Abstract We investigate the expressive power of Higher-Order $Datalog^\neg$ under both the well-founded and the stable model semantics, establishing tight connections with complexity classes. We prove that under the well-founded semantics, for all $k\geq 1$ , $(k+1)$ -Order $Datalog^\neg$ captures $k-\textsf {EXP}$ , a result that holds without explicit ordering of the input database. The proof of this fact can be performed either by using the powerful existential predicate variables of the language or by using partially applied relations and relation enumeration. Furthermore, we demonstrate that this expressive power is retained within a stratified fragment of the language. Under the stable model semantics, we show that $(k+1)$ -Order $Datalog^\neg$ captures $\textsf {co}-(k-\textsf {NEXP})$ using cautious reasoning and $k-\textsf {NEXP}$ using brave reasoning, again with analogous results for the stratified fragment augmented with choice rules. Our results establish a hierarchy of expressive power, highlighting an interesting trade-off between order and non-determinism in the context of higher-order logic programing: increasing the order of programs under the well-founded semantics can surpass the expressive power of lower-order programs under the stable model semantics.

We extend the well-known representation theorem for interlaced bilattices to the broader class of weak interlaced bilattices. Based on this new theorem, we develop a fixpoint theory for non-monotone functions over weak infinitarily interlaced bilattices. Our theory generalizes classical fixpoint constructions introduced by Fitting, as-well-as recent results in the area of approximation fixpoint theory. We argue that the proposed theory has direct practical applications: we develop the semantics of higher-order logic programming with negation under an arbitrary weak infinitarily interlaced bilattice with negation, generalizing in this way recent work on the three-valued semantics of this formalism. We consider a line of research, initiated by Fitting, which investigates the structure of the consistent parts of bilattices in order to obtain natural generalizations of Kleene’s three-valued logic. We demonstrate that the consistent parts of bilattices are closely connected to weak bilattices, generalizing previous results of Fitting and Kondo.

Abstract We propose a stable model semantics for higher-order logic programs. Our semantics is developed using Approximation Fixpoint Theory (AFT), a powerful formalism that has successfully been used to give meaning to diverse non-monotonic formalisms. The proposed semantics generalizes the classical two-valued stable model semantics of Gelfond and Lifschitz as well as the three-valued one of Przymusinski, retaining their desirable properties. Due to the use of AFT, we also get for free alternative semantics for higher-order logic programs, namely supported model, Kripke-Kleene, and well-founded. Additionally, we define a broad class of stratified higher-order logic programs and demonstrate that they have a unique two-valued higher-order stable model which coincides with the well-founded semantics of such programs. We provide a number of examples in different application domains, which demonstrate that higher-order logic programming under the stable model semantics is a powerful and versatile formalism, which can potentially form the basis of novel ASP systems.

Abstract Logic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing preferential disjunctions in the heads of program rules. The initial semantics of LPODs, although simple and quite intuitive, is not purely model-theoretic. As a result, certain properties of programs appear non-trivial to formalize in purely logical terms. For example, the current characterization of strong equivalence for LPODs, does not coincide with logical equivalence in some specific logic. This comes in sharp contrast with the well-known characterization of strong equivalence for classical logic programs, which coincides with logical equivalence in the logic of here-and-there. In this paper we obtain a purely logical characterization of strong equivalence for LPODs as logical equivalence in a four-valued logic. Moreover, we provide a new proof of the coNP-completeness of strong equivalence for LPODs, which has an interest in its own right since it relies on the special structure of such programs. Our results are based on the recent logical semantics of LPODs, a fact which we believe indicates that this new semantics may prove to be a useful tool in the further study of LPODs.

We introduce lexicographic logic, an extension of propositional logic that can represent a variety of preferences, most notably lexicographic ones. The proposed logic supports a simple new connective whose semantics can be defined in terms of finite lists of truth values. We demonstrate that, despite the well-known theoretical limitations that pose barriers to the quantitative representation of lexicographic preferences, there exists a subset of the rational numbers over which the proposed new connective can be naturally defined. Lexicographic logic can be used to define in a simple way some well-known preferential operators, like "A and if possible B", and "A or failing that B". We argue that the new logic is an effective formalism for ranking query results according to the satisfaction level of user preferences.

Logic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing alternatives with decreasing degrees of preference in the heads of program rules. Despite the fact that the operational meaning of ordered disjunction is clear, there exists an important open issue regarding its semantics. In particular, there does not exist a purely model-theoretic approach for determining the most preferred models of an LPOD. At present, the selection of the most preferred models is performed using a technique that is not based exclusively on the models of the program and in certain cases produces counterintuitive results. We provide a novel, model-theoretic semantics for LPODs, which uses an additional truth value in order to identify the most preferred models of a program. We demonstrate that the proposed approach overcomes the shortcomings of the traditional semantics of LPODs. Moreover, the new approach can be used to define the semantics of a natural class of logic programs that can have both ordered and classical disjunctions in the heads of clauses. This allows programs that can express not only strict levels of preferences but also alternatives that are equally preferred. This work is under consideration for acceptance in TPLP.

