Most formalisms for representing common-sense knowledge allow incomplete and potentially inconsistent information. When strong negation is also allowed, contradictory conclusions can arise. A criterion for deciding between them is needed. The aim of this paper is to investigate an inherent and autonomous comparison criterion, based on specificity as defined in [POO 85, SIM 92]. In contrast to other approaches, we consider not only defeasible, but also strict knowledge. Our criterion is

more »

... ensitive, i. e., preference among defeasible rules is determined dynamically during the dialectical analysis. We show how specificity can be defined in terms of two different approaches: activation sets and derivation trees. This allows us to get a syntactic criterion that can be implemented in a computationally attractive way. The resulting definitions may be applied in general rulebased formalisms. We present theorems linking both characterizations. Finally we discuss other frameworks for defeasible reasoning in which preference handling is considered explicitly. sions can arise, which prompts for a criterion for deciding between them. Several extensions of logic programming (LP), default reasoning systems, defeasible logics, and defeasible argumentation formalisms consider priorities over competing rules [ANT 00, COV 88, DIM 95, GEL 97, KAK 94, WAN 97], in order to decide between contradictory conclusions. However, these priorities must be supplied by the programmer, establishing explicitly relations between rules. Another problem, pointed out by Dung and Son in [DUN 96], is that several formalisms "enforce" the principle of reasoning with specificity by first determining a set of priority orders between default rules of a set D, using the information given by a domain knowledge K. The problem is that the resulting semantics is rather weak, in the sense that priorities are defined independently of the set E of evidence. Therefore, if the set E changes, the previous fixed priorities could not behave as expected. On the contrary, this evidence-sensitivity can be naturally captured in argumentationtheoretic approaches as shown in [DUN 96, GAR 98, SIM 92] and also here. In [DUN 96], a transformation from the proposed underlying language into extended logic programming [GEL 90a] is given. However, this transformation encodes the specificity criterion with program rules, forcing re-encoding in the presence of changes in the program. In our approach specificity will be inferred directly from the program rules without any intermediate step. Our approach also takes into consideration the background knowledge B that was assumed empty in [DUN 96]. Dealing with background knowledge (i. e. adding strict rules) is not a trivial matter, because then we have to take into account also the conclusions that are implied by this background knowledge, which has to be considered in the dialectical process when comparing arguments (see also Section 3.2). Motivation The aim of this paper is to investigate beyond explicit comparison between rules, looking forward for a more autonomous comparison criterion, based on specificity as defined in [POO 85, SIM 92]. In contrast to other approaches, we consider not only defeasible, but also strict knowledge. In our setting, arguments will be basically defeasible proofs involving both defeasible and strict knowledge, which may support contradictory conclusions, so that a comparison criterion is needed to decide between them. Our criterion for comparing arguments, namely specificity, is context-sensitive. This means that preference among defeasible rules is determined dynamically during the dialectical analysis (see also the examples in Section 4.1). We show how this criterion can be redefined in terms of two different approaches: activation sets and derivation trees. This allows us to get a syntactic criterion that can be implemented in a computationally attractive way. The resulting definitions may be applied in arbitrary generic rule-based formalisms. As a basis of our presentation we will use Defeasible Logic Programming (DeLP) [GAR 97, GAR 98], where a comparison for arguments based on specificity is given. In DeLP (as in many defeasible Computing Generalized Specificity 89 logics and defeasible argumentation formalisms), there is a distinction between strict rules and defeasible rules. Specificity in DeLP takes into consideration both kinds of rules. Originally, this research has been motivated by the programming of autonomous agents for the RoboCup [MUR 01]. Since agents must be able to cope with contradictory knowledge, defeasible reasoning should be employed for agent programming. Defeasible logic programming is able to extend the logic-based approach for multiagent systems as presented in [MUR 01]. This paper is organized as follows. First, in Section 2 we introduce the fundamentals of DeLP. In Section 3, a definition of generalized specificity will be given, and two computationally attractive ways of comparing arguments by means of specificity in a logic programming framework will be developed. Finally, in Section 4, we discuss other frameworks for defeasible reasoning in which preference handling is considered explicitly, contrasting them with our approach. We will end with concluding remarks in Section 5. Defeasible Logic Programming Defeasible Programs The DeLP language [GAR 97, GAR 98] is defined in terms of two disjoint sets of rules: a set of strict rules for representing strict (sound) knowledge, and a set of defeasible rules for representing tentative information. Rules will be defined using literals. A literal L is an atom p or a negated atom ∼p, where the symbol ∼ represents strong negation. We define this formally as follows: DEFINITION 1 (STRICT RULE). -A strict rule is an ordered pair, conveniently denoted Head ← Body, whose first member, Head, is a literal, and whose second member, Body, is a finite set of literals. A strict rule with the head L 0 and body {L 1 , . . . , L n } can also be written as L 0 ← L 1 , . . . , L n . As usual, if the body is empty, then a strict rule becomes L ← true (or simply L) and it is called a fact. The syntax of strict rules corresponds to basic rules in logic programming [LIF 96], but we call them "strict" in order to emphasize the difference to the "defeasible" ones (see below). There is no contraposition for rules, i. e., a ← b is not equivalent to ∼b ← ∼a. Defeasible rules add a new representational capability for expressing a weaker link between the head and the body in a rule [SIM 92]. DEFINITION 2 (DEFEASIBLE RULE). -A defeasible rule is an ordered pair, conveniently denoted Head -< Body, whose first member, Head, is a literal, and whose second member, Body, is a finite and non-empty set of literals. A defeasible rule with head L 0 and body {L 1 , . . . , L n } can also be written as L 0 -< L 1 , . . . , L n where n ≥ 1. 90 Journal of Applied Non-Classical Logics. Volume 13 -n • 1/2003 1. We use parentheses just for improving the readability of the set of rules.