3. G. Antoniou, D. Billington, G. Governatori and M.J. Maher. A exible framework

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1 3. G. Antoniou, D. Billington, G. Governatori and M.J. Maher. A exible framework for defeasible logics. In Proc. 17th American National Conference on Articial Intelligence (AAAI-2000), G. Antoniou, D. Billington, G. Governatori and M.J. Maher. Representation results for defeasible logic. ACM Transactions on Computational Logic 2 (2001): 255{ D. Billington. Defeasible Logic is Stable. Journal of Logic and Computation 3 (1993): 370{ G. Brewka. On the Relationship between Defeasible Logic and Well-Founded Semantics. In Proc. Logic Programming and Nonmonotonic Reasoning Conference, LNCS 2173, 2001, 121{ J.P. Delgrande, T Schaub and H. Tompits. Logic Programs with Compiled Preferences. In Proc. ECAI'2000, 464{ M. Gelfond and V. Lifschitz. The stable model semantics for logic programming. In Proc. International Conference on Logic Programming, MIT Press 1988, 1070{ M. Gelfond and V. Lifschitz. Classical negation in logic programs and deductive databases. New Generation Computing 9 (1991): 365{ G. Governatori, A. ter Hofstede and P. Oaks. Defeasible Logic for Automated Negotiation. In Proc. Fifth CollECTeR Conference on Electronic Commerce, Brisbane B.N. Grosof. Prioritized conict handling for logic programs. In Proc. International Logic Programming Symposium, MIT Press 1997, 197{ B.N. Grosof, Y. Labrou and H.Y. Chan. A Declarative Approach to Business Rules in Contracts: Courteous Logic Programs in XML. In Proc. 1st ACM Conference on Electronic Commerce (EC-99), ACM Press K. Kunen. Negation in Logic Programming. Journal of Logic Programming 4 (1987): 289{ M.J. Maher. A Denotational Semantics for Defeasible Logic. In Proc. First International Conference on Computational Logic, LNAI 1861, Springer, 2000, M.J. Maher. Propositional Defeasible Logic has Linear Complexity. Theory and Practice of Logic Programming, 1 (6), 691{711, M. Maher and G. Governatori. A Semantic Decomposition of Defeasible Logics. In Proc. American National Conference on Articial Intelligence (AAAI-99), AAAI/MIT Press 1999, 299{ M.J. Maher, A. Rock, G. Antoniou, D. Billington and T. Miller. Ecient Defeasible Reasoning Systems. In Proc. 12th IEEE International Conference on Tools with Articial Intelligence (ICTAI 2000), IEEE 2000, V. Marek and M. Truszczynski. Nonmonotonic Logic. Springer L. Morgenstern. Inheritance Comes of Age: Applying Nonmonotonic Techniques to Problems in Industry. Articial Intelligence, 103 (1998): 1{ D. Nute. Defeasible Logic. In D.M. Gabbay, C.J. Hogger and J.A. Robinson (eds.): Handbook of Logic in Articial Intelligence and Logic Programming Vol. 3, Oxford University Press 1994, 353{ H. Prakken. Logical Tools for Modelling Legal Argument: A Study of Defeasible Reasoning in Law. Kluwer Academic Publishers D.M. Reeves, B.N. Grosof, M.P. Wellman, and H.Y. Chan. Towards a Declarative Language for Negotiating Executable Contracts, Proceedings of the AAAI- 99 Workshop on Articial Intelligence in Electronic Commerce (AIEC-99), AAAI Press / MIT Press, 1999.

