Many planning applications require not only a single solution but benefit substantially from having a set of possible plans from which users can select, for example, when explaining plans. For decades, research in classical AI planning has primarily focused on quickly finding single plans. Only recently researchers have started to investigate preferences, enumerate plans by top-k planning, or count plans to reason about the plan space. Unfortunately, reasoning about the plan space is computationally extremely hard and feeding many similar plans to the user is hardly practical. To circumvent computational shortcomings while still being able to reason about variability in plans, faceted actions have been introduced very recently. These are meaningful actions that can be used by some plan but are not required by all plans. Enforcing or forbidding such facets allows for navigating even large plan spaces while ensuring desired properties quickly and step by step. In this paper, we illustrate an industrial challenge, the Beluga logistics problem of Airbus, where reasoning with facets enables targeted plan space navigation. We present an approach to handle large plan spaces iteratively and interactively and present a tool that we call PlanPilot.

ASPIC+ is one of the main general frameworks for rule-based argumentation for AI. Although first-order rules are commonly used in ASPIC+ examples, most existing approaches to reason over rule-based argumentation only support propositional rules. To enable reasoning over first-order instances, a preliminary grounding step is required. As groundings can lead to an exponential increase in the size of the input theories, intelligent procedures are needed. However, there is a lack of dedicated solutions for ASPIC+. Therefore, we propose an intelligent grounding procedure that keeps the size of the grounding manageable while preserving the correctness of the reasoning process. To this end, we translate the first-order ASPIC+ instance into a Datalog program and query a Datalog engine to obtain ground substitutions to perform the grounding of rules and contraries. Additionally, we propose simplifications specific to the ASPIC+ formalism to avoid grounding of rules that have no influence on the reasoning process. Finally, we performed an empirical evaluation of a prototypical implementation to show scalability.

Answer set programming is a popular declarative paradigm with countless applications for modeling and solving combinatorial problems. We can view a program as a knowledge database compactly representing conditions for solutions. Often we are interested in reasoning about solutions of filtering answer sets. At the heart of these questions is brave and cautious reasoning. For browsing answer sets, we combine both as restricting atoms of answer sets is only meaningful for atoms called facets that belong to some (brave) but not to all answer sets (cautious). Surprisingly, the precise computational complexity of facet problems remained widely open so far. In this paper, we study the complexity of answer set facets. We establish tight results for reasoning with facets, deciding upper and lower bounds as well as the exact number of facets, and comparing facets. Facet reasoning seems to be a natural problem formalism, residing in complexity families Σᴾ, Πᴾ, Dᴾ, and Θᴾ, up to the third level. Moreover, our study considers quantitative importance questions on facets and generalizing from facets to conjunctions, disjunctions, and arbitrary queries. We complete our results by an experimental evaluation.

Abstract Answer set programming is a well-understood and established problem-solving and knowledge representation paradigm. It has become more prominent amongst a wider audience due to its multiple applications in science and industry. The constant development of advanced programming and modeling techniques extends the toolset for developers and users regularly. This paper compiles and demonstrates different techniques to reuse logic program parts (multi-shot) by solving the arcade game snake. This game is particularly interesting because a victory can be assured by solving the NP-hard problem of Hamiltonian Cycles. We will demonstrate five hands-on implementations in clingo and compare their performance in an empirical evaluation. In addition, our implementation utilizes clingraph to generate a simple yet informative image representation of the game’s progress.

Abstract 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.

Answer set programming (ASP) is a popular problem solving paradigm with applications in planning and configuration. In practice, the number of answer sets can be overwhelmingly high, which naturally causes interest in a concise characterisation of the solution space in terms of representative answer sets. We establish a notion of representativeness that refers to the entropy of specified target atoms within a collection of answer sets. Accordingly, we propose different approaches for collecting such representative answer sets, based on answer set navigation. Finally, we conduct experiments using our prototypical implementation, which reveals promising results.

