The Equational Approach to CF2 Semantics

We introduce a family of new equational semantics for argumentation networks which can handle odd and even loops in a uniform manner. We offer one version of equational semantics which is equivalent to CF2 semantics, and a better version which gives the same results as traditional Dung semantics for even loops but can still handle odd loops.Comment: 36 pages, version dated 15 February 201

Probabilistic Argumentation. An Equational Approach

There is a generic way to add any new feature to a system. It involves 1) identifying the basic units which build up the system and 2) introducing the new feature to each of these basic units. In the case where the system is argumentation and the feature is probabilistic we have the following. The basic units are: a. the nature of the arguments involved; b. the membership relation in the set S of arguments; c. the attack relation; and d. the choice of extensions. Generically to add a new aspect (probabilistic, or fuzzy, or temporal, etc) to an argumentation network can be done by adding this feature to each component a-d. This is a brute-force method and may yield a non-intuitive or meaningful result. A better way is to meaningfully translate the object system into another target system which does have the aspect required and then let the target system endow the aspect on the initial system. In our case we translate argumentation into classical propositional logic and get probabilistic argumentation from the translation. Of course what we get depends on how we translate. In fact, in this paper we introduce probabilistic semantics to abstract argumentation theory based on the equational approach to argumentation networks. We then compare our semantics with existing proposals in the literature including the approaches by M. Thimm and by A. Hunter. Our methodology in general is discussed in the conclusion

Equilibrium States in Numerical Argumentation Networks

Given an argumentation network with initial values to the arguments, we look for algorithms which can yield extensions compatible with such initial values. We find that the best way of tackling this problem is to offer an iteration formula that takes the initial values and the attack relation and iterates a sequence of intermediate values that eventually converges leading to an extension. The properties surrounding the application of the iteration formula and its connection with other numerical and non-numerical techniques proposed by others are thoroughly investigated in this paper

Introducing Equational Semantics for Argumentation Networks

This paper provides equational semantics for Dung’s argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the “extensions” of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, etc, etc. The equational approach has its conceptual roots in the 19th century following the algebraic equational approach to logic by George Boole, Louis Couturat and Ernst Schroeder

Theory of Semi-Instantiation in Abstract Argumentation

We study instantiated abstract argumentation frames of the form $(S,R,I)$, where $(S,R)$ is an abstract argumentation frame and where the arguments $x$ of $S$ are instantiated by $I(x)$ as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature

A Comparative Study of Ranking-based Semantics for Abstract Argumentation

Argumentation is a process of evaluating and comparing a set of arguments. A way to compare them consists in using a ranking-based semantics which rank-order arguments from the most to the least acceptable ones. Recently, a number of such semantics have been proposed independently, often associated with some desirable properties. However, there is no comparative study which takes a broader perspective. This is what we propose in this work. We provide a general comparison of all these semantics with respect to the proposed properties. That allows to underline the differences of behavior between the existing semantics.Comment: Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI-2016), Feb 2016, Phoenix, United State

An Imprecise Probability Approach for Abstract Argumentation based on Credal Sets

Some abstract argumentation approaches consider that arguments have a degree of uncertainty, which impacts on the degree of uncertainty of the extensions obtained from a abstract argumentation framework (AAF) under a semantics. In these approaches, both the uncertainty of the arguments and of the extensions are modeled by means of precise probability values. However, in many real life situations the exact probabilities values are unknown and sometimes there is a need for aggregating the probability values of different sources. In this paper, we tackle the problem of calculating the degree of uncertainty of the extensions considering that the probability values of the arguments are imprecise. We use credal sets to model the uncertainty values of arguments and from these credal sets, we calculate the lower and upper bounds of the extensions. We study some properties of the suggested approach and illustrate it with an scenario of decision making.Comment: 8 pages, 2 figures, Accepted in The 15th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2019