2 (c) If D is decisive then the implications (a) and (b) are also true in the opposite direction. That is, if D is decisive, then the stable model semantics of P (D) corresponds to the provability in defeasible logic. However part (c) is not true in the general case, as the following example shows. Example 5. Consider the defeasible theory r 1 : ) :p r 2 : p ) p In defeasible logic, +@:p cannot be proven because we cannot derive?@p. However, blocked(r 2 ) is included in the only stable model of P (D), so def-:p is a sceptical conclusion of P (D) under stable model semantics. If we wish to have an equivalence result without the condition of decisiveness, then we must use a dierent logic programming semantics, namely Kunen semantics. Theorem 3. (a) D ` +p, P (D) `K strict-p. (b) D `?p, P (D) `K not strict-p. (c) D ` +@p, P (D) `K def-p. (d) D `?@p, P (D) `K not def-p. 6 Conclusion We motivated and presented a translation of defeasible theories into logic programs, such that the defeasible conclusions of the former correspond exactly with the sceptical conclusions of the latter under the stable model semantics, if a condition of decisiveness is satised. If decisiveness is not satised, we have to use Kunen semantics instead. This paper closes an important gap in the theory of nonmonotonic reasoning, in that it relates defeasible logic with mainstream semantics of logic programming. This result is particularly important, since defeasible reasoning is one of the most successful nonmonotonic reasoning paradigms in applications. References 1. G. Antoniou, D. Billington and M.J. Maher. On the analysis of regulations using defeasible rules. In Proc. 32nd Hawaii International Conference on Systems Science, G. Antoniou, M.J. Maher and D. Billington. Defeasible Logic versus Logic Programming without Negation as Failure, Journal of Logic Programming, 42 (2000):

3 d 1 (r 1 ) : def-p not strict-:p; ok(r 1 ): d 2 (r 1 ) : ok(r 1 ) ok 0 (r 1 ; r 2 ): d 3 (r 1 ) : ok 0 (r 1 ; r 2 ) blocked(r 2 ): d 1 (r 2 ) : def-:p not strict-p; ok(r 2 ): d 2 (r 2 ) : ok(r 2 ) ok 0 (r 2 ; r 1 ): d 3 (r 2 ) : ok 0 (r 2 ; r 1 ) blocked(r 1 ): d 1 (r 3 ) : def-q not strict-:q; ok(r 3 ): d 2 (r 3 ) : ok(r 3 ) ok 0 (r 3 ; r 4 ): d 3 (r 3 ) : ok 0 (r 3 ; r 4 ) blocked(r 4 ): d 1 (r 4 ) : def-:q def-p; not strict-q; ok(r 4 ): d 2 (r 4 ) : ok(r 4 ) ok 0 (r 4 ; r 3 ): d 3 (r 4 ) : ok 0 (r 4 ; r 3 ) blocked(r 3 ): d 4 (r 4 ) : blocked(r 4 ) not def-p: fblocked(r 4 ); ok 0 (r 3 ; r 4 ); ok(r 3 ); def-qg is the only stable model. 5 Properties of the Translation We begin with an observation on the size of the translation. By the size of a defeasible theory, we mean the number of rules. Proposition 1. The size of P (D) is bound by L + n (3 + L) + n 2, where n is the number of rules in D and L the number of literals occurring in D. Next we establish relationships between D and its translation P (D). To do so we must select appropriate logic program semantics to interpret not. First we consider stable model semantics. Theorem 1. (a) D ` +p, strict-p is included in all stable models of P (D). (b) D `?p ) strict-p is not included in any stable model of P (D). (c) If D is decisive on denite conclusions then the implication (b) is also true in the opposite direction. A defeasible theory D is decisive on denite conclusions if, for every literal p, either D ` +p or D `?p. Recall that D is decisive if, for every literal p, either D ` +@p or D `?@p. Theorem 2. Suppose D is decisive on denite conclusions. (a) D ` +@p ) def-p is included in all stable models of P (D). (b) D `?@p ) def-p is not included in any stable model of P (D).

