Gabriele Kern-Isberner

Andre Thevapalan, Gabriele Kern-Isberner

Answer set programs used in real-world applications often require that the program is usable with different input data. This, however, can often lead to contradictory statements and consequently to an inconsistent program. Causes for potential contradictions in a program are conflicting rules. In this paper, we show how to ensure that a program $\mathcal{P}$ remains non-contradictory given any allowed set of such input data. For that, we introduce the notion of conflict-resolving $λ$- extensions. A conflict-resolving $λ$-extension for a conflicting rule $r$ is a set $λ$ of (default) literals such that extending the body of $r$ by $λ$ resolves all conflicts of $r$ at once. We investigate the properties that suitable $λ$-extensions should possess and building on that, we develop a strategy to compute all such conflict-resolving $λ$-extensions for each conflicting rule in $\mathcal{P}$. We show that by implementing a conflict resolution process that successively resolves conflicts using $λ$-extensions eventually yields a program that remains non-contradictory given any allowed set of input data.

Christoph Beierle, Alexander Hahn, Diana Howey et al.

Forgetting as a knowledge management operation deliberately ignores parts of the knowledge and beliefs of an agent, for various reasons. Forgetting has many facets, one may want to forget parts of the syntax, a proposition, or a conditional. In the literature, two main operators suitable for performing forgetting have been proposed and investigated in depth: First, variable elimination is a syntactical method that blends out certain atomic variables to focus on the rest of the language. It has been mainly used in the area of logic programming and answer set programming. Second, contraction in AGM belief revision theory effectively removes propositions from belief sets under logical deduction. Both operations rely mainly on classical logics. In this article, we take an epistemic perspective and study forgetting operations in epistemic states with richer semantic structures, but with clear links to propositional logic. This allows us to investigate what forgetting in the epistemic background means, thereby lifting well-known and novel forgetting operations to the epistemic level. We present five general types of epistemic forgetting and instantiate them with seven concrete forgetting operations for Spohn's ranking functions. We take inspiration from postulates of forgetting both from logic programming and AGM theory to propose a rich landscape of axioms for evaluating forgetting operations. Finally, we evaluate all concrete forgetting operations according to all postulates, leading to a novel comprehensive overview highlighting differences and commonalities among the forgetting operators.

Kenneth Skiba, Tjitze Rienstra, Matthias Thimm et al.

In this paper, we present a general framework for ranking sets of arguments in abstract argumentation based on their plausibility of acceptance. We present a generalisation of Dung's extension semantics as extension-ranking semantics, which induce a preorder over the power set of all arguments, allowing us to state that one set is "closer" to being acceptable than another. To evaluate the extension-ranking semantics, we introduce a number of principles that a well-behaved extension-ranking semantics should satisfy. We consider several simple base relations, each of which models a single central aspect of argumentative reasoning. The combination of these base relations provides us with a family of extension-ranking semantics. We also adapt a number of approaches from the literature for ranking extensions to be usable in the context of extension-ranking semantics, and evaluate their behaviour.

Kai Sauerwald, Gabriele Kern-Isberner, Christoph Beierle

In this article, we consider iteration principles for contraction, with the goal of identifying properties for contractions that respect conditional beliefs. Therefore, we investigate and evaluate four groups of iteration principles for contraction which consider the dynamics of conditional beliefs. For all these principles, we provide semantic characterization theorems and provide formulations by postulates which highlight how the change of beliefs and of conditional beliefs is constrained, whenever that is possible. The first group is similar to the syntactic Darwiche-Pearl postulates. As a second group, we consider semantic postulates for iteration of contraction by Chopra, Ghose, Meyer and Wong, and by Konieczny and Pino Pérez, respectively, and we provide novel syntactic counterparts. Third, we propose a contraction analogue of the independence condition by Jin and Thielscher. For the fourth group, we consider natural and moderate contraction by Nayak. Methodically, we make use of conditionals for contractions, so-called contractionals and furthermore, we propose and employ the novel notion of $ α$-equivalence for formulating some of the new postulates.

Marco Wilhelm, Diana Howey, Gabriele Kern-Isberner et al.

Activation-based conditional inference applies conditional reasoning to ACT-R, a cognitive architecture developed to formalize human reasoning. The idea of activation-based conditional inference is to determine a reasonable subset of a conditional belief base in order to draw inductive inferences in time. Central to activation-based conditional inference is the activation function which assigns to the conditionals in the belief base a degree of activation mainly based on the conditional's relevance for the current query and its usage history. Therewith, our approach integrates several aspects of human reasoning into expert systems such as focusing, forgetting, and remembering.

A. Becker, G. Kern-Isberner, K. Sauerwald et al.

Delgrande's knowledge level account of forgetting provides a general approach to forgetting syntax elements from sets of formulas with links to many other forgetting operations, in particular, to Boole's variable elimination. On the other hand, marginalisation of epistemic states is a specific approach to actively reduce signatures in more complex semantic frameworks, also aiming at forgetting atoms that is very well known from probability theory. In this paper, we bring these two perspectives of forgetting together by showing that marginalisation can be considered as an extension of Delgrande's approach to the level of epistemic states. More precisely, we generalize Delgrande's axioms of forgetting to forgetting in epistemic states, and show that marginalisation is the most specific and informative forgetting operator that satisfies these axioms. Moreover, we elaborate suitable phrasings of Delgrande's concept of forgetting for formulas by transferring the basic ideas of the axioms to forgetting formulas from epistemic states. However, here we show that this results in trivial approaches to forgetting formulas. This finding supports the claim that forgetting syntax elements is essentially different from belief contraction, as e.g. axiomatized in the AGM belief change framework.

Kai Sauerwald, Gabriele Kern-Isberner, Christoph Beierle

The research on non-prioritized revision studies revision operators which do not accept all new beliefs. In this paper, we contribute to this line of research by introducing the concept of dynamic-limited revision, which are revisions expressible by a total preorder over a limited set of worlds. For a belief change operator, we consider the scope, which consists of those beliefs which yield success of revision. We show that for each set satisfying single sentence closure and disjunction completeness there exists a dynamic-limited revision having the union of this set with the beliefs set as scope. We investigate iteration postulates for belief and scope dynamics and characterise them for dynamic-limited revision. As an application, we employ dynamic-limited revision to studying belief revision in the context of so-called inherent beliefs, which are beliefs globally accepted by the agent. This leads to revision operators which we call inherence-limited. We present a representation theorem for inherence-limited revision, and we compare these operators and dynamic-limited revision with the closely related credible-limited revision operators.

Kai Sauerwald, Gabriele Kern-Isberner, Christoph Beierle

According to Boutillier, Darwiche, Pearl and others, principles for iterated revision can be characterised in terms of changing beliefs about conditionals. For iterated contraction a similar formulation is not known. This is especially because for iterated belief change the connection between revision and contraction via the Levi and Harper identity is not straightforward, and therefore, characterisation results do not transfer easily between iterated revision and contraction. In this article, we develop an axiomatisation of iterated contraction in terms of changing conditional beliefs. We prove that the new set of postulates conforms semantically to the class of operators like the ones given by Konieczny and Pino Pérez for iterated contraction.

Matthias Thimm, Gabriele Kern-Isberner

We introduce stratified labelings as a novel semantical approach to abstract argumentation frameworks. Compared to standard labelings, stratified labelings provide a more fine-grained assessment of the controversiality of arguments using ranks instead of the usual labels in, out, and undecided. We relate the framework of stratified labelings to conditional logic and, in particular, to the System Z ranking functions.