In this note, we introduce the notion of support graph to define explanations for any model of a logic program. An explanation is an acyclic support graph that, for each true atom in the model, induces a proof in terms of program rules represented by labels. A classical model may have zero, one or several explanations: when it has at least one, it is called a justified model. We prove that all stable models are justified, whereas, for disjunctive programs, some justified models may not be stable. We also provide a meta-programming encoding in Answer Set Programming that generates the explanations for a given stable model of some program. We prove that the encoding is sound and complete, that is, there is a one-to-one correspondence between each answer set of the encoding and each explanation for the original stable model.

Answer set programming (ASP) is a popular declarative programming paradigm with various applications. Programs can easily have many answer sets that cannot be enumerated in practice, but counting still allows quantifying solution spaces. If one counts under assumptions on literals, one obtains a tool to comprehend parts of the solution space, so-called answer set navigation. However, navigating through parts of the solution space requires counting many times, which is expensive in theory. Knowledge compilation compiles instances into representations on which counting works in polynomial time. However, these techniques exist only for conjunctive normal form (CNF) formulas, and compiling ASP programs into CNF formulas can introduce an exponential overhead. This paper introduces a technique to iteratively count answer sets under assumptions on knowledge compilations of CNFs that encode supported models. Our anytime technique uses the inclusion–exclusion principle to improve bounds by over- and undercounting systematically. In a preliminary empirical analysis, we demonstrate promising results. After compiling the input (offline phase), our approach quickly (re)counts.

We present the Answer Set Programming (ASP)-based visualization tool clingraph, which aims at visualizing various concepts of ASP by means of ASP itself. This idea traces back to the aspviz tool and clingraph redevelops and extends it in the context of modern ASP systems. More precisely, clingraph takes graph specifications in terms of ASP facts and hands them over to the graph visualization system graphviz. The use of ASP provides a great interface between logic programs and/or answer sets and their visualization. Also, clingraph offers a Python application programming interface (API) that extends this ease of interfacing to clingo’s API and in turn to connect and monitor various aspects of the solving process.

Epistemic logic programs (ELPs), extend answer set programming (ASP) with epistemic operators. The semantics of such programs is provided in terms of world views, which are sets of belief sets, that is, syntactically, sets of sets of atoms. Different semantic approaches propose different characterizations of world views. Recent work has introduced semantic properties that should be met by any semantics for ELPs, like the Epistemic Splitting Property, that, if satisfied, allows to modularly compute world views in a bottom-up fashion, analogously to “traditional” ASP. We analyze the possibility of changing the perspective, shifting from a bottom-up to a top-down approach to splitting. We propose a basic top-down approach, which we prove to be equivalent to the bottom-up one. We then propose an extended approach, where our new definition: (i) is provably applicable to many of the existing semantics; (ii) operates similarly to “traditional” ASP; (iii) provably coincides under any semantics with the bottom-up notion of splitting at least on the class of Epistemically Stratified Programs (which are, intuitively, those where the use of epistemic operators is stratified); (iv) better adheres to common ASP programming methodology.

As the popularity of adhesive joints in industry increases, so does the need for tools to support the process of selecting a suitable adhesive. While some such tools already exist, they are either too limited in scope or offer too little flexibility in use. This work presents a more advanced tool, that was developed together with a team of adhesive experts. We first extract the experts’ knowledge about this domain and formalize it in a Knowledge Base (KB). The IDP-Z3 reasoning system can then be used to derive the necessary functionality from this KB. Together with a user-friendly interactive interface, this creates an easy-to-use tool capable of assisting the adhesive experts. To validate our approach, we performed user testing in the form of qualitative interviews. The experts are very positive about the tool, stating that, among others, it will help save time and find more suitable adhesives.

