This paper develops a logical language for representing probabilistic causal laws. Our interest in such a language is two-fold. First, it can be motivated as a fundamental study of the representation of causal knowledge. Causality has an inherent dynamic aspect, which has been studied at the semantical level by Shafer in his framework of probability trees. In such a dynamic context, where the evolution of a domain over time is considered, the idea of a causal law as something which guides this evolution is quite natural. In our formalization, a set of probabilistic causal laws can be used to represent a class of probability trees in a concise, flexible and modular way. In this way, our work extends Shafer's by offering a convenient logical representation for his semantical objects. Second, this language also has relevance for the area of probabilistic logic programming. In particular, we prove that the formal semantics of a theory in our language can be equivalently defined as a probability distribution over the well-founded models of certain logic programs, rendering it formally quite similar to existing languages such as ICL or PRISM. Because we can motivate and explain our language in a completely self-contained way as a representation of probabilistic causal laws, this provides a new way of explaining the intuitions behind such probabilistic logic programs: we can say precisely which knowledge such a program expresses, in terms that are equally understandable by a non-logician. Moreover, we also obtain an additional piece of knowledge representation methodology for probabilistic logic programs, by showing how they can express probabilistic causal laws.

On the one hand, termination analysis of logic programs is now a fairly established research topic within the logic programming community. On the other hand, non-termination analysis seems to remain a much less attractive subject. If we divide this line of research into two kinds of approaches, dynamic versus static analysis, this paper belongs to the latter. It proposes a criterion for detecting non-terminating atomic queries with respect to binary constraint logic programming (CLP) rules, which strictly generalizes our previous works on this subject. We give a generic operational definition and an implemented logical form of this criterion. Then we show that the logical form is correct and complete with respect to the operational definition.

This paper presents a computational model for the cooperation of constraint domains and an implementation for a particular case of practical importance. The computational model supports declarative programming with lazy and possibly higher-order functions, predicates, and the cooperation of different constraint domains equipped with their respective solvers, relying on a so-called constraint functional logic programming (CFLP) scheme. The implementation has been developed on top of the CFLP system , supporting the cooperation of the three domains ℋ, ℛ, and ℱ , which supply equality and disequality constraints over symbolic terms, arithmetic constraints over the real numbers, and finite domain constraints over the integers, respectively. The computational model has been proved sound and complete w.r.t. the declarative semantics provided by the CFLP scheme, while the implemented system has been tested with a set of benchmarks and shown to behave quite efficiently in comparison to the closest related approach we are aware of.

The paper introduces the notion of offline justification for answer set programming (ASP). Justifications provide a graph-based explanation of the truth value of an atom with respect to a given answer set. The paper extends also this notion to provide justification of atoms during the computation of an answer set (on-line justification) and presents an integration of online justifications within the computation model of Smodels. Offline and online justifications provide useful tools to enhance understanding of ASP, and they offer a basic data structure to support methodologies and tools for debugging answer set programs. A preliminary implementation has been developed in – .

We present the CIFF proof procedure for abductive logic programming with constraints, and we prove its correctness. CIFF is an extension of the IFF proof procedure for abductive logic programming, relaxing the original restrictions over variable quantification (allowedness conditions) and incorporating a constraint solver to deal with numerical constraints as in constraint logic programming. Finally, we describe the CIFF system, comparing it with state-of-the-art abductive systems and answer set solvers and showing how to use it to program some applications.

Equilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs into quantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to as quantified Boolean formulas (QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are the consistency problem, brave reasoning and skeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz. ordinary, strong and uniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds. Hereby, our encodings yield upper bounds in a direct manner. Besides this useful feature, our approach has the following benefits: First, our encodings yield a uniform axiomatisation for a variety of problems in a common language. Second, extant solvers for QBFs can be used as back-end inference engines to realise implementations of the encoded task in a rapid prototyping manner. Third, our axiomatisations also allow us to straightforwardly relate equilibrium logic with circumscription.

This paper investigates the relationship between the Logical Algorithms (LA) language of Ganzinger and McAllester and Constraint Handling Rules (CHR). We present a translation schema from LA to CHRrp: CHR with rule priorities, and show that the meta-complexity theorem for LA can be applied to a subset of CHRrp via inverse translation. Inspired by the high-level implementation proposal for Logical Algorithm by Ganzinger and McAllester and based on a new scheduling algorithm, we propose an alternative implementation for CHRrp that gives strong complexity guarantees and results in a new and accurate meta-complexity theorem for CHRrp. It is furthermore shown that the translation from Logical Algorithms to CHRrp combined with the new CHRrp implementation satisfies the required complexity for the Logical Algorithms meta-complexity result to hold.

This paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond–Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al. (2007), which is based on a fixpoint approach. One advantage of our approach is that it can be applied, in a natural way, to define stable models for disjunctive logic programs with constraint atoms, which may appear in the disjunctive head as well as in the body of a rule. As a result, our approach to the stable model semantics for logic programs with constraint atoms generalizes a number of previous approaches. Second, we show that our abstract representation of constraint atoms provides a means to characterize dependencies of atoms in a program with constraint atoms, so that some standard characterizations and properties relying on these dependencies in the past for logic programs with ordinary atoms can be extended to logic programs with constraint atoms.

This paper develops a declarative language, P-log, that combines logical and probabilistic arguments in its reasoning. Answer Set Prolog is used as the logical foundation, while causal Bayes nets serve as a probabilistic foundation. We give several non-trivial examples and illustrate the use of P-log for knowledge representation and updating of knowledge. We argue that our approach to updates is more appealing than existing approaches. We give sufficiency conditions for the coherency of P-log programs and show that Bayes nets can be easily mapped to coherent P-log programs.

