We propose a new declarative semantics for logic programs with negation. Its formulation is quite simple; at the same time, it is more general than the iterated fixed point semantics for stratied programs, and is applicable to some useful programs that are not stratified.

The Gelfond-Lifschitz operator [GL88] associated with a logic program (and likewise the operator associated with default theories by Reiter) exhibits oscillating behavior. In the case of logic programs, there is always at least one finite, non-empty collection of Herbrand interpretations around which the Gelfond-Lifschitz [GL88] operator "bounces around". The same phenomenon occurs with default logic when Reiter's operator \Gamma \Delta is considered. Based on this, a "stable class" semantics and "extension class" semantics was proposed in [BS90]. The main advantage of this semantics was that it was defined for all logic programs (and default theories), and that this definition was modelled using the standard operators existing in the literature such as Reiter's \Gamma \Delta operator. In this paper, our primary aim is to prove that there is a very interesting duality between stable class theory and the well founded semantics for logic programming. In the stable class semantics, class...

Research on nonmonotonic and default reasoning has identified several important criteria for preferring alternative default inferences. The theories of reasoning based on each of these criteria may uniformly be viewed as theories of rational inference, in which the reasoner selects maximally preferred states of belief. Though researchers have noted some cases of apparent conflict between the preferences supported by different theories, it has been hoped that these special theories of reasoning may be combined into a universal logic of nonmonotonic reasoning. We show that the different categories of preferences conflict more than has been realized, and adapt formal results from social choice theory to prove that every universal theory of default reasoning will violate at least one reasonable principle of rational reasoning. Our results can be interpreted as demonstrating that, within the preferential framework, we cannot expect much improvement on the rigid lexicographic priority mechanisms that have been proposed for conflict resolution.

The class of logic programs with negation as failure in the head is a subset of the logic of MBNF introduced by Lifschitz and is an extension of the class of extended disjunctive programs. An interesting feature of such programs is that the minimality of answer sets does not hold. This paper considers the class of {\em general extended disjunctive programs\/} (GEDPs) as logic programs with negation as failure in the head. First, we discuss that the class of GEDPs is useful for representing knowledge in various domains in which the principle of minimality is too strong. In particular, the class of abductive programs is properly included in the class of GEDPs. Other applications include the representation of inclusive disjunctions and circumscription with fixed predicates. Secondly, the semantic nature of GEDPs is analyzed by the syntax of programs. In acyclic programs, negation as failure in the head can be shifted to the body without changing the answer sets of the program. On the other hand, supported sets of any program are always preserved by the same transformation. Thirdly, the computational complexity of the class of GEDPs is shown to remain in the same complexity class as normal disjunctive programs. Through the simulation of negation as failure in the head, computation of answer sets and supported sets is realized using any proof procedure for extended or positive disjunctive programs. Finally, a simple translation of GEDPs into autoepistemic logic is presented.

A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequent-like tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cut rules, used by sequent-like tableaux, are totally eliminated. A strong completeness theorem without cut is also given for symmetric and euclidean logics. The system gains the same modularity of Hilbert-style formulations, where the addition or deletion of rules is the way to change logic. Since each rule has to consider only adjacent possible worlds, the calculus also gains efficiency. Moreover, the rules satisfy the strong Church Rosser property and can thus be fully parallelized. Termination properties and a general algorithm are devised. The propositional modal logics thus treated are K, D, T, KB, K4, K5, K45, KDB, D4, KD5, KD45, B, S4, S5, OM, OB, OK4, OS4, OM + , OB + , OK4 + ,...

This paper surveys methods for representing and reasoning with imperfect information. It opens with an attempt to classify the different types of imperfection that may pervade data, and a discussion of the sources of such imperfections. The classification is then used as a framework for considering work that explicitly concerns the representation of imperfect information, and related work on how imperfect information may be used as a basis for reasoning. The work that is surveyed is drawn from both the field of databases and the field of artificial intelligence. Both of these areas have long been concerned with the problems caused by imperfect information, and this paper stresses the relationships between the approaches developed in each.

In the paper, we investigate the way in which nonmonotonic modal logics depend on their underlying monotonic modal logics. Most notably, we study when different monotonic modal logics define the same nonmonotonic system. In particular, we show that for an important class of the so called stratified theories all nonmonotonic logics considered in the paper, with the exception of S5, coincide. It turns out that in some cases, nonstandard (that is, non-normal) logics have interesting nonmonotonic counterparts. Two such systems are investigated in the paper in detail. For the case of finite theories, all nonmonotonic logics considered are shown to be decidable and an appropriate algorithm is presented.

For many commonsense reasoning tasks associated with action domains, only a relatively simple kind of causal knowledge is required - knowledge of the conditions under which facts are caused. This note introduces a modal nonmonotonic logic for representing causal knowledge of this kind, relates it to other nonmonotonic formalisms, and shows that a variety of causal theories of action can be expressed in it, including the recently proposed causal action theories of Lin. The new logic extends the causal theories formalism of McCain and Turner, and provides a more adequate semantic account of it. A useful subset of the logic has a concise translation into classical propositional logic, and so can be used for automated planning and reasoning about action. A larger subset is closely related to logic programming under the answer set semantics, yielding another approach to automated reasoning.

In Reiter's default logic, the parameters of a default are treated as metavariables for ground terms. We propose an alternative definition of an extension for a default theory, which handles parameters as genuine object variables. The new form of default logic may be preferable when the domain closure assumption is not postulated. It stands in a particularly simple relation to circumscription. Like circumscription, it can be viewed as a syntactic transformation of formulas of higher order logic. 1 Introduction Default logic [Reiter, 1980] is one of the most expressive and most widely used nonmonotonic formalisms. In one respect, however, the main definition of default logic, that of an extension, is not entirely satisfactory. Recall that a default ff : fi 1 ; : : : ; fi m =fl (1) is open if it contains free variables, and closed otherwise. The concept of an extension is defined in two steps: It is first introduced, by means of a fixpoint construction, for default theories without op...

this paper. Here, drawing on all the research mentioned above for inspiration, we present a coherent unified theory of nonmonotonic formal systems. At the level of abstraction we achieve, we are finally able to see that nonmonotone systems pervade ordinary mathematical practice. There is no sign of any realization of the existence of such mathematical examples in the previous nonmonotonic logic literature. Perhaps these connections can only be seen by having a common abstract notion. What this commonality does for us is to make available known mathematical techniques from other areas of conventional mathematics for constructing and classifying belief sets (extensions) and, simultaneously, to make evident a common thread among disparate parts of mathematics and disparate nonmonotonic systems from artificial intelligence and computer science. On the level of Mathematical Philosophy there is a connection worth stating as well. Non-monotone reasoning takes place during the process of discovery of mathematical theorems, when one posits temporarily some proposition on the basis of no evidence against it, and explores the consequences of such a belief until new mathematical facts force their abandonment. These nonmonotone belief sets have their traces eradicated when final belief sets are achieved and demonstrative proofs are finished and published. The only hint of provisional belief sets left in mathematical papers is in the motivational remarks explaining what obstacles were overcome and by what changes in viewpoint the proof was achieved. Here is the main definition. A nonmonotone rule system consists of a set U and a set of triples (ff; fi; fl) called rules. Here ff = (ff 1 ; : : : ; ff n ) is a finite sequence of elements of U , called premises, and fi = (fi 1 ; : : : ...