4 b(p) : def-p strict-p for every literal p. Intuitively, strict-p means that p is strictly provable, and defp that p is defeasibly provable. And the clause b(p) corresponds to the condition (1) in the +@ inference condition: a literal p is defeasibly provable if it is strictly provable. Next we turn our attention to defeasible rules and consider r : fq 1 ; : : : ; q n g ) p r is translated into the following set of clauses: d 1 (r) : def-p def-q 1 ; : : : ; def-q n ; not strict- p; ok(r): d 2 (r) : ok(r) ok 0 (r; s 1 ); : : : ; ok 0 (r; s m ), where R[ p] = fs 1 ; : : : ; s m g. d 3 (r; s) : ok 0 (r; s) blocked(s), for all s 2 R[ p]. d 4 (r; q i ) : blocked(r) not def-q i, for all i 2 f1; : : :; ng. In the above, the predicates ok; ok 0 and blocked are new and pairwise disjoint. { d 1 (r) says that to prove p defeasibly by applying r, we must prove all the antecedents of r, the negation of p should not be strictly provable, and it must be ok to apply r. { The clause d 2 (r) says when it is ok to apply a rule r with head p: we must check that it is ok to apply r w.r.t. every rule with head p. { d 3 (r; s) says that it is ok to apply r w.r.t. s if s is blocked. Obviously this clause would look more complicated if we had considered priorities, instead of compiling them into the defeasible theory prior to the translation. Indeed, in the present framework we could have used a somewhat simpler translation, replacing d 1, d 2, and d 3 by def-p def-q 1 ; : : : ; def-q n ; not strict- p; blocked(s 1 ); : : : ; blocked(s m ) but chose to maintain the intuitive nature of the translation in its present form. { Finally, d 4 species the only way a rule r can be blocked: it must be impossible to prove one of its antecedents. For a defeasible theory D we dene P (D) to be the union of all clauses a(r); b(p); d 1 (r); d 2 (r); d 3 (r; s) and d 4 (r; q i ). Example 4. We consider the defeasible theory from Example 1: r 1 : ) p r 2 : ) :p r 3 : ) q r 4 : p ) :q Its translation looks as follows:

5 Finally, there is a aw in the use of explicit (or classical) negation in the translated program to represent explicit negation in the defeasible theory. Logic programs, under the answer set semantics, react to an inconsistency by inferring all literals whereas defeasible logic is paraconsistent. As a consequence, the translated program does not reect the behavior of defeasible logic when an inconsistency is involved, as in the following example. Example 3. Consider the defeasible theory! p! :p! q The translation is p :p q The only answer set of this program is fp; :p; q; :qg which does not agree with defeasible logic: the literal :q is included in the answer set but is not strictly provable in defeasible logic. 4.2 A Translation Using Control Literals Above we outlined the reasons why a direct translation of a defeasible theory into a logic program must fail. Here we propose a dierent translation which uses \control literals" that carry meaning regarding the applicability status of rules. First we translate strict rules. In defeasible logic, strict rules play a twofold role: on one hand they can be used to derive undisputed conclusions if all their antecedents have been strictly proved. And on the other hand they can be used essentially as defeasible rules, if their antecedents are defeasibly provable. These two roles can be clearly seen in the inference condition +@ is section 2. To capture both uses we introduce mutually disjoint copies strict-p and defp, for all literals p. Note that this way the logic program we get does not have classical negation, as in the previous section. Among others, this solution avoids the problem illustrated by Example 3. Given a strict rule r : fq 1 ; : : : ; q n g! p we translate it into the program clause a(r) : strict-p strict-q 1 ; : : : ; strict-q n. Additionally, we introduce the clause

6 p q 1 ; : : :; q n ; not p. Unfortunately this translation does not lead to a correspondence between the defeasible conclusions and the sceptical conclusions in answer set semantics, as the following example demonstrates. Example 1. Consider the defeasible theory ) p ) :p ) q p ) :q Here q is defeasibly provable because the only rule with head :q is not applicable, because?@p. However, the translated logic program p not :p: :p not p: q not :q: :q p; not q: has three answer sets, fp; qg, fp; :qg and f:p; qg. Thus none of p; :p; q; :q is included in all stable models. The example above demonstrates that the translation does not capture the ambiguity blocking behaviour of defeasible logic (ambiguity of p is not propagated to the dependent atom q). But even if we try to overcome this problem by considering an ambiguity propagating defeasible logic instead [3], there remains the problem of oating conclusions, as the following example demonstrates. Example 2. Consider the defeasible theory ) p ) :p p ) q :p ) q In defeasible logic, q is not defeasibly provable because neither p nor :p are defeasibly provable. However, the translation p not :p: :p not p: q p; not :q: q :p; not :q: has two answer sets, fp; qg and f:p; qg, so q is a sceptical conclusion under the answer set semantics.

