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 study adding aggregate operators, such as summing up elements of a column of a relation, to logics with counting mechanisms. The primary motivation comes from database applications, where aggregate operators are present in all real life query languages. Unlike other features of query languages, aggregates are not adequately captured by the existing logical formalisms. Consequently, all previous approaches to analyzing the expressive power of aggregation were only capable of producing partial results, depending on the allowed class of aggregate and arithmetic operations. We consider a powerful counting logic, and extend it with the set of all aggregate operators. We show that the resulting logic satis es analogs of Hanf's and Gaifman's theorems, meaning that it can only express local properties. We consider a database query language that expresses all the standard aggregates found in commercial query languages, and show how it can be translated into the aggregate logic, thereby pro...

Most proofs showing limitations of expressive power of first-order logic rely on Ehrenfeucht-Fraisse games. Playing the game often involves a nontrivial combinatorial argument, so it was proposed to find easier tools for proving expressivity bounds. Most of those known for first-order logic are based on its "locality", that is defined in different ways. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressivity bounds. These results apply beyond the first-order case. We use them to derive expressivity bounds for first-order logic with unary quantifiers and counting. Finally, we apply these results to relational database...

We examine the power of incremental evaluation systems that use an SQL-like language for maintaining recursively-defined views. We show that recursive queries such as transitive closure, and "alternating paths" can be incrementally maintained in a nested relational language, when some auxiliary relations are allowed. In the presence of aggregate functions, even more queries can be maintained, for example, the "same generation" query. In contrast, it is still an open problem whether such queries are maintainable in relational calculus. We then restrict the language so that no nested relations are involved (but wekeep the aggregate functions). Such a language captures the capability of most practical relational database systems. We prove that this restriction does not reduce the incremental computational power; that is, any query that can be maintained in a nested language with aggregates, is still maintainable using only flat relations. We also show that one does not need auxiliar...

The expressive power of first-order logic over finite structures is limited in two ways: it lacks a recursion mechanism, and it cannot count. Overcoming the first limitation has been a subject of extensive study. A number of fixpoint logics have been introduced, and shown to be subsumed by an infinitary logic L ! 1! . This logic is easier to analyze than fixpoint logics, and it still lacks counting power, as it has a 0-1 law. On the counting side, there is no analog of L ! 1! . There are a number of logics with counting power, usually introduced via generalized quantifiers. Most known expressivity bounds are based on the fact that counting extensions of first-order logic preserve the locality properties. This paper has three main goals. First, we introduce a new logic L 1! (C) that plays the same role for counting as L ! 1! does for recursion -- it subsumes a number of extensions of first-order logic with counting, and has nice properties that make it easy to study. Second, we ...

Views are a central component of both traditional database systems and new applications such as data warehouses. Very often the desired views (e.g. the transitive closure) cannot be defined in the standard language of the underlying database system. Fortu-nately, it is often possible to incrementally maintain these views using the standard language. For exam-ple, transitive closure of acyclic graphs, and of undi-rected graphs, can be maintained in relational cal-culus after both single edge insertions and deletions. Many such results have been published in the theoret-ical database community. The purpose of this survey is to make these useful results known to the wider database research and development community. There are many interesting issues involved in the maintenance of recursive views. A maintenance al-gorithm may be applicable to just one view, or to a class of views specified by a view definition language such as Datalog. The maintenance algorithm can be specified in a maintenance language of different ex-pressiveness, such as the conjunctive queries, the re-lational calculus or SQL. Ideally, this maintenance language should be less expensive than the view def-inition language. The maintenance algorithm may allow updates of different kinds, such as just single tu-ple insertions, just single tuple deletions, special set-based insertions and/or deletions, or combinations thereof. The view maintenance algorithms may also need to maintain auxiliary relations to help maintain the views of interest. It is of interest to know the minimal arity necessary for these auxiliary relations

A query is local if...

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This survey article introduces into the essential concepts and methods underlying rule-based query languages. It covers four complementary areas: declarative semantics based on adaptations of mathematical logic, operational semantics, complexity and expressive power, and optimisation of query evaluation. The treatment of these areas is foundation-oriented, the foundations having resulted from over four decades of research in the logic programming and database communities on combinations of query languages and rules. These results have later formed the basis for conceiving, improving, and implementing several Web and Semantic Web technologies, in particular query languages such as XQuery or SPARQL for querying relational, XML, and RDF data, and rule languages like the “Rule Interchange Framework (RIF) ” currently being developed in a working group of the W3C. Coverage of the article is deliberately limited to declarative languages in a classical setting: issues such as query answering in F-Logic or in description logics, or the relationship of query answering to reactive rules and events, are not addressed.

ing with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or permissions@acm.org. 2 \Delta 1. INTRODUCTION The expressive power of first-order logic (FO) on finite structures is rather limited. Two main limitations of first-order logic are its inability to count and the lack of a recursion mechanism. Since first-order logic over finite structures plays an important role in several areas of computer science (e.g., databases and complexity), various extensions have been proposed to deal with these shortcomings. On the recursion side, a beautiful theory has been developed over the past decade. Various fixpoint extensions of first-order logic have been introduced, including least, inflationary and partial fixpoint...