Program completion is a translation from the language of logic programs into the language of first-order theories. Its original definition has been extended to programs that include integer arithmetic, accept input, and distinguish between output predicates and auxiliary predicates. For tight programs, that generalization of completion is known to match the stable model semantics, which is the basis of answer set programming. We show that the tightness condition in this theorem can be replaced by a less restrictive “local tightness” requirement. From this fact we conclude that the proof assistant anthem-p2p can be used to verify equivalence between locally tight programs.

Answer set programming (ASP) is a popular declarative programming language for solving hard combinatorial problems. Although ASP has gained widespread acceptance in academic and industrial contexts, there are certain user groups who may find it more advantageous to employ a higher-level language that closely resembles natural language when specifying ASP programs. In this paper, we propose a novel tool, called CNL2ASP, for translating English sentences expressed in a controlled natural language (CNL) form into ASP. In particular, we first provide a definition of the type of sentences allowed by our CNL and their translation as ASP rules and then exemplify the usage of the CNL for the specification of both synthetic and real-world combinatorial problems. Finally, we report the results of an experimental analysis conducted on the real-world problems to compare the performance of automatically generated encodings with the ones written by ASP practitioners, showing that our tool can obtain satisfactory performance on these benchmarks.

Given a conditional sentence “${\varphi}\Rightarrow \psi$" (if ${\varphi}$ then $\psi$) and respective facts, four different types of inferences are observed in human reasoning: Affirming the antecedent (AA) (or modus ponens) reasons $\psi$ from ${\varphi}$; affirming the consequent (AC) reasons ${\varphi}$ from $\psi$; denying the antecedent (DA) reasons $\neg\psi$ from $\neg{\varphi}$; and denying the consequent (DC) (or modus tollens) reasons $\neg{\varphi}$ from $\neg\psi$. Among them, AA and DC are logically valid, while AC and DA are logically invalid and often called logical fallacies. Nevertheless, humans often perform AC or DA as pragmatic inference in daily life. In this paper, we realize AC, DA and DC inferences in answer set programming. Eight different types of completion are introduced, and their semantics are given by answer sets. We investigate formal properties and characterize human reasoning tasks in cognitive psychology. Those completions are also applied to commonsense reasoning in AI.

To answer database queries over incomplete data, the gold standard is finding certain answers: those that are true regardless of how incomplete data is interpreted. Such answers can be found efficiently for conjunctive queries and their unions, even in the presence of constraints. With negation added, the problem becomes intractable however. We concentrate on the complexity of certain answers under constraints and on effficiently answering queries outside the usual classes of (unions) of conjunctive queries by means of rewriting as Datalog and first-order queries. We first notice that there are three different ways in which query answering can be cast as a decision problem. We complete the existing picture and provide precise complexity bounds on all versions of the decision problem, for certain and best answers. We then study a well-behaved class of queries that extends unions of conjunctive queries with a mild form of negation. We show that for them, certain answers can be expressed in Datalog with negation, even in the presence of functional dependencies, thus making them tractable in data complexity. We show that in general, Datalog cannot be replaced by first-order logic, but without constraints such a rewriting can be done in first order.

Tau Prolog is a client-side Prolog interpreter fully implemented in JavaScript, which aims at implementing the ISO Prolog Standard. Tau Prolog has been developed to be used with either Node.js or a browser seamlessly, and therefore, it has been developed following a non-blocking, callback-based approach to avoid blocking web browsers. Taking the best from JavaScript and Prolog, Tau Prolog allows the programmer to handle browser events and manipulate the Document Object Model (DOM) of a web using Prolog predicates. In this paper we describe the architecture of Tau Prolog and its main packages for interacting with the Web, and we present its programming environment.

In previous work, summarized in this paper, we proposed an operation of parallel composition for rewriting-logic theories, allowing compositional specification of systems and reusability of components. The present paper focuses on compositional verification. We show how the assume/guarantee technique can be transposed to our setting, by giving appropriate definitions of satisfaction based on transition structures and path semantics. We also show that simulation and equational abstraction can be done componentwise. Appropriate concepts of fairness and deadlock for our composition operation are discussed, as they affect satisfaction of temporal formulas. We keep in parallel a distributed and a global view of composed systems. We show that these views are equivalent and interchangeable, which may help our intuition and also has practical uses as, for example, it allows global-style verification of a modularly specified system. Under consideration in Theory and Practice of Logic Programming (TPLP).