7 3.2 Kunen Semantics Kunen semantics [13] is a 3-valued semantics for logic programs. An interpretation is a mapping from ground atoms to one of the three truth values t, f and u, which denote true, false and unknown, respectively. This mapping can be extended to arbitrary formulas using Kleene's 3-valued logic. Kleene's truth tables can be summarized as follows. If ' is a boolean combination of atoms with truth values t, f or u, its truth value is t i all possible ways of putting t or f for the various u-values lead to a value t being computed in ordinary (2-valued) logic; ' gets the value f i not ' gets the value t; and ' gets the value u otherwise. These truth values can be extended in the obvious way to predicate logic, thinking of the quantiers as innite conjunctions or disjunctions. The Kunen semantics of a program P is obtained from a sequence fi n g of interpretations, dened as follows: 1. I 0 () = u for every atom. 2. I n+1 () = t i for some clause ' in the program, I n (') = t. 3. I n+1 () = f i for all clauses ' in the program, I n (') = f. 4. I n+1 () = u if neither 2. nor 3. applies. We shall say that the Kunen semantics of P supports, written P `K, i there is an interpretation I n, for some nite n, such that I n () = t. 4 A Translation of Defeasible Theories into Logic Programs 4.1 A Direct Translation that Fails Here we consider the most natural translation of a defeasible theory into logic programs. Since in defeasible logic both positive and negative literals are used, the translation in this section yields an extended logic program. We will consider the answer set semantics for extended logic programs [9], which is a generalisation of the stable model semantics. A natural translation of a defeasible theory into a logic program would look as follows. A strict rule fq 1 ; : : : ; q n g! p is translated into the program clause p q 1 ; : : :; q n. And a defeasible rule fq 1 ; : : : ; q n g ) p is translated into

8 If P (i + 1) = +@q then either (1) +q 2 P (1::i) or (2) (2.1) 9r 2 R[q] 8a 2 A(r) : +@a 2 P (1::i) and (2.2)? q 2 P (1::i) and (2.3) 8s 2 R[ q]9a 2 A(s) :?@a 2 P (1::i) Let us illustrate this denition. To show that q is provable defeasibly we have two choices: (1) We show that q is already denitely provable; or (2) we need to argue using the defeasible part of D as well. In particular, we require that there must be a strict or defeasible rule with head q which can be applied (2.1). But now we need to consider possible \counterattacks", that is, reasoning chains in support of q. To be more specic: to prove q defeasibly we must show that q is not denitely provable (2.2). Also (2.3) we must consider the set of all rules which are not known to be inapplicable and which have head q. Essentially each such rule s attacks the conclusion q. For q to be provable, each such rule s must have been established as non-applicable. A defeasible theory D is called decisive i for every literal p, either D `?@p or D ` +@p. Not every defeasible theory satises this property. For example, in the theory consisting of the single rule p ) p neither?@p nor +@p is provable. However, decisiveness is guaranteed in defeasible theories with an acyclic atom dependency graph [5]. 3 Semantics of Logic Programs A logic program P is a nite set of program clauses. A program clause r has the form A B 1 ; : : : ; B n ; not C 1 ; : : : ; not C m where A; B 1 ; : : : B n ; C 1 ; : : : ; C m are positive literals. 3.1 Stable Model Semantics Let M be a subset of the Herbrand base. We call a ground program clause A B 1 ; : : : ; B n ; not C 1 ; : : : ; not C m irrelevant w.r.t. M if at least one C i is included in M. Given a logic program P, we dene the reduct of P w.r.t. M, denoted by P M, to be the logic program obtained from ground(p ) by 1. removing all clauses that are irrelevant w.r.t. M, and 2. removing all premises not C i from all remaining program clauses. Note that the reduct P M is a denite logic program, and we are no longer faced with the problem of assigning semantics to negation, but can use the least Herbrand model instead. M is a stable model of P i M = M P M.

