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.

yosikawaQkyoto-su.ac.jp Abstract: This paper introduces ILOG ( a declarative language in the style of (stratified) datalog ( which can be used for querying, schema translation, and schema augmentation in the context of object-based data models. The semantics of ILOG is based on the use of Skolem functors, and is closely related to semantics for object-based data manipulation languages which provide mechanisms for explicit creation of object identifiers (OIDs). A normal form is presented for ILOG ’ programs not involving recursion through OID creation, which identifies a precise correspondence between OIDs created in the target, and values and OIDs in the source. The expressive power of various sublanguages of ILOG ’ is shown to range from a natural generalization of the conjunctive queries to the object-based context, to a language which can specify all computable database translat.ions (up to duplicate copies). The issue of testing vuliilityof ILOG programs translat.ing one semantic schema to another is studied: cases are presented for which several-validity issues (e.g., functional and/or subset relationships in the

. The declarative programming paradigms used in constraint languages can lead to powerful extensions of Codd's relational data model. The development of constraint database query languages from logical database query languages has many similarities with the development of constraint logic programming from logic programming, but with the additional requirements of data efficient, set-at-a-time, and bottomup evaluation. In this overview of constraint query languages (CQLs) we first present the framework of [41]. The principal idea is that: "the k-tuple (or record) data type can be generalized by a conjunction of quantifier-free constraints over k variables". The generalization must preserve various language properties of the relational data model, e.g., the calculus/algebra equivalence, and have time complexity polynomial in the size of the data. We next present an algebra for dense order constraints that is simpler to evaluate than the calculus described in [41], and we sharpen some of...

The extensions of first-order logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abiteboul and Vianu, 1991b] investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ! 1! (see, for instance, [Kolaitis and Vardi, 1990]). We investigate this logic of finite structures and provide a normal form for it. We also present a treatment of the results in [Abiteboul and Vianu, 1991b] from this point of view. In particular, we show that we can write a formula of FO + LFP that defines ...

: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove that in this context the basic properties of queries (satisfiability, containment, equivalence, etc.) are nonrecursive. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples. We then mainly concentrate on dense order databases, and exhibit...

We study two important extensions of first-order logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...

... constructs that can be used to realize a variety of different semantics for rule application in active database systems. The primary novel feature introduced is the "delayed update", or delta, which is a first-class value representing a set of proposed modifications to the underlying persistent store. Deltas can be created, inspected, and combined without committing to the given modifications. The utility of these concepts for expressing the semantics of active databases is demonstrated through a series of examples, including the presentation of the essential features of rule application in the AP5 system of USC/Information Sciences Institute and the Starburst Rule System being developed at IBM Almaden. Technical results concerning the simulatability of certain fundamental constructs by other fundamental constructs are also presented. The

These two rule-oriented paradigms of databases have been the focus of extensive research and are now coming of age in the commercial DBMS world. However, the systems developed so far support well only one of the two paradigms---thus limiting the effectiveness of such systems in many applications that require complete integration of both kinds of rules. In this paper, we discuss the technical problems that make such an integration difficult, and trace their roots to a lack of a unified underlying semantics. Then, we review recent advances in the semantics of non-monotonic logic and show that they can be used to unify the foundations of active databases and deductive databases. Finally, we outline the design a new rule language for databases that integrates a deductive system with a trigger-based DBMS. 1 Introduction Rules provide the main paradigm for expressing computation in active databases and deductive databases. The unification of the two paradigms represents a research problem o...

: We propose an extension of classical predicate calculus, called Transaction Logic, which provides a logical foundation for the phenomenon of state changes in logic programs and databases. Transaction Logic comes with a natural model theory and a sound and complete proof theory. The proof theory not only verifies programs, but also executes them, which makes this logic an ideal tool for declarative programming of database transactions and state-modifying logic programs. The semantics of Transaction Logic leads naturally to features whose amalgamation in a single logic has proved elusive in the past. These features include hypothetical and committed updates, dynamic constraints on transaction execution, non-determinism, and bulk updates. Finally, Transaction Logic holds promise as a logical model of hitherto non-logical phenomena, including so-called procedural knowledge in AI, and the behavior of object-oriented databases, especially methods with side effects. This paper presents the...

The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as the number of variables, quantifiers, operators, etc. A close correspondence has been observed between these two, with many natural logics corresponding exactly to independently defined complexity classes. For the complexity classes that are generally identified with feasible computation, such characterizations require the presence of a linear order on the domain of every structure, in which case the class PTIME is characterized by an extension of first-order logic by means of an inductive operator. No logical characterization of feasible computation is known for unordered structures. We approach this question from two directions. On the one hand, we seek to accurately characterize the expre...