Existential rules form an expressive ${{\textsf{Datalog}}}$-based language to specify ontological knowledge. The presence of existential quantification in rule-heads, however, makes the main reasoning tasks undecidable. To overcome this limitation, in the last two decades, a number of classes of existential rules guaranteeing the decidability of query answering have been proposed. Unfortunately, only some of these classes fully encompass ${{\textsf{Datalog}}}$ and, often, this comes at the price of higher computational complexity. Moreover, expressive classes are typically unable to exploit tools developed for classes exhibiting lower expressiveness. To mitigate these shortcomings, this paper introduces a novel general syntactic condition that allows us to define, systematically and in a uniform way, from any decidable class $\mathcal{C}$ of existential rules, a new class called ${{\textsf{Dyadic-}\mathcal{C}}}$ enjoying the following properties: (i) it is decidable; (ii) it generalizes ${{\textsf{Datalog}}}$; (iii) it generalizes $\mathcal{C}$; (iv) it can effectively exploit any reasoner for query answering over $\mathcal{C}$; and (v) its computational complexity does not exceed the highest between the one of $\mathcal{C}$ and the one of ${{\textsf{Datalog}}}$.

Data exchange, the problem of transferring data from a source schema to a target schema, has been studied for several years. The semantics of answering positive queries over the target schema has been defined in early work, but little attention has been paid to more general queries. A few proposals of semantics for more general queries exist but they either do not properly extend the standard semantics under positive queries, giving rise to counterintuitive answers, or they make query answering undecidable even for the most important data exchange settings, for example, with weakly-acyclic dependencies.

The goal of this paper is to provide a new semantics for data exchange that is able to deal with general queries. At the same time, we want our semantics to coincide with the classical one when focusing on positive queries, and to not trade-off too much in terms of complexity of query answering. We show that query answering is undecidable in general under the new semantics, but it is $\text{co}\text{NP}\text{-complete}$ when the dependencies are weakly-acyclic. Moreover, in the latter case, we show that exact answers under our semantics can be computed by means of logic programs with choice, thus exploiting existing efficient systems. For more efficient computations, we also show that our semantics allows for the construction of a representative target instance, similar in spirit to a universal solution, that can be exploited for computing approximate answers in polynomial time.

Many practical problems can be understood as the search for a state of affairs that extends a fixed partial state of affairs, the environment, while satisfying certain conditions that are formally specified. Such problems are found in, for example, engineering, law or economics. We study this class of problems in a context where some of the relevant information about the environment is not known by the user at the start of the search. During the search, the user may consider tentative solutions that make implicit hypotheses about these unknowns. To ensure that the solution is appropriate, these hypotheses must be verified by observing the environment. Furthermore, we assume that, in addition to knowledge of what constitutes a solution, knowledge of general laws of the environment is also present. We formally define partial solutions with enough verified facts to guarantee the existence of complete and appropriate solutions. Additionally, we propose an interactive system to assist the user in their search by determining (1) which hypotheses implicit in a tentative solution must be verified in the environment, and (2) which observations can bring useful information for the search. We present an efficient method to over-approximate the set of relevant information, and evaluate our implementation.

We introduce negation under the stable model semantics in DatalogMTL – a temporal extension of Datalog with metric temporal operators. As a result, we obtain a rule language which combines the power of answer set programming with the temporal dimension provided by metric operators. We show that, in this setting, reasoning becomes undecidable over the rational timeline, and decidable in ${{\rm E}{\small\rm XP}{\rm S}{\small\rm PACE}}$ in data complexity over the integer timeline. We also show that, if we restrict our attention to forward-propagating programs, reasoning over the integer timeline becomes ${{\rm PS}{\small\rm PACE}}$-complete in data complexity, and hence, no harder than over positive programs; however, reasoning over the rational timeline in this fragment remains undecidable.