Answer set programming (ASP) is a popular declarative programming paradigm with a wide range of applications in artificial intelligence. Oftentimes, when modeling an AI problem with ASP, and in particular when we are interested beyond simple search for optimal solutions, an actual solution, differences between solutions, or number of solutions of the ASP program matter. For example, when a user aims to identify a specific answer set according to her needs, or requires the total number of diverging solutions to comprehend probabilistic applications such as reasoning in medical domains. Then, there are only certain problem specific and handcrafted encoding techniques available to navigate the solution space of ASP programs, which is oftentimes not enough. In this paper, we propose a formal and general framework for interactive navigation toward desired subsets of answer sets analogous to faceted browsing. Our approach enables the user to explore the solution space by consciously zooming in or out of sub-spaces of solutions at a certain configurable pace. We illustrate that weighted faceted navigation is computationally hard. Finally, we provide an implementation of our approach that demonstrates the feasibility of our framework for incomprehensible solution spaces.

Dialectical Frameworks [1] (ADF) are a generalisation of Dung's Argumentation frameworks [2].Multiple approaches for reasoning under various semantics have been proposed over the last decade [3,4,5,6].We present "Abstract Dialectical Frameworks solved by Binary Decision Diagrams, developed in Dresden" (ADF-BDD) 2 , a novel approach that relies on the translation of the acceptance conditions of a given ADF into reduced ordered binary decision diagrams (roBDD) [7].Our system is based on the consideration that many otherwise hard to decide problems in ADF semantics (e. g., answering SAT-questions) can be solved in polynomial time on roBDDs (see [8] for an in-depth analysis).Our novel approach differs to the currently used systems, like the SAT-based approach K++ADF [5] or the wide spectrum of answer set programming (ASP) focused approaches like the DIAMOND family (e.g., DIAMOND [3] or GODIA-MOND [4]) and YADF [6].ADF-BDD is written in RUST [9] to provide good performance while enforcing a high amount of memory-and type-safety.In addition the rust-compiler produces highly optimised machine code, while keeping the whole tech stack simple.ADF-BDD accepts the established input format, introduced first in [10].There statements are unary predicates s, defining the labels and the acceptance conditions are binary predicates ac, relating the label to a formula.It allows to enumerate the grounded and complete interpretations, and stable models of the given input instance.The set of statements is the shared signature of all acceptance conditions, hence our implementation uses a single structure to store the nodes of all the roBDDs, which represent each acceptance condition.This allows for efficient caching of nodes and to eliminate duplicate node candidates.Another side-effect is that shared sub-BDDs are computed only once.ADF-BDD provides the explained implementation of roBDDs as the representation of the acceptance conditions.As the instantiation of roBDDs is a computational hard task, it is possible to utilise another state-of-the art competitive library called Biodivine/LibBDD 3 .It is part of the Biodivine software in the AEON project [11].While LibBDD is faster in 1 This work is partly supported by the BMBF, Grant 01IS20056 NAVAS, by the Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI), and by the DFG through the Collaborative Research Center, Grant TRR 248 project ID 389792660.2

Recent developments on solvers for abstract argumentation frameworks (AFs) made them capable to compute extensions for many semantics efficiently. However, for many input instances these solution spaces can become very large and incomprehensible. So far, for the further exploration and investigation of the AF solution space the user needs to use post-processing methods or handcrafted tools. To compare and explore the solution spaces of two selected semantics, we propose an approach that visually supports the user, via a combination of dimensionality reduction of argumentation extensions and a projection of extensions to sets of accepted or rejected arguments. We introduce the novel web-based visualization tool NEXAS that allows for an interactive exploration of the solution space together with a statistical analysis of the acceptance of individual arguments for the selected semantics, as well as provides an interactive correlation matrix for the acceptance of arguments. We validate the tool with a walk-through along three use cases.

We study the extension of non-monotonic disjunctive logic programs with terms that represent sets of constants, called DLP(S), under the stable model semantics. This strictly increases expressive power, but keeps reasoning decidable, though cautious entailment is coNEXPTIME^NP-complete, even for data complexity. We present two new reasoning methods for DLP(S): a semantics-preserving translation of DLP(S) to logic programming with function symbols, which can take advantage of lazy grounding techniques, and a ground-and-solve approach that uses non-monotonic existential rules in the grounding stage. Our evaluation considers problems of ontological reasoning that are not in scope for traditional ASP (unless EXPTIME =ΠP2 ), and we find that our new existential-rule grounding performs well in comparison with native implementations of set terms in ASP.

