dialectical frameworks (ADFs) have recently been proposed as a versatile generalization of Dung's abstract argumentation frameworks (AFs). In this paper, we present a comprehensive analysis of the computational complexity of ADFs. Our results show that while ADFs are one level up in the polynomial hierarchy compared to AFs, there is a useful subclass of ADFs which is as complex as AFs while arguably offering more modeling capacities. As a technical vehicle, we employ the approximation fixpoint theory of Denecker, Marek and Truszczyński, thus showing that it is also a useful tool for complexity analysis of operator-based semantics.

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 decomposition-based 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.

By expressiveness we mean the ability to encode a desired set of two-valued interpretations over a given propositional vocabulary A using only atoms from A. We also compare ADFs' expressiveness with that of (the two-valued semantics of) abstract argumentation frameworks, normal logic programs and propositional logic. While the computational complexity of the two-valued model existence problem for all these languages is (almost) the same, we show that the languages form a neat hierarchy with respect to their expressiveness. We then demonstrate that this hierarchy collapses once we allow to introduce a linear number of new vocabulary elements. We finally also analyse and compare the representational succinctness of ADFs (for two-valued model semantics), that is, their capability to represent two-valued interpretation sets in a space-efficient manner.

We investigate fundamental properties of three-valued semantics for abstract dialectical frameworks (ADFs). In particular, we deal with realizability, i.e., the question whether there exists an ADF that has a given set of interpretations as its semantics. We provide necessary and sufficient conditions that hold for a set of three-valued interpretations whenever there is an ADF realizing it under admissible, complete, grounded, or preferred semantics. Moreover, we discuss how to construct such an ADF in case of realizability. Our results lay the ground for studying the expressiveness of ADFs under three-valued semantics. As a first application we study implications of our results on the existence of certain join operators on ADFs.

Abstract dialectical frameworks (ADFs) are a powerful generalization of Dung’s abstract argumentation frameworks. ADFs allow to model argumentation scenarios such that ADF semantics then provide interpretations of the scenarios. Among the considerable number of ADF semantics, the naive-based ones are built upon the fundamental concept of conflict-freeness. Intuitively, a three-valued interpretation of an ADF’s statements is conflict-free iff all true statements can possibly be accepted, and all false statements cannot possibly be accepted. In this paper, we perform an exhaustive analysis of the computational complexity of 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.

This paper relates two extensively studied formalisms: abstract dialectical frameworks and logic programs with generalized atoms or similar constructs. While the syntactic similarity is easy to see, also a strong relation between various stable model semantics proposed for these formalisms is shown by means of a unifying framework in which these semantics are restated in terms of program reducts and an immediate consequence operator, where program reducts have only minimal differences. This approach has advantages for both formalisms, as for example implemented systems for one formalism are usable for the other, and properties such as computational complexity do not have to be rediscovered. As a first, concrete result of this kind, one stable model semantics based on program reducts and subset-minimality that reached a reasonable consensus for logic programs with generalized atoms provides a novel, alternative semantics for abstract dialectical frameworks.