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.

We investigate the decidability and computational complexity of conservative extensions and the related notions of inseparability and entailment in Horn description logics (DLs) with inverse roles. We consider both query conservative extensions, defined by requiring that the answers to all conjunctive queries are left unchanged, and deductive conservative extensions, which require that the entailed concept inclusions, role inclusions, and functionality assertions do not change. Upper bounds for query conservative extensions are particularly challenging because characterizations in terms of unbounded homomorphisms between universal models, which are the foundation of the standard approach to establishing decidability, fail in the presence of inverse roles. We resort to a characterization that carefully mixes unbounded and bounded homomorphisms and enables a decision procedure that combines tree automata and a mosaic technique. Our main results are that query conservative extensions are 2ExpTime-complete in all DLs between ELI and Horn-ALCHIF and between Horn-ALC and Horn-ALCHIF, and that deductive conservative extensions are 2ExpTime-complete in all DLs between ELI and ELHIF_\bot. The same results hold for inseparability and entailment.

We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the guarded negation fragment GNF, and the DL $\mathcal{ALCFIO}$, and that strong separability is decidable in $\mathcal{ALCI}$, GF, and GNF. We also provide (mostly tight) complexity bounds.

We investigate the decidability and computational complexity of conservative extensions and the related notions of inseparability and entailment in Horn description logics (DLs) with inverse roles. We consider both query conservative extensions, defined by requiring that the answers to all conjunctive queries are left unchanged, and deductive conservative extensions, which require that the entailed concept inclusions, role inclusions, and functionality assertions do not change. Upper bounds for query conservative extensions are particularly challenging because characterizations in terms of unbounded homomorphisms between universal models, which are the foundation of the standard approach to establishing decidability, fail in the presence of inverse roles. We resort to a characterization that carefully mixes unbounded and bounded homomorphisms and enables a decision procedure that combines tree automata and a mosaic technique. Our main results are that query conservative extensions are 2ExpTime-complete in all DLs between ELI and Horn-ALCHIF and between Horn-ALC and Horn-ALCHIF, and that deductive conservative extensions are 2ExpTime-complete in all DLs between ELI and ELHIF_bot. The same results hold for inseparability and entailment.

Answer set programming is a prominent declarative programming paradigm used in formulating combinatorial search problems and implementing distinct knowledge representation formalisms. It is common that several related and yet substantially different answer set programs exist for a given problem. Sometimes these encodings may display significantly different performance. Uncovering {\em precise formal} links between these programs is often important and yet far from trivial. This paper claims the correctness of a number of interesting program rewritings.

Dependence is an important concept for many tasks in artificial intelligence. A task can be executed more efficiently by discarding something independent from the task. In this paper, we propose two novel notions of dependence in propositional logic: formula-formula dependence and formula forgetting. The first is a relation between formulas capturing whether a formula depends on another one, while the second is an operation that returns the strongest consequence independent of a formula. We also apply these two notions in two well-known issues: belief update and conservative extension. Firstly, we define a new update operator based on formula-formula dependence. Furthermore, we reduce conservative extension to formula forgetting.

The question whether an ontology can safely be replaced by another, possibly simpler, one is fundamental for many ontology engineering and maintenance tasks. It underpins, for example, ontology versioning, ontology modularization, forgetting, and knowledge exchange. What safe replacement means depends on the intended application of the ontology. If, for example, it is used to query data, then the answers to any relevant ontology-mediated query should be the same over any relevant data set; if, in contrast, the ontology is used for conceptual reasoning, then the entailed subsumptions between concept expressions should coincide. This gives rise to different notions of ontology inseparability such as query inseparability and concept inseparability, which generalize corresponding notions of conservative extensions. We survey results on various notions of inseparability in the context of description logic ontologies, discussing their applications, useful model-theoretic characterizations, algorithms for determining whether two ontologies are inseparable (and, sometimes, for computing the difference between them if they are not), and the computational complexity of this problem.

Dependence is an important concept for many tasks in artificial intelligence. A task can be executed more efficiently by discarding something independent from the task. In this paper, we propose two novel notions of dependence in propositional logic: formula-formula dependence and formula forgetting. The first is a relation between formulas capturing whether a formula depends on another one, while the second is an operation that returns the strongest consequence independent of a formula. We also apply these two notions in two well-known issues: belief update and conservative extension. Firstly, we define a new update operator based on formula-formula dependence. Furthermore, we reduce conservative extension to formula forgetting.

Causal models defined in terms of structural equations have proved to be quite a powerful way of representing knowledge regarding causality. However, a number of authors have given examples that seem to show that the Halpern-Pearl (HP) definition of causality gives intuitively unreasonable answers. Here it is shown that, for each of these examples, we can give two stories consistent with the description in the example, such that intuitions regarding causality are quite different for each story. By adding additional variables, we can disambiguate the stories. Moreover, in the resulting causal models, the HP definition of causality gives the intuitively correct answer. It is also shown that, by adding extra variables, a modification to the original HP definition made to deal with an example of Hopkins and Pearl may not be necessary. Given how much can be done by adding extra variables, there might be a concern that the notion of causality is somewhat unstable. Can adding extra variables in a "conservative" way (i.e., maintaining all the relations between the variables in the original model) cause the answer to the question "Is X=x a cause of Y=y" to alternate between "yes" and "no"? It is shown that we can have such alternation infinitely often, but if we take normality into consideration, we cannot. Indeed, under appropriate normality assumptions. adding an extra variable can change the answer from "yes" to "no", but after that, it cannot cannot change back to "yes".

An important subclass of social choice functions, so-called majoritarian (or C1) functions, only take into account the pairwise majority relation between alternatives. In the absence of majority ties--e.g., when there is an odd number of agents with linear preferences--the majority relation is antisymmetric and complete and can thus conveniently be represented by a tournament. Tournaments have a rich mathematical theory and many formal results for majoritarian functions assume that the majority relation constitutes a tournament. Moreover, most majoritarian functions have only been defined for tournaments and allow for a variety of generalizations to unrestricted preference profiles, none of which can be seen as the unequivocal extension of the original function. In this paper, we argue that restricting attention to tournaments is justified by the existence of a conservative extension, which inherits most of the commonly considered properties from its underlying tournament solution.