An important limitation of traditional logic programming as a knowledge representation tool, in comparison with classical logic, is that logic programming does not allow us to deal directly with incomplete information. In order to overcome this limitation, we extend the class of general logic programs by including classical negation, in addition to negation-as-failure. The semantics of such extended programs is based on the method of stable models. The concept of a disjunctive database can be extended in a similar way. We show that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available. Computationally, classical negation can be eliminated from extended programs by a simple preprocessor. Extended programs are identical to a special case of default theories in the sense of Reiter. 1 Introduction An important limitation of traditional logic programming as a knowledge representation tool, in comp...

The idea of circumscription can be explained on a simple example. We would like to represent information about the locations of blocks in a blocks world, using the "default":

In this paper, we review recent work aimed at the application of declarative logic programming to knowledge representation in artificial intelligence. We consider exten- sions of the language of definite logic programs by classical (strong) negation, disjunc- tion, and some modal operators and show how each of the added features extends the representational power of the language.

The alternating fixpoint of a logic program with negation is defined constructively. The underlying idea is monotonically to build up a set of negative conclusions until the least fixpoint is reached, using a transformation related to the one that defines stable models. From a fixed set of negative conclusions, the positive conclusions follow (without deriving any further negative ones), by traditional Horn clause semantics. The union of positive and negative conclusions is called the alternating xpoint partial model. The name "alternating" was chosen because the transformation runs in two passes; the first pass transforms an underestimate of the set of negative conclusions into an (intermediate) overestimate; the second pass transforms the overestimate into a new underestimate; the composition of the two passes is monotonic. The principal contributions of this work are (1) that the alternating fixpoint partial model is identical to the well-founded partial model, and (2) that alternating xpoint logic is at least as expressive as xpoint logic on all structures. Also, on finite structures, fixpoint logic is as expressive as alternating fixpoint logic.

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.

We introduce 3-valued extensions of major non-monotonic formalisms and we prove that the recently proposed well-founded semantics of logic programs is equivalent, for arbitrary logic programs, to 3-valued forms of McCarthy's circumscription, Reiter's closed world assumption, Moore's autoepistemic logic and Reiter's default theory. This result not only provides a further justification of the well-founded semantics, as a natural extension of the perfect model semantics from the class of stratified programs to the class of all logic programs, but it also establishes the class of all logic programs as a large class of theories, for which natural forms of all four non-monotonic formalisms coincide. It also paves the way for using efficient computation methods, developed for logic programming, as inference mechanisms for non-monotonic reasoning. 1 Introduction A precise meaning or semantics must be associated with any logic program or a deductive database in order to provide its declarative...

The focus of this paper is nonmonotonic reasoning as it relates to logic programming. I discuss the pre-history of nonmonotonic reasoning starting from approximately 1958. I then review the research that has been accomplished in the areas of circumscription, default theory, modal theories and logic programming. The overview includes the major results developed including complexity results that are known about the various theories. I then provide a summary which includes an assessment of the field and what must be done to further research in nonmonotonic reasoning and logic programming. 1 Introduction Classical logic has played a major role in computer science. It has been an important tool both for the development of architecture and of software. Logicians have contended that reasoning, as performed by humans, is also amenable to analysis using classical logic. However, workers in the field of artificial 1 This paper is an updated version of an invited Banquet Address, First Interna...

. After a short analysis of the requirements that a knowledge representation language must satisfy, we introduce Description Logics, Modal Logics, and Nonmonotonic Logics as formalisms for representing terminological knowledge, time-dependent or subjective knowledge, and incomplete knowledge respectively. At the end of each section, we briefly comment on the connection to Logic Programming. 1 Introduction This section is concerned with the question under which conditions one may rightfully claim to have represented knowledge about an application domain, and not just stored data occurring in this domain. 1 In the early days of Artificial Intelligence and Knowledge Representation, there was a heated discussion on whether logic can at all be used as a formalism for Knowledge Representation (see e.g. [135, 91, 92]). One aspect of the requirements on knowledge representation formalisms that can be derived from the considerations in this section is very well satisfied by logical for...

Moore's autoepistemic logic AEL proved to be a very successful approach to formalizing non-monotonic reasoning and logic programming. However, AEL also has some important drawbacks, e.g., quite "reasonable" theories are often inconsistent in AEL, it does not always lead to the expected, intended semantics and it cannot be effectively computed even for very simple classes of theories. In this paper we propose a more general approach to autoepistemic reasoning by introducing Autoepistemic Logics of Closed Beliefs AEL cl , where j= cl denotes a specific negative introspection inference operator ("closed world assumption") on which negative introspection in this logic is based. Negative introspection determines which formulae in autoepistemic logic are disbelieved, or, putting it differently, negation of which formulae can be assumed by default. It determines therefore the set of closed world beliefs derivable in a given autoepistemic logic. Moore's autoepistemic logic AEL is a specia...

This article discusses the problems and difficulties, the results so far, and some improvements in logic and logical languages that may be required to formalize common sense. Fundamental conceptual advances are almost certainly required. The object of the paper is to get more help for AI from philosophical logicians. Some of the requested help will be mostly philosophical and some will be logical. Likewise the concrete AI approach may fertilize philosophical logic as physics has repeatedly fertilized mathematics.