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.

The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies.

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...

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence to

The class of locally stratified logic programs is shown to be Π 1 1-complete by the construction of a reducibility of the class of infinitely branching nondeterministic finite register machines. 1

We characterize the expressiveness and data complexity of nonrecursive logic programming. Our main results show that nonrecursive logic programming has the same expressive power as nonrecursive Datalog with negation (under a natural representation of inputs). Thus, the use of recursive data structures, namely trees, in nonrecursive Datalog : gives no gain in the expressive power. It also follows from our results that nonrecursive logic programming, like nonrecursive Datalog : , has polynomial data complexity. This contrasts with a huge difference between these query languages in the program complexity. y Copyright c fl Evgeny Dantsin and Andrei Voronkov, 1998. This technical report and other technical reports in this series can be obtained at http://www.csd.uu.se/papers/reports.html or at ftp://ftp.csd.uu.se/pub/papers/reports. Some reports can be updated, check one of these addresses for the latest version. y Evgeny Dantsin is partially supported by grants from TFR, INTAS a...

. In general, the least fixed point of a positive elementary inductive definition over the Herbrand universe is # 1 1 and has no computational meaning. The finite stages, however, are computable, since validity of equality formulas in the Herbrand universe is decidable. We set up a formal system BID for the finite stages of positive elementary inductive definitions over the Herbrand universe and show that the provably total functions of the system are exactly that of Peano arithmetic. The formal system BID contains the so-called inductive extension of a logic program as a special case. This first-order theory can be used to prove termination and correctness properties of pure Prolog programs, since notions like negation-as-failure and left-termination can be turned into positive inductive definitions. 1 Why inductive definitions over the Herbrand universe? In traditional logic programming, the semantics of a program is always given by the least fixed point of a monotonic operator over the Herbrand universe. The first example is the well-known van Emden-Kowalski operator for definite Horn clause programs in [25]. This operator is defined by a purely existential formula and is therefore continuous. The least fixed point of the operator is recursively enumerable. Moreover, the finite stages of the inductive definition are exactly what is computed by SLD-resolution. In [11], Fitting has generalized the van Emden-Kowalski operator using threevalued logic to programs which may also contain negation in the bodies of the clauses. Although Fitting's operator is still monotonic it is no longer continuous. It follows from Blair [2] and Kunen [14] that the least fixed point of this operator can be # 1 1 -complete and that the closure ordinal can be # CK 1 even for definite Horn...

. This paper surveys various complexity and expressiveness results on different forms of

This paper surveys various complexity and expressiveness results on dierent 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 dierent forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming