In temporal extensions of answer set programming (ASP) based on linear time, the behavior of dynamic systems is captured by sequences of states. While this representation reflects their relative order, it abstracts away the specific times associated with each state. However, timing constraints are important in many applications like, for instance, when planning and scheduling go hand in hand. We address this by developing a metric extension of linear-time temporal equilibrium logic, in which temporal operators are constrained by intervals over natural numbers. The resulting Metric Equilibrium Logic (MEL) provides the foundation of an ASP-based approach for specifying qualitative and quantitative dynamic constraints. To this end, we define a translation of metric formulas into monadic first-order formulas and give a correspondence between their models in MEL and Monadic Quantified Equilibrium Logic, respectively. Interestingly, our translation provides a blue print for implementation in terms of ASP modulo difference constraints.

In this work, we show that both logic programming and abstract argumentation frameworks can be interpreted in terms of Nelson’s constructive logic N4. We do so by formalising, in this logic, two principles that we call noncontradictory inference and strengthened closed world assumption: the first states that no belief can be held based on contradictory evidence while the latter forces both unknown and contradictory evidence to be regarded as false. Using these principles, both logic programming and abstract argumentation frameworks are translated into constructive logic in a modular way and using the object language. Logic programming implication and abstract argumentation supports become, in the translation, a new implication connective following the noncontradictory inference principle. Attacks are then represented by combining this new implication with strong negation. Under consideration in Theory and Practice of Logic Programming (TPLP).

In this survey, we present an overview on (Modal) Temporal Logic Programming in view of its application to Knowledge Representation and Declarative Problem Solving. The syntax of this extension of logic programs is the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). In the paper, we focus on the main recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL but involves a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). As a result, we obtain a proper extension of the stable models semantics for the general case of temporal formulas in the syntax of LTL. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between finite and infinite trace length. We also provide further useful results, such as the translation into other formalisms like Quantified Equilibrium Logic and Second-order LTL, and some techniques for computing temporal stable models based on automata constructions. In the remainder of the paper, we focus on practical aspects, defining a syntactic fragment called (modal) temporal logic programs closer to ASP, and explaining how this has been exploited in the construction of the solver telingo, a temporal extension of the well-known ASP solver clingo that uses its incremental solving capabilities.

A common feature in Answer Set Programming is the use of a second negation, stronger than default negation and sometimes called explicit, strong or classical negation. This explicit negation is normally used in front of atoms, rather than allowing its use as a regular operator. In this paper we consider the arbitrary combination of explicit negation with nested expressions, as those defined by Lifschitz, Tang and Turner. We extend the concept of reduct for this new syntax and then prove that it can be captured by an extension of Equilibrium Logic with this second negation. We study some properties of this variant and compare to the already known combination of Equilibrium Logic with Nelson’s strong negation.

In this paper, we propose a variant of Answer Set Programming (ASP) with evaluable functions that extends their application to sets of objects, something that allows a fully logical treatment of aggregates. Formally, we start from the syntax of First Order Logic with equality and the semantics of Quantified Equilibrium Logic with evaluable functions (${\rm QEL}^=_{\cal F}$). Then, we proceed to incorporate a new kind of logical term, intensional set (a construct commonly used to denote the set of objects characterised by a given formula), and to extend ${\rm QEL}^=_{\cal F}$ semantics for this new type of expression. In our extended approach, intensional sets can be arbitrarily used as predicate or function arguments or even nested inside other intensional sets, just as regular first-order logical terms. As a result, aggregates can be naturally formed by the application of some evaluable function (count, sum, maximum, etc) to a set of objects expressed as an intensional set. This approach has several advantages. First, while other semantics for aggregates depend on some syntactic transformation (either via a reduct or a formula translation), the ${\rm QEL}^=_{\cal F}$ interpretation treats them as regular evaluable functions, providing a compositional semantics and avoiding any kind of syntactic restriction. Second, aggregates can be explicitly defined now within the logical language by the simple addition of formulas that fix their meaning in terms of multiple applications of some (commutative and associative) binary operation. For instance, we can use recursive rules to define sum in terms of integer addition. Last, but not least, we prove that the semantics we obtain for aggregates coincides with the one defined by Gelfond and Zhang for the ${\cal A}\mathit{log}$ language, when we restrict to that syntactic fragment.

In this paper we provide an alternative semantics for Equilibrium Logic and its monotonic basis, the logic of Here-and-There (also known as Gödel's G3 logic) that relies on the idea of denotation of a formula, that is, a function that collects the set of models of that formula. Using the three-valued logic G3 as a starting point and an ordering relation (for which equilibrium/stable models are minimal elements) we provide several elementary operations for sets of interpretations. By analysing structural properties of the denotation of formulas, we show some expressiveness results for G3 such as, for instance, that conjunction is not expressible in terms of the other connectives. Moreover, the denotational semantics allows us to capture the set of equilibrium models of a formula with a simple and compact set expression. We also use this semantics to provide several formal definitions for entailment relations that are usual in the literature, and further introduce a new one called strong entailment. We say that α strongly entails β when the equilibrium models of α ∧ γ are also equilibrium models of β ∧ γ for any context γ. We also provide a characterisation of strong entailment in terms of the denotational semantics, and give an example of a sufficient condition that can be applied in some cases.

In this paper we propose an extension of Answer Set Programming (ASP) to deal with (possibly partial) evaluable functions. To this aim, we start from the most general logical counterpart of ASP, Quantified Equilibrium Logic (QEL), and propose a variant QEL=ℱ where the set of functions is partitioned into Herbrand functions (or constructors) and evaluable functions (or operations). We show how this extension has a direct connection to Scott's Logic of Existence, and introduce several useful derived operators, some of them directly borrowed from Scott's formalisation. Using this general framework for arbitrary theories, we proceed to focus on a syntactic subclass that corresponds to normal logic programs with evaluable functions and equality. We provide a translation of this class into function-free normal programs and consider a safety condition so that the resulting program is also safe, under the usual meaning in ASP. Finally, we also establish a formal comparison to Lin and Wang's approach (FASP) dealing with evaluable total functions.

Equilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs into quantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to as quantified Boolean formulas (QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are the consistency problem, brave reasoning and skeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz. ordinary, strong and uniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds. Hereby, our encodings yield upper bounds in a direct manner. Besides this useful feature, our approach has the following benefits: First, our encodings yield a uniform axiomatisation for a variety of problems in a common language. Second, extant solvers for QBFs can be used as back-end inference engines to realise implementations of the encoded task in a rapid prototyping manner. Third, our axiomatisations also allow us to straightforwardly relate equilibrium logic with circumscription.

This paper presents a property of propositional theories under the answer sets semantics (called Equilibrium Logic for this general syntax): any theory can always be reexpressed as a strongly equivalent disjunctive logic program, possibly with negation in the head. We provide two different proofs for this result: one involving a syntactic transformation, and one that constructs a program starting from the countermodels of the theory in the intermediate logic of here-and-there.