We present a heuristic framework for attacking the undecidable termination problem of logic programs, as an alternative to current termination/nontermination proof approaches. We introduce an idea of termination prediction, which predicts termination of a logic program in case that neither a termination nor a non-termination proof is applicable. We establish a necessary and sufficient characterization of infinite (generalized) SLDNF-derivations with arbitrary (concrete or moded) queries, and develop an algorithm that predicts termination of general logic programs with arbitrary nonfloundering queries. We have implemented a termination prediction tool and obtained quite satisfactory experimental results. Except for five programs which break the experiment time limit, our prediction is 100% correct for all 296 benchmark programs of the Termination Competition 2007, of which 18 programs cannot be proved by any of the existing state-of-the-art analyzers like AProVE07, NTI, Polytool, and TALP.

We face the problems of correctness, optimality, and precision for the static analysis of logic programs, using the theory of abstract interpretation. We propose a framework with a denotational, goal-dependent semantics equipped with two unification operators for forward unification (calling a procedure) and backward unification (returning from a procedure). The latter is implemented through a matching operation. Our proposal clarifies and unifies many different frameworks and ideas on static analysis of logic programming in a single, formal setting. On the abstract side, we focus on the domain sharing by Jacobs and Langen (The Journal of Logic Programming, 1992, vol. 13, nos. 2–3, pp. 291–314) and provide the best correct approximation of all the primitive semantic operators, namely, projection, renaming, and forward and backward unifications. We show that the abstract unification operators are strictly more precise than those in the literature defined over the same abstract domain. In some cases, our operators are more precise than those developed for more complex domains involving linearity and freeness.

The paper introduces fuzzy linguistic logic programming, which is a combination of fuzzy logic programming, introduced by P. Vojtáš, and hedge algebras in order to facilitate the representation and reasoning on human knowledge expressed in natural languages. In fuzzy linguistic logic programming, truth values are linguistic ones, e.g., VeryTrue, VeryProbablyTrue and LittleFalse, taken from a hedge algebra of a linguistic truth variable, and linguistic hedges (modifiers) can be used as unary connectives in formulae. This is motivated by the fact that humans reason mostly in terms of linguistic terms rather than in terms of numbers, and linguistic hedges are often used in natural languages to express different levels of emphasis. The paper presents: (a) the language of fuzzy linguistic logic programming; (b) a declarative semantics in terms of Herbrand interpretations and models; (c) a procedural semantics which directly manipulates linguistic terms to compute a lower bound to the truth value of a query, and proves its soundness; (d) a fixpoint semantics of logic programs, and based on it, proves the completeness of the procedural semantics; (e) several applications of fuzzy linguistic logic programming; and (f) an idea of implementing a system to execute fuzzy linguistic logic programs.

A recent framework of relativized hyperequivalence of programs offers a unifying generalization of strong and uniform equivalence. It seems to be especially well suited for applications in program optimization and modular programming due to its flexibility that allows us to restrict, independently of each other, the head and body alphabets in context programs. We study relativized hyperequivalence for the three semantics of logic programs given by stable, supported, and supported minimal models. For each semantics, we identify four types of contexts, depending on whether the head and body alphabets are given directly or as the complement of a given set. Hyperequivalence relative to contexts where the head and body alphabets are specified directly has been studied before. In this paper, we establish the complexity of deciding relativized hyperequivalence with respect to the three other types of context programs.

Disjunctive finitary programs are a class of logic programs admitting function symbols and hence infinite domains. They have very good computational properties; for example, ground queries are decidable, while in the general case the stable model semantics are Π11-hard. In this paper we prove that a larger class of programs, called finitely recursive programs, preserve most of the good properties of finitary programs under the stable model semantics, which are as follows: (i) finitely recursive programs enjoy a compactness property; (ii) inconsistency checking and skeptical reasoning are semidecidable; (iii) skeptical resolution is complete for normal finitely recursive programs. Moreover, we show how to check inconsistency and answer skeptical queries using finite subsets of the ground program instantiation. We achieve this by extending the splitting sequence theorem by Lifschitz and Turner: we prove that if the input program P is finitely recursive, then the partial stable models determined by any smooth splitting ω-sequence converge to a stable model of P.

Traditional algorithms for description logic (DL) instance retrieval are inefficient for large amounts of underlying data. As DL is becoming more and more popular in areas such as the Semantic Web and information integration, it is very important to have systems which can reason efficiently over large data sets. In this paper we present an approach to transform DL axioms, formalised in the DL language, into a Prolog program under the unique name assumption. This transformation is performed with no knowledge about particular individuals: they are accessed dynamically during the normal Prolog execution of the generated program. This technique, together with the top-down Prolog execution, implies that only those pieces of data are accessed that are indeed important for answering the query. This makes it possible to store the individuals in a database instead of memory, which results in better scalability and helps in using DL ontologies directly on top of existing information sources. The transformation process consists of two steps: (1) the DL axioms are converted to first-order clauses of a restricted form, and (2) a Prolog program is generated from these clauses. Step (2), which is the focus of the present paper, actually works on more general clauses than those obtainable by applying step (1) to a knowledge base. We first present a base transformation, the output of which can be either executed using a simple interpreter or further extended to executable Prolog code. We then discuss several optimisation techniques, applicable to the output of the base transformation. Some of these techniques are specific to our approach, while others are general enough to be interesting for DL reasoner implementors not using Prolog. We give an overview of DLog, a DL reasoner in Prolog, which is an implementation of the techniques outlined above. We evaluate the performance of DLog and compare it to some widely used DL reasoners, such as RacerPro, Pellet and KAON2.