9 A rule r : A(r),! C(r) consists of its unique label r, its antecedent A(r) (A(r) may be omitted if it is the empty set) which is a nite set of literals, an arrow,! (which is a placeholder for concrete arrows to be introduced in a moment), and its head (or consequent) C(r) which is a literal. In writing rules often we omit set notation for antecedents and sometimes we omit the label when it is not relevant for the context. There are two kinds of rules, each represented by a dierent arrow. Strict rules use! and defeasible rules use ). Given a set R of rules, we denote the set of all strict rules in R by R s, and the set of defeasible rules in R by R d. R[q] denotes the set of rules in R with consequent q. A defeasible theory D is a nite set of rules R. 2.3 Proof Theory A conclusion of a defeasible theory D is a tagged literal. A conclusion has one of the following four forms: { +q, which is intended to mean that the literal q is denitely provable, using only strict rules. {?q, which is intended to mean that q is provably not strictly provable (nite failure). { +@q, which is intended to mean that q is defeasibly provable in D. {?@q which is intended to mean that we have proved that q is not defeasibly provable in D. Provability is dened below. It is based on the concept of a derivation (or proof) in D = R. A derivation is a nite sequence P = P (1); : : :; P (n) of tagged literals satisfying the following conditions. The conditions are essentially inference rules phrased as conditions on proofs. P (1::i) denotes the initial part of the sequence P of length i. +: If P (i + 1) = +q then 9r 2 R s [q] 8a 2 A(r) : +a 2 P (1::i) That means, to prove +q we need to establish a proof for q using strict rules only. This is a deduction in the classical sense { no proofs for the negation of q need to be considered (in contrast to defeasible provability below, where opposing chains of reasoning must be taken into account, too).?: If P (i + 1) =?q then 8r 2 R s [q] 9a 2 A(r) :?a 2 P (1::i) The denition of? is the so-called strong negation of +: normal negation rules like De-Morgan rules are applied to the denition, + is replaced by?, and vice versa. Therefore in the following we may omit giving inference conditions of both + and?.

10 Strict rules are rules in the classical sense: whenever the premises are indisputable (e.g. facts) then so is the conclusion. An example of a strict rule is \Emus are birds". Written formally: emu(x)! bird(x): Defeasible rules are rules that can be defeated by contrary evidence. An example of such a rule is \Birds typically y"; written formally: bird(x) ) flies(x): The idea is that if we know that something is a bird, then we may conclude that it ies, unless there is other, not inferior, evidence suggesting that it may not y. The superiority relation among rules is used to dene priorities among rules, that is, where one rule may override the conclusion of another rule. For example, given the defeasible rules r : bird(x) ) flies(x) r 0 : brokenw ing(x) ) :flies(x) which contradict one another, no conclusive decision can be made about whether a bird with broken wings can y. But if we introduce a superiority relation > with r 0 > r, with the intended meaning that r 0 is strictly stronger than r, then we can indeed conclude that the bird cannot y. It is worth noting that, in defeasible logic, priorities are local in the following sense: Two rules are considered to be competing with one another only if they have complementary heads. Thus, since the superiority relation is used to resolve conicts among competing rules, it is only used to compare rules with complementary heads; the information r > r 0 for rules r; r 0 without complementary heads may be part of the superiority relation, but has no eect on the proof theory. [4] showed that there is a constructive, conclusion-preserving transformation which takes an arbitrary defeasible theory and translates it into a theory which has only strict rules and defeasible rules. For the sake of simplicity, we will assume in this paper that indeed a defeasible theory consists only of strict rules and defeasible rules. 2.2 Formal Denition In this paper we restrict attention to essentially propositional defeasible logic. Rules with free variables are interpreted as rule schemas, that is, as the set of all ground instances; in such cases we assume that the Herbrand universe is nite. We assume that the reader is familiar with the notation and basic notions of propositional logic. If q is a literal, q denotes the complementary literal (if q is a positive literal p then q is :p; and if q is :p, then q is p). Rules are dened over a language (or signature), the set of propositions (atoms) and labels that may be used in the rule.