The tension between deduction and induction is perhaps the most fundamental issue in areas such as philosophy, cognition, and artificial intelligence. In an influential paper, Valiant recognized that the challenge of learning should be integrated with deduction. In particular, he proposed a semantics to capture the quality possessed by the output of probably approximately correct (PAC) learning algorithms when formulated in a logic. Although weaker than classical entailment, it allows for a powerful model-theoretic framework for answering queries. In this paper, we provide a new technical foundation to demonstrate PAC learning with multi-agent epistemic logics. To circumvent the negative results in the literature on the difficulty of robust learning with the PAC semantics, we consider so-called implicit learning where we are able to incorporate observations to the background theory in service of deciding the entailment of an epistemic query. We prove correctness of the learning procedure and discuss results on the sample complexity, that is how many observations we will need to provably assert that the query is entailed given a user-specified error bound. Finally, we investigate under what circumstances this algorithm can be made efficient. On the last point, given that reasoning in epistemic logics especially in multi-agent epistemic logics is PSPACE-complete, it might seem like there is no hope for this problem. We leverage some recent results on the so-called Representation Theorem explored for single-agent and multi-agent epistemic logics with the only knowing operator to reduce modal reasoning to propositional reasoning.

In answer set programming (ASP), the user can define declaratively a problem and solve it with efficient solvers; practical applications of ASP are countless and several constraint problems have been successfully solved with ASP. On the other hand, solution time usually grows in a superlinear way (often, exponential) with respect to the size of the instance, which is impractical for large instances. A widely used approach is to split the optimization problem into subproblems (SPs) that are solved in sequence, some committing to the values assigned by others, and reconstructing a valid assignment for the whole problem by juxtaposing the solutions of the single SPs. On the one hand, this approach is much faster due to the superlinear behavior; on the other hand, it does not provide any guarantee of optimality: committing to the assignment of one SP can rule out the optimal solution from the search space. In other research areas, logic-Based Benders decomposition (LBBD) proved effective; in LBBD, the problem is decomposed into a master problem (MP) and one or several SPs. The solution of the MP is passed to the SPs that can possibly fail. In case of failure, a no-good is returned to the MP that is solved again with the addition of the new constraint. The solution process is iterated until a valid solution is obtained for all the SPs or the MP is proven infeasible. The obtained solution is provably optimal under very mild conditions. In this paper, we apply for the first time LBBD to ASP, exploiting an application in health care as case study. Experimental results show the effectiveness of the approach. We believe that the availability of LBBD can further increase the practical applicability of ASP technologies.

We establish a novel relation between delete-free planning, an important task for the AI planning community also known as relaxed planning, and logic programming. We show that given a planning problem, all subsets of actions that could be ordered to produce relaxed plans for the problem can be bijectively captured with stable models of a logic program describing the corresponding relaxed planning problem. We also consider the supported model semantics of logic programs, and introduce one causal and one diagnostic encoding of the relaxed planning problem as logic programs, both capturing relaxed plans with their supported models. Our experimental results show that these new encodings can provide major performance gain when computing optimal relaxed plans, with our diagnostic encoding outperforming state-of-the-art approaches to relaxed planning regardless of the given time limit when measured on a wide collection of STRIPS planning benchmarks.

Fairness in machine learning is of considerable interest in recent years owing to the propensity of algorithms trained on historical data to amplify and perpetuate historical biases. In this paper, we argue for a formal reconstruction of fairness definitions, not so much to replace existing definitions but to ground their application in an epistemic setting and allow for rich environmental modeling. Consequently we look into three notions: fairness through unawareness, demographic parity and counterfactual fairness, and formalize these in the epistemic situation calculus.