Probabilistic argumentation allows reasoning about argumentation problems in a way that is well-founded by probability theory. However, in practice, this approach can be severely limited by the fact that probabilities are defined by adding an exponential number of terms. We show that this exponential blowup can be avoided in an interesting fragment of epistemic probabilistic argumentation and that some computational problems that have been considered intractable can be solved in polynomial time. We give efficient convex programming formulations for these problems and explore how far our fragment can be extended without loosing tractability.
@cite_36 recently introduced a very general probabilistic argumentation framework that generalizes many ideas that have been considered before in the literature. The authors consider probability functions over subsets of defeasible theories or over subgraphs. The latter approach can then be seen as a generalization of the former, which abstracts from the structure of arguments. The authors discuss probabilistic labellings that should not be confused with probability labellings that we considered here. Roughly speaking, in @cite_36 , a probabilistic labelling frame corresponds to a probability function over subsets of possible classical labellings over an argumentation framework. These probabilistic labelling frames can then be used to assign probabilities to arguments. In this sense, a probabilistic labelling considered in @cite_36 induces a probability labelling as considered here. However, the focus in @cite_36 is on conceptual questions and computational problems are not discussed.
@cite_36
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We propose a new graded semantics for abstract argumentation frameworks that is based on the constellations approach to probabilistic argumentation. Given an abstract argumentation framework, our approach assigns uniform probability to all arguments and then ranks arguments according to the probability of acceptance wrt. some classical semantics. Albeit relying on a simple idea this approach (1) is based on the solid theoretical foundations of probability theory, and (2) complies with many rationality postulates proposed for graded semantics. We also investigate an application of our approach for inconsistency measurement in argumentation frameworks and show that the measure induced by the probabilistic graded semantics also complies with the basic rationality postulates from that area. Combining computational models of argumentation with probability theory has recently gained increasing attention, in particular with respect to abstract argumentation frameworks. Approaches following this idea can be categorised into the constellations and the epistemic approach. While the former considers probability functions on the subgraphs of abstract argumentation frameworks, the latter uses probability theory to represent degrees of belief in arguments, given a fixed framework. In this paper, we investigate the case where probability functions are given on the extensions of abstract argumentation frameworks. This generalises classical semantics in a straightforward fashion and we show that our approach also complies with many postulates for epistemic probabilistic argumentation. The combination of argumentation and probability paves the way to new accounts of qualitative and quantitative uncertainty, thereby offering new theoretical and applicative opportunities. Due to a variety of interests, probabilistic argumentation is approached in the literature with different frameworks, pertaining to structured and abstract argumentation, and with respect to diverse types of uncertainty, in particular the uncertainty on the credibility of the premises, the uncertainty about which arguments to consider, and the uncertainty on the acceptance status of arguments or statements. Towards a general framework for probabilistic argumentation, we investigate a labelling-oriented framework encompassing a basic setting for rule-based argumentation and its (semi-) abstract account, along with diverse types of uncertainty. Our framework provides a systematic treatment of various kinds of uncertainty and of their relationships and allows us to back or question assertions from the literature. Probabilistic abstract argumentation combines Dung’s abstract argumentation framework with probability theory in order to model uncertainty in argumentation. In this setting, we address the fundamental problem of computing the probability that a set of arguments is an extension according to a given semantics. We focus on the most popular semantics (i.e., admissible, stable, complete, grounded, preferred, ideal-set, ideal, stage, and semistable) and show the following dichotomy result: computing the probability that a set of arguments is an extension is either FP or FP#P-complete depending on the semantics adopted. Our polynomial-time results are particularly interesting, as they hold for some semantics for which no polynomial-time technique was known so far.