Ranking-based Argumentation Semantics Applied to Logical Argumentation (full version)

In formal argumentation, a distinction can be made between extension-based semantics, where sets of arguments are either (jointly) accepted or not, and ranking-based semantics, where grades of acceptability are assigned to arguments. Another important distinction is that between abstract approaches, that abstract away from the content of arguments, and structured approaches, that specify a method of constructing argument graphs on the basis of a knowledge base. While ranking-based semantics have been extensively applied to abstract argumentation, few work has been done on ranking-based semantics for structured argumentation. In this paper, we make a systematic investigation into the behaviour of ranking-based semantics applied to existing formalisms for structured argumentation. We show that a wide class of ranking-based semantics gives rise to so-called culpability measures, and are relatively robust to specific choices in argument construction methods.

Argumentative Characterizations of (Extended) Disjunctive Logic Programs

This paper continues an established line of research about the relations between argumentation theory, particularly assumption-based argumentation, and different kinds of logic programs. In particular, we extend known result of Caminada, Schultz and Toni by showing that assumption-based argumentation can represent not only normal logic programs, but also disjunctive logic programs and their extensions. For this, we consider some inference rules for disjunction that the core logic of the argumentation frameworks should respect, and show the correspondence to the handling of disjunctions in the heads of the logic programs' rules.

Non-deterministic approximation operators: ultimate operators, semi-equilibrium semantics and aggregates (full version)

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, i.e.\ operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of non-deterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola, et al., and (3) we generalize the characterisations of disjunctive logic programs to disjunctive logic programs with aggregates.

Non-Deterministic Approximation Fixpoint Theory and Its Application in Disjunctive Logic Programming

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to dealing with non-deterministic constructs that allow to handle indefinite information, represented e.g. by disjunctive formulas. This is done by generalizing the main constructions and corresponding results of AFT to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

On Nested Justification Systems (full version)

Justification theory is a general framework for the definition of semantics of rule-based languages that has a high explanatory potential. Nested justification systems, first introduced by Denecker et al. (2015), allow for the composition of justification systems. This notion of nesting thus enables the modular definition of semantics of rule-based languages, and increases the representational capacities of justification theory. As we show in this paper, the original semantics for nested justification systems lead to the loss of information relevant for explanations. In view of this problem, we provide an alternative characterization of semantics of nested justification systems and show that this characterization is equivalent to the original semantics. Furthermore, we show how nested justification systems allow representing fixpoint definitions (Hou and Denecker 2009).

Assumption-Based Approaches to Reasoning with Priorities

This paper maps out the relation between different approaches for handling preferences in argumentation with strict rules and defeasible assumptions by offering translations between them. The systems we compare are: non-prioritized defeats i.e. attacks, preference-based defeats, and preference-based defeats extended with reverse defeat.

Reasoning by Cases in Structured Argumentation

We extend the $ASPIC^+$ framework for structured argumentation so as to allow applications of the reasoning by cases inference scheme for defeasible arguments. Given an argument with conclusion `$A$ or $B$', an argument based on $A$ with conclusion $C$, and an argument based on $B$ with conclusion $C$, we allow the construction of an argument with conclusion $C$. We show how our framework leads to different results than other approaches in non-monotonic logic for dealing with disjunctive information, such as disjunctive default theory or approaches based on the OR-rule (which allows to derive a defeasible rule `If ($A$ or $B$) then $C$', given two defeasible rules `If $A$ then $C$' and `If $B$ then $C$'). We raise new questions regarding the subtleties of reasoning defeasibly with disjunctive information, and show that its formalization is more intricate than one would presume.

Relations between assumption-based approaches in nonmonotonic logic and formal argumentation

In this paper we make a contribution to the unification of formal models of defeasible reasoning. We present several translations between formal argumentation frameworks and nonmonotonic logics for reasoning with plausible assumptions. More specifically, we translate adaptive logics into assumption-based argumentation and ASPIC+, ASPIC+ into assumption-based argumentation and a fragment of assumption-based argumentation into adaptive logics. Adaptive logics are closely related to Makinson's default assumptions and to a significant class of systems within the tradition of preferential semantics in the vein of KLM and Shoham. Thus, our results also provide close links between formal argumentation and the latter approaches.