Introduction

This issue of the Annals of Mathematics and Artificial Intelligence brings together papers from prominent researchers in foundations of logic programming and deductive databases in which complexity and expressive power are investigated.

Issues of complexity and expressive power arise predominantly by altering the core fragment of first-order logic used in logic programs, namely Horn clauses in the form A ~ B 1 A ... /k Bn, to allow negation, disjunction, restricting the usage of function symbols, and incorporating operations involving conceptually higher-order notions, for example, Prolog's setof. There are derivative or attendant sources of issues of complexity and expressive power. Negation as failure, for example, is a principal form of nonmonotonic reasoning. Also, the idiosyncrasies and pragmatics of a logic programming language implementation play an obviously important role in its expressive power. Each of the papers in this issue treats complexity or expressive power arising from one or more of these sources.

John Schlipf's paper surveys much of what was known until recently about expressive power and fills in gaps in our previous knowledge with substantial new results.

Thomas Eiter and Georg Gottlob treat complexity issues arising with disjunc- tive propositional logic programs, which in a suitable sense are identifiable with disjunctive function symbol-free programs. In this setting, complexities reach into level 2 of the polynomial hierarchy. They also show how various results transfer to nonmonotonic formalisms.

Audrey Ferry observes that while the complexity of the set of stable models of a program is understood, not every set of interpretations of this level of complexity is the set of stable models of some program. The proper characterization turns out to be topological and is related to notions that arise in the study of continuous lattices.

Victor Marek, Arcot Rajasekar and Miroslaw Truszczyfiski study the complexity of computing with programs built upon the idea of replacing the atomic formulas of a Horn clause program with formulas from a designated class .Y'. Such programs bear a strong relationship to certain kinds of theories in default logic.

Yael Moscowitz and Ehud Shapiro introduce structural simplicity and compare families of abstract machines by algebraic methods.

Inderpal Singh Mumick and Oded Shmueli study the expressive power of various deductive database languages extended with stratified aggregation, showing that extending only by stratified aggregation is insufficient for expressing all computable queries, and present additional extensions to Datalog that are sufficient.

Andrei Voronkov reconsiders the computational universality of logic programs as Prolog programs taking account of the fact that the expressive

�9 J.C. Baltzer AG, Science Publishers

Introduction

power of a strictly logical fragment of the implemented language may fail to account for much of the language's expressive power as well as its means of expression.

Each of these papers was presented in preliminary form at the first workshop on Structural Complexity and Recursion Theoretic Methods in Logic Programming held in conjunction with the Joint International Conference and Symposium on Logic Programming in November, 1992. The papers have been recently updated and remain current. Open problems discussed in several papers remain open.

I would like to express my thanks to my co-organizers of the original work- shop: Victor Marek, Anil Nerode, and Jeffrey Remmel. Without their assistance and encouragement these papers would not have been brought together to achieve the unique focus of this special issue.

Howard Blair Syracuse University February 2, 1995