The goal of this article is threefold. First, we trace the history of the development of answer set solvers, by accounting for more than a dozen of them. Second, we discuss development tools and environments that facilitate the use of answer set programming technology in practical applications. Last, we present the evolution of the answer set programming competitions, prime venues for tracking advances in answer set solving technology.

Answer set programming (ASP) has emerged as an approach to declarative problem solving based on the stable model semantics for logic programs. The basic idea is to represent a computational problem by a logic program, formulating constraints in terms of rules, such that its answer sets correspond to problem solutions. To this end, ASP combines an expressive language for high-level modeling with powerful low-level reasoning capacities, provided by off-the-shelf tools. Compact problem representations take advantage of genuine modeling features of ASP, including (first-order) variables, negation by default, and recursion. In this article, we demonstrate the ASP methodology on two example scenarios, illustrating basic as well as advanced modeling and solving concepts. We also discuss mechanisms to represent and implement extended kinds of preferences and optimization. An overview of further available extensions concludes the article.

At first, a problem is expressed as a logic program. ASP's success is largely due to the availability of a rich modeling language (Gebser and Schaub 2016) along with effective systems. Early ASP solvers SModels (Simons, Niemelä, and Soininen 2002) and DLV (Leone et al. 2006) were followed by SAT DLV (Faber, Leone, and Perri 2012) or GrinGo (Gebser ground rules, corresponding to the number of net al. 2011) are based on seminaive database evaluation tuples, over a set of two elements. For more details techniques (Ullman 1988) for avoiding duplicate about complexity of ASP the reader may refer to work during grounding. Grounding is seen as an iterative Dantsin et al. (2001).

In addition, we illustrate the potential of ASP including molecular biology (Gebser et computational hardness of our application problem al. 2010a, 2010b), decision support system for space by explaining its connection to the NPcomplete shuttle controllers (Balduccini, Gelfond, and decision problem Exact-3-SAT.

Its main ideas are described in the article by Janhunen and Niemelä (2016) and in other contributions to this special issue. In this introductory article my goal is to discuss the concept of an answer set, or stable model, which defines the semantics of ASP languages. The answer sets of a logic program are sets of atomic formulas without variables ("ground atoms"), and they were introduced in the course of research on the semantics of negation in Prolog. For this reason, I will start with examples illustrating the relationship between answer sets and Prolog and the relationship between answer set solvers and Prolog systems. Then I will review the mathematical definition of an answer set and discuss some extensions of the basic language of ASP.

What distinguishes ASP from other declarative paradigms, like satisfiability (SAT) or constraint solving (CSP), is its underlying modeling language and the semantics involved. Problems are specified using logic programminglike rules, with some convenient extensions facilitating compact and readable problem descriptions. Sets of such rules, or answer set programs, come with an intuitive, well-defined and, by now, well-accepted semantics. This semantics has its roots in research in knowledge representation, in particular nonmonotonic reasoning, and avoids the pitfalls of earlier attempts such as the procedural semantics of Prolog based on negation as finite failure. This semantics was originally called the stable-model semantics and was defined for normal logic programs only, that is, programs consisting of rules with a single atom in the head and any finite number of atoms, possibly preceded by default negation, not, in the body. Stable models were later generalized to broader classes of programs, where the semantics can no longer be defined in terms of sets of atoms, which is a natural representation of classical models. Instead, it was defined by means of some sets of literals. For this reason the term answer set was adopted as more adequate (although answer sets also have a straightforward interpretation as models, albeit three-valued ones). Over the last decade or so, ASP has evolved into a vibrant and active research area that produced not only theoretical insights, but also highly effective and useful software tools and interesting and promising applications.

While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool for reasoning about computer programs, we consider this problem from the perspective of integrating probability and logic. Recent work on assigning probabilities to mathematical statements has used the concept of coherent distributions, which satisfy logical constraints such as the probability of a sentence and its negation summing to one. Although there are algorithms which converge to a coherent probability distribution in the limit, this yields only weak guarantees about finite approximations of these distributions. In our setting, this is a significant limitation: Coherent distributions assign probability one to all statements provable in a specific logical theory, such as Peano Arithmetic, which can prove what the output of any terminating computation is; thus, a coherent distribution must assign probability one to the output of any terminating computation. To model uncertainty about computations, we propose to work with approximations to coherent distributions. We introduce inductive coherence, a strengthening of coherence that provides appropriate constraints on finite approximations, and propose an algorithm which satisfies this criterion.

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic $\mathcal{EL}$ to disunification since negative constraints can be used to avoid unwanted unifiers. While decidability of the solvability of general $\mathcal{EL}$-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we consider only solutions that are constructed from terms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of disunification problems.

In the summer of 2015 a team of hackers attempted to take control of an unmanned military helicopter known as Little Bird. The helicopter, which is similar to the piloted version long-favored for US special operations missions, was stationed at a Boeing facility in Arizona. The hackers had a head start: At the time they began the operation, they already had access to one part of the drone's computer system. From there, all they needed to do was hack into Little Bird's onboard flight-control computer, and the drone was theirs. When the project started, a "Red Team" of hackers could have taken over the helicopter almost as easily as it could break into your home Wi-Fi.

Graph aggregation is the process of computing a single output graph that constitutes a good compromise between several input graphs, each provided by a different source. One needs to perform graph aggregation in a wide variety of situations, e.g., when applying a voting rule (graphs as preference orders), when consolidating conflicting views regarding the relationships between arguments in a debate (graphs as abstract argumentation frameworks), or when computing a consensus between several alternative clusterings of a given dataset (graphs as equivalence relations). In this paper, we introduce a formal framework for graph aggregation grounded in social choice theory. Our focus is on understanding which properties shared by the individual input graphs will transfer to the output graph returned by a given aggregation rule. We consider both common properties of graphs, such as transitivity and reflexivity, and arbitrary properties expressible in certain fragments of modal logic. Our results establish several connections between the types of properties preserved under aggregation and the choice-theoretic axioms satisfied by the rules used. The most important of these results is a powerful impossibility theorem that generalises Arrow's seminal result for the aggregation of preference orders to a large collection of different types of graphs.