11 Markup Language and for current W3C activities on rules. Therefore defeasible reasoning is arguably the most successful subarea in nonmonotonic reasoning as far as applications and integration to mainstram IT is concerned. Recent theoretical work on defeasible logics has: (i) established some relationships to other logic programming approaches without negation as failure [2]; (ii) analysed the formal properties of these logics [4, 14, 15], and (iii) has delivered ecient implementations [17]. However the problem remains that defeasible logic is not rmly linked to the mainstream of nonmonotonic reasoning, in particular the semantics of logic programs. This paper aims at resolving this problem. Our initial approach is to consider answer set semantics of logic programs [9] and use a natural, direct translation (defeasible rules translated into \normal defaults"). We discuss why this translation cannot be successful. Then we dene a second translation which makes use of control literals, similar to those used in [7]. Under this translation of a defeasible theory D into a logic program P (D) we can show that p is defeasibly provable in D () p is included in all stable models of P (D). () However this result can only be shown under the additional condition of decisiveness: for every literal q, either +@q or?@q can be derived. A sucient condition for decisiveness is the absence of cycles in the atom dependency graph. If we wish to drop decisiveness, () holds only in one direction, from left to right. We show that if we wish the equivalence in the general case, we need to use another semantics for logic programs, namely Kunen semantics [13]. There is previous work relating defeasible logic and logic programs. [16] showed that the notion of failure in defeasible logic corresponds to Kunen semantics. That work used a metaprogram to express defeasible logic in logic programming terms. The translation we present here is more direct. [6] provided a translation of a dierent defeasible logic to logic programs with well-founded semantics, but that translation does not provide a characterization of the defeasible logic. The paper is organised as follows. Sections 2 and 3 present the basics of defeasible logic and logic programming semantics, respectively. Section 4 presents our translation and its ideas, while section 5 contains the main results. 2 Defeasible Logic 2.1 A Language for Defeasible Reasoning A defeasible theory (a knowledge base in defeasible logic) consists of three dierent kinds of knowledge: strict rules, defeasible rules, and a superiority relation. (Fuller versions of defeasible logic also have facts and defeaters, but [4] shows that they can be simulated by the other ingredients).

12 Embedding Defeasible Logic into Logic Programs G. Antoniou 1 and M.J. Maher 2 1 Department of Computer Science, University of Bremen ga@tzi.de 2 Department of Mathematical and Computer Sciences, Loyola University Chicago mjm@math.luc.edu Abstract. Defeasible reasoning is a simple but ecient approach to nonmonotonic reasoning that has recently attracted considerable interest and that has found various applications. Defeasible logic and its variants are an important family of defeasible reasoning methods. So far no relationship has been established between defeasible logic and mainstream nonmonotonic reasoning approaches. In this paper we establish close links to known semantics of extended logic programs. In particular, we give a translation of a defeasible theory D into a program P (D). We show that under a condition of decisiveness, the defeasible consequences of D correspond exactly to the sceptical conclusions of P (D) under the stable model semantics. Without decisiveness, the result holds only in one direction (all defeasible consequences of D are included in all stable models of P (D)). If we wish a complete embedding for the general case, we need to use the Kunen semantics of P (D), instead. 1 Introduction Defeasible reasoning is a nonmonotonic reasoning [18] approach in which the gaps due to incomplete information are closed through the use of defeasible rules that are usually appropriate. Defeasible logics were introduced and developed by Nute over several years [20]. These logics perform defeasible reasoning, where a conclusion supported by a rule might be overturned by the eect of another rule. Roughly, a proposition p can be defeasibly proved (+@p) only when a rule supports it, and it has been demonstrated that no applicable rule supports :p; this demonstration makes use of statements?@q which mean intuitively that an attempt to prove q defeasibly has failed nitely. These logics also have a monotonic reasoning component, and a priority on rules. One advantage of Nute's design was that it was aimed at supporting ecient reasoning, and in our work we follow that philosophy. Defeasible reasoning has recently attracted considerable interest. Its use in various application domains has been advocated, including the modelling of regulations and business rules [19, 12, 1], modelling of contracts [22], legal reasoning [21] and agent negotiations [10]. In fact, defeasible reasoning (in the form of courteous logic programs [11]) provides a foundation for IBM's Business Rules