Abstract argumentation is a prominent reasoning framework. It comes with a variety of semantics and has lately been enhanced by probabilities to enable a quantitative treatment of argumentation. While admissibility is a fundamental notion for classical reasoning in abstract argumentation frameworks, it has barely been reflected so far in the probabilistic setting. In this paper, we address the quantitative treatment of abstract argumentation based on probabilistic notions of admissibility. Our approach follows the natural idea of defining probabilistic semantics for abstract argumentation by systematically imposing constraints on the joint probability distribution on the sets of arguments, rather than on probabilities of single arguments. As a result, there might be either a uniquely defined distribution satisfying the constraints, but also none, many, or even an infinite number of satisfying distributions are possible. We provide probabilistic semantics corresponding to the classical complete and stable semantics and show how labeling schemes provide a bridge from distributions back to argument labelings. In relation to existing work on probabilistic argumentation, we present a taxonomy of semantic notions. Enabled by the constraint-based approach, standard reasoning problems for probabilistic semantics can be tackled by SMT solvers, as we demonstrate by a proof-of-concept implementation.

Abstract argumentation is a prominent reasoning framework. It comes with a variety of semantics, and has lately been enhanced by probabilities to enable a quantitative treatment of argumentation. While admissibility is a fundamental notion in the classical setting, it has been merely reflected so far in the probabilistic setting. In this paper, we address the quantitative treatment of argumentation based on probabilistic notions of admissibility in a way that they form fully conservative extensions of classical notions. In particular, our building blocks are not the beliefs regarding single arguments. Instead we start from the fairly natural idea that whatever argumentation semantics is to be considered, semantics systematically induces constraints on the joint probability distribution on the sets of arguments. In some cases there might be many such distributions, even infinitely many ones, in other cases there may be one or none. Standard semantic notions are shown to induce such sets of constraints, and so do their probabilistic extensions. This allows them to be tackled by SMT solvers, as we demonstrate by a proof-of-concept implementation. We present a taxonomy of semantic notions, also in relation to published work, together with a running example illustrating our achievements.

Abstract dialectical frameworks (ADFs) are a recently introduced powerful generalization of Dung’s popular abstract argumentation frameworks (AFs). Inspired by similar work for AFs, we introduce a decomposition scheme for ADFs, which proceeds along the ADF’s strongly connected components. We find that, for several semantics, the decompositionbased version coincides with the original semantics, whereas for others, it gives rise to a new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs. We perform an exhaustive analysis of the computational complexity of these new, so-called naive-based semantics. The results are quite interesting, for some of them involve little-known classes of the so-called Boolean hierarchy (another hierarchy in between classes of the polynomial hierarchy). Furthermore, in credulous and sceptical entailment, the complexity can be different depending on whether we check for truth or falsity of a specific statement.

After decades of fruitful research, description logics (DLs) have evolved into a de facto standard in logic-based knowledge representation. In particular, they serve as the formal basis of the standardized and very popular web ontology language (OWL), which also comes with the advantage of readily available user-friendly modeling tools and optimized reasoning engines. In the course of the wide-spread adoption of OWL and DLs, situations have been observed where logically less skilled practitioners are (ab)using these formalisms as constraint languages adopting a closed-world assumption, contrary to the open-world semantics imposed by the classical definitions and the standards. To provide a clear theoretical basis and inferencing support for this often practically reasonable “off-label use” we propose an alternative formal semantics reflecting the intuitive understanding of such scenarios. To that end, we introduce the fixed-domain semantics and argue that this semantics gives rise to an interesting new inferencing task: model enumeration. We describe how the new semantics can be axiomatized in very expressive DLs. We thoroughly investigate the complexities for standard reasoning as well as query answering under the fixed-domain semantics for a wide range of DLs. Further, we present an implementation of a fixed-domain DL reasoner based on a translation into answer set programming (ASP) which is competitive with alternative approaches for standard reasoning tasks and provides the added functionality of model enumeration.

Recently, stage and cf 2 semantics for abstract argumentation attracted specific attention. By distancing from the notion of defence, they are capable to select arguments out of odd-length cycles. In case of cf 2 semantics, the SCC-recursive schema guarantees that important evaluation criteria for argumentation semantics, like directionality, weak- and CF -reinstatement, are fulfilled. Beside several desirable properties, both stage and cf 2 semantics still have some drawbacks. The stage semantics does not satisfy the above mentioned evaluation criteria, whereas cf 2 semantics produces some questionable results on frameworks with cycles of length ≥ 6. Therefore, we suggest to combine stage semantics with the SCC-recursive schema of cf 2 semantics. The resulting stage 2 semantics overcomes the problems regarding cf 2 and stage semantics. We study properties of stage 2 semantics and its relations to existing semantics, show that it fulfills the mentioned evaluation criteria, study strong equivalence for stage 2 semantics and provide a comprehensive complexity analysis of the associated reasoning problems. Besides the analysis of stage 2 semantics, we also complement existing complexity results for cf 2 by an analysis of tractable fragments and fixed parameter tractability. Furthermore, we provide answer-set programming (ASP) encodings for stage 2 semantics and labelling-based algorithms for cf 2 and stage 2 semantics.

