Checking the acceptability of a set of arguments

Considering Dung’s argumentation framework and semantics, we are interested in the problem which consists in deciding whether a set of arguments is acceptable under a given semantics. We look at three approaches. The first one consists in testing whether the set satisfies an equation; In particular, we look at the equations presented in (Dung 1995; Besnard & Doutre 2004). The second approach consists in testing whether the set is a model of a propositional formula and the third one consists in testing the satisfiability of a propositional formula. Introduction Argumentation is a reasoning model which amounts to building and evaluating arguments, generally conflicting. Dung’s argumentation framework ((Dung 1995)) constitutes an adequate formal framework to study this reasoning model. Its abstract structure makes it possible to unify many other approaches proposed for argumentation on the one hand (see (Prakken & Vreeswijk 2002) for a synthesis of these approaches) and formalisms modelling nonmonotonic reasoning on the other hand (see (Bondarenko et al. 1997; Doutre 2002)). From a set of arguments and a binary relation between arguments representing the notion of a conflict, arguments are evaluated in order to determine the most acceptable ones. Among the semantics given to acceptability, those of Dung define sets of arguments jointly acceptable called extensions. Algorithms to compute extensions are presented in (Doutre & Mengin 2001). The credulous decision problem (does an argument belong to at least one extension?) and the skeptical one (does a given argument belong to any extension?) associated to each semantics are studied in various works: Algorithms to answer these problems are presented in (Cayrol, Doutre, & Mengin 2003), dialectical proof procedures are introduced in (Jakobovits & Vermeir 1999; Vreeswijk & Prakken 2000; Cayrol, Doutre, & Mengin 2003) and algorithms computing some of these proof procedures are established in (Cayrol, Doutre, & Mengin 2003). In this article, we are interested in the problem which consists in deciding whether a set is an extension of a given semantics. For his well-known stable semantics, Dung answers this problem by giving in (Dung 1995) a simple equation that a set satisfies if and only if it is a stable extension. Equations of this kind for two other semantics and new equations for the stable semantics are established in (Besnard & Doutre 2004); These equations aim at exhibiting the sameness of the various notions of an extension as introduced by Dung. We are going to briefly study if some of them could be efficient to check if a set is an extension under a given semantics. Then, to answer the problem at hand, we will turn to another technique already considered in other formalisms used to represent knowledge: it consists in associating to the formalism a formula in propositional logic whose models correspond to the acceptable sets of the formalism. For example, in (Ben-Eliyahu & Dechter 1996) a formula of propositional logic is given whose models correspond to the extensions of a default theory. A similar correspondence is established for circumscription in (Gelfond, Przymusinska, & Przymusinski 1989) and for disjunctive logic programs in (Ben-Eliyahu & Dechter 1994). In the same way, we attempt to associate to an argument system a propositional formula whose models correspond to the extensions under a given semantics. For the stable semantics, the work of (Creignou 1995) completed in graph theory can be used. Notice that in (Dung 1995) a logic program is associated to an argument system and that the stable models of this logic program correspond to the stable extensions of the argument system; A translation of this logic program into propositional logic would associate to the argument system a propositional formula whose models would be the stable extensions of the system. This association is in two steps and could lead to formulas containing more informations than necessary; In this article we want to make such an association directly from a Dung argument system. Exploiting again propositional logic, we will study a third way to check if a set is an extension. This way consists in associating to the set a propositional formula which is satisfiable if and only if the set is an extension of the considered semantics. This method and the previous one would make possible the use of existing constraint satisfaction and satisfiability techniques possible for argumentation. This article presents preliminary ideas in each of the three approaches explicited above. The outline of the paper is as follows: In the next section we present Dung’s argumentation framework and semantics. In the section “Equation checking”, we study the equations characterizing extensions established in (Dung 1995; Besnard & Doutre 2004) to decide if a set is an extension of a given semantics. In the section “Model checking”, we show how to associate to an argument system a propositional formula such that the models of the formula correspond to the extensions of the system under a given semantics. In the section “Satisfiability checking”, we attempt to associate to the set which one wants to know if it is acceptable a propositional formula satisfiable if and only if the set is acceptable. Argumentation and extensions The argument system defined by Dung in (Dung 1995) is an abstract system in which arguments and conflicts between arguments are primitives. Definition 1 (Dung 1995) An argument system is a pair (A,R) where A is a set of arguments and R is a binary relation over A which represents a notion of attack between arguments (R ⊆ A × A). Given two arguments a and b, (a, b) ∈ R or equivalently aRb, means that a attacks b or that a is an attacker of b. A set of arguments S attacks an argument a if a is attacked by an argument of S. A set of arguments S attacks a set of arguments S ′ if there is an argument a ∈ S which attacks an argument b ∈ S . In all the definitions and notations which follow, we assume that an argument system (A,R) is given. An argument system can be represented in a very simple way by a directed graph whose vertices are the arguments and edges correspond to the elements of R. Dung gave several semantics to acceptability. These various semantics produce none, one or several acceptable sets of arguments, called extensions. One of these semantics, the stable semantics, is only defined via the notion of an attack: Definition 2 (Dung 1995) A set S ⊆ A is conflict-free iff it does not exist two arguments a and b in S such that a attacks b. A conflict-free set S ⊆ A is a stable extension iff for each argument which is not in S, there exists an argument in S that attacks it. The other semantics for acceptability rely upon the concept of defense: Definition 3 An argument a is defended by a set S ⊆ A (or S defends a) iff for any argument b ∈ A, if b attacks a then S attacks b. An acceptable set of arguments according to Dung must be a conflict-free set which defends all its elements. Formally: Definition 4 (Dung 1995) A conflict-free set S ⊆ A is admissible iff each argument in S is defended by S. Even if the definition of a stable extension does not rely upon the notion of defense, a stable extension is an admissible set. Admissibility has an advantage over stable semantics: given an argument system, there need not be any stable extension but there always exists at least one admissible set (the empty set is always admissible). A drawback of admissibility is that an argument system may have a large number of admissible sets. This is why other notions of acceptability which select only some admissible sets were introduced. Besides the stable semantics, three semantics refining admissibility have been introduced by Dung: Definition 5 (Dung 1995) A preferred extension is a maximal (wrt set inclusion) admissible subset of A. An admissible S ⊆ A is a complete extension iff each argument which is defended by S is in S. The least (wrt set inclusion) complete extension is the grounded extension. Notice that a stable extension is also a preferred extension and a preferred extension is also a complete extension. Stable, preferred and complete semantics admit multiple extensions whereas the grounded semantics ascribes a single extension to a given argument system. Deciding if a set is a stable extension or an admissible set can be computed in polynomial time, but deciding if a set is a preferred extension is a CO-NP-complete problem. These results of complexity were given in (Dimopoulos & Torres 1996) in the context of graph theory; given that the concepts of kernel, semi-kernel and maximum semi-kernel correspond respectively to the concepts of stable extension, admissible set and preferred extension (see (Dunne & BenchCapon 2002; Doutre 2002)), these results are transposable to argumentation. Example 1 Let (A,R) be the argument system such that A = {a, b, c, d, e} and R = {(a, b), (c, b), (c, d), (d, c), (d, e), (e, e)}. The graph representation of (A,R) is the following: