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.

This paper presents a new notion of typing for logic programs which generalizes the notion of directional types. The generation of type dependencies for a logic program is fully automatic with respect to a given domain of types. The analysis method is based on a novel combination of program abstraction and ACI-unification which is shown to be correct and optimal. Type dependencies are obtained by abstracting programs, replacing concrete terms by their types, and evaluating the meaning of the abstract programs using a standard semantics for logic programs enhanced by ACI-unification. This approach is generic and can be used with any standard semantics. The method is both theoretically clean and easy to implement using general purpose tools. The proposed domain of types is condensing which means that analyses can be carried out in both top-down or bottom-up frameworks with no loss of precision for goal-independent analyses. The proposed method has been fully implemented within a bottom-up approach and the experimental results are promising.

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Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1 ; : : : ; E n in order to obtain a unification algorithm for the union E 1 [ : : : [ E n of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E 1 [ : : : [E n . Our first result says that solvability of disunification problems in the free algebra of the combined theory E 1 [ : : : [E n is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E i (i = 1; : : : ; n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E 1 [ : : : [ E n we have to consider a new kind of subproblem for the particular theories E i , namely solvability (in the free algebra) of disunification problems with linear constant restricti...

A new algorithm for computing a complete set of unifiers for two terms involving associative-commutative function symbols is presented. The algorithm is based on a non-deterministic algorithm given by the authors in 1986 to show the NP-completeness of associative-commutative unifiability. The algorithm is easy to understand, its termination can be easily established. More importantly, its complexity can be easily analyzed and is shown to be doubly exponential in the size of the input terms. The analysis also shows that there is a double-exponential upper bound on the size of a complete set of unifiers of two input terms. Since there is a family of simple associative-commutative unification problems which have complete sets of unifiers whose size is doubly exponential, the algorithm is optimal in its order of complexity in this sense. This is the first associative-commutative unification algorithm whose complexity has been completely analyzed. The approach can also be used to show a singl...

This paper describes an algebraic approach to the sharing analysis of logic programs based on an abstract domain of set logic programs. Set logic programs are logic programs in which the terms are sets of variables and unification is based on an associative, commutative, and idempotent equality theory. All of the basic operations required for sharing analyses, as well as their formal justification, are based on simple algebraic properties of set substitutions and set-based atoms. An ordering on set-based syntactic objects, similar to "less general" on concrete syntactic objects, is shown to reflect the notion of "less sharing" information. The (abstract) unification of a pair of set-based terms corresponds to finding their most general ACI1 unifier with respect to this ordering. The unification of a set of equations between set-based terms is defined exactly as in the concrete case, by solving the equations one by one and repeatedly applying their solutions to the remaini...

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.

We review and compare the main techniques considered to represent finite sets in logic languages. We present a technique that combines the benefits of the previous techniques, avoiding their drawbacks. We show how to verify satisfiability of any conjunction of (positive and negative) literals based on =, ⊆, ∈, and ∪, ∩, \, and ||, viewed as predicate symbols, in a (hybrid) universe of finite sets. We also show that ∪ and | | (i.e., disjointness of two sets) are sufficient to represent all the above mentioned operations. 1

We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only non-negative coefficients is shown to be undecidable. Applications to undecidability proofs of several unification problems are illustrated, one of which, unification modulo one associative-commutative function and one endomorphism, has been a long-standing open problem. The problem of solving multiset constraints is also shown to be undecidable.

A unification algorithm is said to be minimal for a unification problem if it generates exactly a (minimal) complete set of most-general unifiers, without instances, and without repetitions. The aim of this paper is to present a combinatorial minimality study for a significant collection of sample problems that can be used as benchmarks for testing any set-unification algorithm. Based on this combinatorial study, a new Set-Unification Algorithm (named SUA) is also described and proved to be minimal for all the analyzed problems. Furthermore, an existing nave set-unification algorithm has also been tested to show its bad behavior for most of the sample problems.