This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.

We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the proof-theoretic and model-theoretic issues and the relationships between them.

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.

In this paper we reexamine the place and role of stable model semantics in logic programming and contrast it with a least Herbrand model approach to Horn programs. We demonstrate that inherent features of stable model semantics naturally lead to a logic programming system that offers an interesting alternative to more traditional logic programming styles of Horn logic programming, stratified logic programming and logic programming with well-founded semantics. The proposed approach is based on the interpretation of program clauses as constraints. In this setting programs do not describe a single intended model, but a family of stable models. These stable models encode solutions to the constraint satisfaction problem described by the program. Our approach imposes restrictions on the syntax of logic programs. In particular, function symbols are eliminated from the language. We argue that the resulting logic programming system is well-attuned to problems in the class NP, has a well-defined domain of applications, and an emerging methodology of programming. We point out that what makes the whole approach viable is recent progress in implementations of algorithms to compute stable models of propositional logic programs. 1

This paper surveys the main results appeared in the literature on the computational complexity of non-monotonic inference tasks. We not only give results about the tractability/intractability of the individual problems but we also analyze sources of complexity and explain intuitively the nature of easy/hard cases. We focus mainly on non-monotonic formalisms, like default logic, autoepistemic logic, circumscription, closed-world reasoning and abduction, whose relations with logic programming are clear and well studied. Complexity as well as recursion-theoretic results are surveyed. Work partially supported by the ESPRIT Basic Research Action COMPULOG and the Progetto Finalizzato Informatica of the CNR (Italian Research Council). The first author is supported by a CNR scholarship 1 Introduction Non-monotonic logics and negation as failure in logic programming have been defined with the goal of providing formal tools for the representation of default information. One of the ideas und...

We study the expressive powers of two semantics for deductive databases and logic programming: the well-founded semantics and the stable semantics. We compare them especially to two older semantics, the two-valued and three-valued program completion semantics. We identify the expressive power of the stable semantics, and in fairly general circumstances that of the well-founded semantics. In particular, over infinite Herbrand universes, the four semantics all have the same expressive power. We discuss a feature of certain logic programming semantics, which we call the Principle of Stratification, a feature allowing a program to be built easily in modules. The three-valued program completion and well-founded semantics satisfy this principle. Over infinite Herbrand models, we consider a notion of translatability between the three-valued program completion and well-founded semantics which is in a sense uniform in the strata. In this sense of uniform translatability we show the well-founded semantics to be more expressive than the three-valued program completion. The proof is a corollary of our result that over non-Herbrand infinite models, the well-founded semantics is more expressive than the three-valued program completion semantics. 1

. At a workshop held in Toulouse, France in 1977, Gallaire, Minker and Nicolas stated that logic and databases was a field in its own right (see [131]). This was the first time that this designation was made. The impetus for this started approximately twenty years ago in 1976 when I visited Gallaire and Nicolas in Toulouse, France, which culminated in a workshop held in Toulouse, France in 1977. It is appropriate, then to provide an assessment as to what has been achieved in the twenty years since the field started as a distinct discipline. In this retrospective I shall review developments that have taken place in the field, assess the contributions that have been made, consider the status of implementations of deductive databases and discuss the future of work in this area. 1 Introduction As described in [234], the use of logic and deduction in databases started in the late 1960s. Prominent among the developments was the work by Levien and Maron [202, 203, 199, 200, 201] and Kuhns [1...

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 ; : : : ...

Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a non-monotonic reasoning formalism strongly related to Default Logic and Auto-epistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent non-monotone inductive definitions. It is argued that this thesis results in an alternative justification of the well-founded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy to comprehend meaning

In this tutorial-overview, which resulted from a lecture course given by the authors at