Answer-set programming (ASP) is a successful problem-solving approach in logic-based AI. In ASP, problems are represented as declarative logic programs, and solutions are identified through their answer sets. Equilibrium logic (EL) is a general-purpose nonmonotonic reasoning formalism, based on a monotonic logic called here-and-there logic. EL was basically proposed by Pearce as a foundational framework of ASP. Epistemic specifications (ES) are extensions of ASP-programs with subjective literals. These new modal constructs in the ASP-language make it possible to check whether a regular literal of ASP is true in every (or some) answer-set of a program. ES-programs are interpreted by world-views, which are essentially collections of answer-sets. (Reflexive) autoepistemic logic is a nonmonotonic formalism, modeling self-belief (knowledge) of ideally rational agents. A relatively new semantics for ES is based on a combination of EL and (reflexive) autoepistemic logic. In this paper, we first propose an overarching framework in the epistemic ASP domain. We then establish a correspondence between existing (reflexive) (auto)epistemic equilibrium logics and our easily-adaptable comprehensive framework, building on Pearce's characterisation of answer-sets as equilibrium models. We achieve this by extending Ferraris' work on answer sets for propositional theories to the epistemic case and reveal the relationship between some ES-semantic proposals.

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. In many applications, however, timing constraints are important like, for instance, when planning and scheduling go hand in hand. We address this by developing a metric extension of linear-time Dynamic Equilibrium Logic, in which dynamic operators are constrained by intervals over integers. The resulting Metric Dynamic Equilibrium Logic provides the foundation of an ASP-based approach for specifying qualitative and quantitative dynamic constraints. As such, it constitutes the most general among a whole spectrum of temporal extensions of Equilibrium Logic. In detail, we show that it encompasses Temporal, Dynamic, Metric, and regular Equilibrium Logic, as well as its classic counterparts once the law of the excluded middle is added.

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 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 Metric Equilibrium Logic and Monadic Quantified Equilibrium Logic, respectively. Interestingly, our translation provides a blue print for implementation in terms of ASP modulo difference constraints.

Possibilistic logic is a well-known logic for reasoning under uncertainty, which is based on the idea that the epistemic state of an agent can be modeled by assigning to each possible world a degree of possibility, taken from a totally ordered, but essentially qualitative scale. Recently, a generalization has been proposed that extends possibilistic logic to a meta-epistemic logic, endowing it with the capability of reasoning about epistemic states, rather than merely constraining them. In this paper, we further develop this generalized possibilistic logic (GPL). We introduce an axiomatization showing that GPL is a fragment of a graded version of the modal logic KD, and we prove soundness and completeness w.r.t. a semantics in terms of possibility distributions. Next, we reveal a close link between the well-known stable model semantics for logic programming and the notion of minimally specific models in GPL. More generally, we analyze the relationship between the equilibrium logic of Pearce and GPL, showing that GPL can essentially be seen as a generalization of equilibrium logic, although its notion of minimal specificity is slightly more demanding than the notion of minimality underlying equilibrium logic.

We present an overview on Temporal Logic Programming under the perspective of its application for Knowledge Representation and declarative problem solving. Such programs are the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). We focus on recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL, but performs a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). We obtain a proper extension of the stable models semantics for the general case of arbitrary temporal formulas. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between infinite and finite traces. We also provide other useful results, such as the translation into other formalisms like Quantified Equilibrium Logic or Second-order LTL, and some techniques for computing temporal stable models based on automata. In a second part, we focus on practical aspects, defining a syntactic fragment called temporal logic programs closer to ASP, and explain how this has been exploited in the construction of the solver TELINGO.

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. Under consideration for acceptance in TPLP.

In a recent line of research, two familiar concepts from logic programming semantics (unfounded sets and splitting) were extrapolated to the case of epistemic logic programs. The property of epistemic splitting provides a natural and modular way to understand programs without epistemic cycles but, surprisingly, was only fulfilled by Gelfond's original semantics (G91), among the many proposals in the literature. On the other hand, G91 may suffer from a kind of self-supported, unfounded derivations when epistemic cycles come into play. Recently, the absence of these derivations was also formalised as a property of epistemic semantics called foundedness. Moreover, a first semantics proved to satisfy foundedness was also proposed, the so-called Founded Autoepistemic Equilibrium Logic (FAEEL). In this paper, we prove that FAEEL also satisfies the epistemic splitting property something that, together with foundedness, was not fulfilled by any other approach up to date. To prove this result, we provide an alternative characterisation of FAEEL as a combination of G91 with a simpler logic we called Founded Epistemic Equilibrium Logic (FEEL), which is somehow an extrapolation of the stable model semantics to the modal logic S5.

Defined by Gelfond in 1991 (G91), epistemic specifications (or programs) are an extension of logic programming under stable models semantics that introduces subjective literals. A subjective literal allows checking whether some regular literal is true in all (or in some of) the stable models of the program, being those models collected in a set called world view. One epistemic program may yield several world views but, under the original G91 semantics, some of them resulted from selfsupported derivations. During the last eight years, several alternative approaches have been proposed to get rid of these self-supported world views. Unfortunately, their success could only be measured by studying their behaviour on a set of common examples in the literature, since no formal property of "self-supportedness" had been defined. To fill this gap, we extend in this paper the idea of unfounded set from standard logic programming to the epistemic case. We define when a world view is founded with respect to some program and propose the foundedness property for any semantics whose world views are always founded. Using counterexamples, we explain that the previous approaches violate foundedness, and proceed to propose a new semantics based on a combination of Moore's Autoepistemic Logic and Pearce's Equilibrium Logic. The main result proves that this new semantics precisely captures the set of founded G91 world views.

Possibilistic logic is a well-known logic for reasoning under uncertainty, which is based on the idea that the epistemic state of an agent can be modeled by assigning to each possible world a degree of possibility, taken from a totally ordered, but essentially qualitative scale. Recently, a generalization has been proposed that extends possibilistic logic to a meta-epistemic logic, endowing it with the capability of reasoning about epistemic states, rather than merely constraining them. In this paper, we further develop this generalized possibilistic logic (GPL). We introduce an axiomatization showing that GPL is a fragment of a graded version of the modal logic KD, and we prove soundness and completeness w.r.t. a semantics in terms of possibility distributions. Next, we reveal a close link between the well-known stable model semantics for logic programming and the notion of minimally specific models in GPL. More generally, we analyze the relationship between the equilibrium logic of Pearce and GPL, showing that GPL can essentially be seen as a generalization of equilibrium logic, although its notion of minimal specificity is slightly more demanding than the notion of minimality underlying equilibrium logic.

Interpolation is an important property of classical and many non-classical logics that has been shown to have interesting applications in computer science and AI. Here we study the Interpolation Property for the the non-monotonic system of equilibrium logic, establishing weaker or stronger forms of interpolation depending on the precise interpretation of the inference relation. These results also yield a form of interpolation for ground logic programs under the answer sets semantics. For disjunctive logic programs we also study the property of uniform interpolation that is closely related to the concept of variable forgetting. The first-order version of equilibrium logic has analogous Interpolation properties whenever the collection of equilibrium models is (first-order) definable. Since this is the case for so-called safe programs and theories, it applies to the usual situations that arise in practical answer set programming.