Within the last decade, abstract argumentation has emerged as a central field in Artificial Intelligence. Besides providing a core formalism for many advanced argumentation systems, abstract argumentation has also served to capture several non-monotonic logics and other AI related principles. Although the idea of abstract argumentation is appealingly simple, several reasoning problems in this formalism exhibit high computational complexity. This calls for advanced techniques when it comes to implementation issues, a challenge which has been recently faced from different angles. In this survey, we give an overview on different methods for solving reasoning problems in abstract argumentation and compare their particular features. Moreover, we highlight available state-of-the-art systems for abstract argumentation, which put these methods to practice.

Abstract The design of efficient solutions for abstract argumentation problems is a crucial step towards advanced argumentation systems. One of the most prominent approaches in the literature is to use Answer-Set Programming (ASP) for this endeavor. In this paper, we present new encodings for three prominent argumentation semantics using the concept of conditional literals in disjunctions as provided by the ASP-system clingo. Our new encodings are not only more succinct than previous versions, but also outperform them on standard benchmarks.

We introduce a decomposition scheme for abstract dialectical frameworks (ADFs). The decomposition proceeds along the ADF's strongly connected components. For several semantics, the decomposition-based version coincides with the original semantics. For others, the scheme defines new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs.

Abstract argumentation frameworks nowadays provide the most popular formalization of argumentation on a conceptual level. Numerous semantics for this paradigm have been proposed, whereby the cf2 semantics has shown to solve particular problems concerned with odd-length cycles in such frameworks. Due to the complicated definition of this semantics it has somehow been neglected in the literature. In this article, we introduce an alternative characterization of the cf2 semantics which, roughly speaking, avoids the recursive computation of subframeworks. This facilitates further investigation steps, like a complete complexity analysis. Furthermore, we show how the notion of strong equivalence can be characterized in terms of the cf2 semantics. In contrast to other semantics, it turns out that for the cf2 semantics strong equivalence coincides with syntactical equivalence. We make this particular behaviour more explicit by defining a new property for argumentation semantics, called the succinctness property. If a semantics σ satisfies the succinctness property, then for every framework F, all its attacks contribute to the evaluation of at least one framework F′ containing F. We finally characterize strong equivalence also for the stage and the naive semantics. Together with known results these characterizations imply that none of the prominent semantics for abstract argumentation, except the cf2 semantics, satisfies the succinctness property.

We consider two instantiations of the SCC-recursive schema for argumentation semantics, cf2, using maximal conflict-free sets as base semantics, and stage2, using stage extensions as base semantics. Both of them have been shown to be in general of high complexity. We provide a detailed analysis of possible tractable fragments for these semantics. Moreover we present a labeling based algorithm for computing cf2 extension, which is complexity-sensitive w.r.t. one of the tractable fragments.

Abstract argumentation frameworks nowadays provide the most popular formalization of argumentation on a conceptual level. Numerous semantics for this paradigm have been proposed, whereby cf2 semantics has shown to nicely solve particular problems concernend with odd-length cycles in such frameworks. In order to compare different semantics not only on a theoretical basis, it is necessary to provide systems which implement them within a uniform platform. Answer-Set Programming (ASP) turned out to be a promising direction for this aim, since it not only allows for a concise representation of concepts inherent to argumentation semantics, but also offers sophisticated off-the-shelves solvers which can be used as core computation engines. In fact, many argumentation semantics have meanwhile been encoded within the ASP paradigm, but not all relevant semantics, among them cf2 semantics, have yet been considered. The contributions of this work are thus twofold. Due to the particular nature of cf2 semantics, we first provide an alternative characterization which, roughly speaking, avoids the recursive computation of sub-frameworks. Then, we provide the concrete ASP-encodings, which are incorporated within the ASPARTIX system, a platform which already implements a wide range of semantics for abstract argumentation.