"Minimal defence": a refinement of the preferred semantics for argumentation frameworks

Abstract Dung’s abstract framework for argumenta-tion enables a study of the interactions be-tween arguments based solely on an “attack”binary relation on the set of arguments. Var-ious ways to solve conﬂicts between contra-dictory pieces of information have been pro-posed in the context of argumentation, non-monotonic reasoning or logic programming,and can be captured by appropriate seman-tics within Dung’s framework. A commonfeature of these semantics is that one canalways maximize in some sense the set ofacceptable arguments. We propose in thispaper to extend Dung’s framework in orderto allow for the representation of what wecall “restricted” arguments: these argumentsshould only be used if absolutely necessary,that is, in order to support other argumentsthat would otherwise be defeated. We mod-ify Dung’s preferred semantics accordingly: aset ofargumentsbecomesacceptable onlyif itcontains a minimum of restricted arguments,for a maximum of unrestricted arguments. 1 INTRODUCTION Argumentation is a form of reasoning that has recentlycaptured the interest of many researchers in the Arti-ﬁcial Intelligence community, among others. An ab-stract framework for studying it has been proposedby Dung [Dung, 1995]. It is based on the assumptionthat the process of argumentation is built from twotypes of jobs: the formation of arguments, associatedwith the interactions that exist between them; and theevaluation of the validity of the arguments. A form ofinteraction between arguments that is often diﬃcultto ignore is contradiction, since a usual basic measureof the validity of a piece of reasoning is the level ofself-contradiction it contains. In Dung’s approach, anargumentation framework consists of two elements: aset of arguments, and a binary relation between ar-guments, so that one can represent an “attack”, orcontradiction, relation between arguments. Thus, theset of arguments and the attack relation form a graph.If one accepts that the graph contains all needed in-formation, it is possible to deﬁne which arguments arevalid, or acceptable, by considering solely this graph.Our intuition tells us that an argument which is not at-tacked at all should be acceptable. More generally, onecan deﬁne that an acceptable argument is one whichis defended, that is, whose attackers are all attackedby arguments which are themselves acceptable.When the graph contains cycles, this deﬁnition is cir-cular, and one needs a more global approach. A solu-tion is to deﬁne acceptable sets of arguments. Argu-ments in such a set do usually not attack each other,and defend each other: every attacker of an argumentin the set is itself attacked by an argument in the set.This deﬁnition leads to Dung’s “admissible” sets ofarguments. Maximal sets of arguments that are ac-ceptable are then called “preferred extensions” of theframework.These deﬁnitions of acceptability make no distinctionbetween arguments, in the sense that defenders areall equally desirable: if an argument can be indepen-dently defended by two diﬀerent arguments, then onemay choose any of the two, or even the two argumentstogether if they do not attack each other, to defendthe ﬁrst one. However, there are problems where adistinction between several levels of desirability of de-fenders is needed. Suppose for example that our argu-ments “belong” to two agents: one, the proponent, istrying to make a point, while the other one, the oppo-nent, is trying to refute it. The proponent may have