2V can be extended in various ways without compromising these complexity results; however, relaxing our policy on anaphora resolution renders the satisfiability problem for E2V undecidable. Finally, we argue that our results have a bearing on the broader philosophical issue of the relationship between natural and formal languages.

Two variable first-order logic over ordered domains

AbstractThe satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations.It is shown that FO2 over ordered, respectively wellordered. domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO2. namely non-deterministic exponential time.In contrast FO2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings. wellorderings. or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO2 and computation tree logic CTL.

On logics with two variables

This paper is a survey and systematic presentation of decidability and complexity issues for modal and non-modal two-variable logics. A classical result due to Mortimer says that the two-variable fragment of first-order logic, denoted FO 2, has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2. Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points, etc. Examples of such logics are computation tree logic CTL, the modal μ-calculus L μ , or popular description logics used in artificial intelligence. Although the additional features are usually not first-order constructs, the resulting logics can still be seen as two-variable logics that are embedded in suitable extensions of FO 2. Typically, the applications call for an analysis of the satisfiability and model checking problems of the logics employed. The decidability and complexity issues for modal and non-modal two-variables logics have been studied quite intensively in the last years. It has turned out that the satisfiability problems for two-variable logics with full first-order quantification are usually much harder (and indeed highly undecidable in many cases) than the satisfiability problems for corresponding modal logics. On the other side, the situation is different for model checking problems. The model checking problem of a modal logic has essentially the same complexity as the model checking problem of the corresponding two variable logic with full quantification.