Abstract A classical result in descriptive complexity theory states that Datalog expresses exactly the class of polynomially computable queries on ordered databases (Papadimitriou 1985; Grädel 1992; Vardi 1982; Immerman 1986; Leivant 1989). In this paper we extend this result to the case of higher-order Datalog. In particular, we demonstrate that on ordered databases, for all k ≥ 2, k-order Datalog captures (k − 1)-EXPTIME. This result suggests that higher-order extensions of Datalog possess superior expressive power and they are worthwhile of further investigation both in theory and in practice.

In this paper we examine the problem of providing a purely extensional three-valued semantics for higher-order logic programs with negation. We demonstrate that a technique that was proposed by M. Bezem for providing extensional semantics to positive higher-order logic programs, fails when applied to higher-order logic programs with negation. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved by the technique. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting can not distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

Abstract We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotone-monotone. By deriving an extension ofconsistent approximation fixpoint theory(Deneckeret al.2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually aminimalone. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal an appealing formulation for capturing thewell-founded semantics for higher-order logic programs.

Abstract M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

We consider the problem of concisely representing and handling preferences in logic programming and relational databases. Our starting point is a well-known proposal [8] which advocates the embedding of first-order preference formulas into relational algebra through a single winnow operator that is parameterized by a database relation and a preference formula. We argue that despite its elegance, the framework of [8] has a number of shortcomings: only intrinsic preference formulas are supported, the preference relations and preference queries are expressed in two different languages, and there is no direct way to define alternative operators beyond winnow. We propose the use of higher-order logic programming as a logical framework that remedies all the above deficiencies. In particular, the proposed framework supports both intrinsic and extrinsic preference formulas, it can represent both preference relations as-well-as queries, and it can be used to define a variety of interesting alternative operators beyond winnow. We demonstrate the feasibility of our approach by presenting an implementation of all the proposed concepts in the higher-order logic programming language Hilog.

We propose the new logic programming language PrefLog, which is based on an infinite-valued logic in order to support operators for expressing preferences. We demonstrate that if the operators used are continuous over the infinite-valued underlying domain, then the resulting logic programming language retains the well-known properties of classical logic programming (and most notably the existence of a least Herbrand model). We argue that one can define simple and natural new continuous operators by using a small set of operators that are easily shown to be continuous. Finally, we demonstrate that despite the fact that the underlying truth domain and the set of possible interpretations of a PrefLog program are infinite, we can define a terminating bottom-up proof procedure for implementing a significant and useful fragment of the language.

Abstract Extensional higher-order logic programming has been introduced as a generalization of classical logic programming. An important characteristic of this paradigm is that it preserves all the well-known properties of traditional logic programming. In this paper we consider the semantics of negation in the context of the new paradigm. Using some recent results from non-monotonic fixed-point theory, we demonstrate that every higher-order logic program with negation has a unique minimum infinite-valued model. In this way we obtain the first purely model-theoretic semantics for negation in extensional higher-order logic programming. Using our approach, we resolve an old paradox that was introduced by W. W. Wadge in order to demonstrate the semantic difficulties of higher-order logic programming.

We propose a purely extensional semantics for higher-order logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixed-point of an immediate consequence operator. We also propose an SLD-resolution proof system which is proven sound and complete with respect to the minimum Herbrand model semantics. In other words, we provide a purely extensional theoretical framework for higher-order logic programming which generalizes the familiar theory of classical (first-order) logic programming.

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We give a purely model-theoretic characterization of the semantics of logic programs with negation-as-failure allowed in clause bodies. In our semantics, the meaning of a program is, as in the classical case, the unique minimum model in a program-independent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between False (the minimum element) and True (the maximum), with a Zero element in the middle. The truth values below Zero are ordered like the countable ordinals. The values above Zero have exactly the reverse order. Negation is interpreted as reflection about Zero followed by a step towards Zero ; the only truth value that remains unaffected by negation is Zero . We show that every program has a unique minimum model M P , and that this model can be constructed with a T P iteration which proceeds through the countable ordinals. Furthermore, we demonstrate that M P can alternatively be obtained through a construction that generalizes the well-known model intersection theorem for classical logic programming. Finally, we show that by collapsing the true and false values of the infinite-valued model M P to (the classical) True and False , we obtain a three-valued model identical to the well-founded one.

In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M -dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.

One of the most serious shortcomings of multidimensional languages is their inability to collaborate with conventional programming languages and systems. Multidimensional languages are used to define (potentially infinite) streams, grids, cubes, and so on, concepts which resemble in nature the familiar imperative arrays. The main difference is that the former entities are lazy while the latter are generally eager. This paper proposes the embedding of multidimensional languages into conventional ones as a form of definitional lazy arrays. The paper describes the details of an implementation of the proposed idea, as well as the expressibility and the performance of the resulting system. The main advantage of the new approach is that multidimensional languages can now benefit from the advanced features that have been developed for conventional languages. Moreover, multidimensionality adds to conventional languages the idea of lazy arrays, which in many cases offer significant advantages compared to the classical imperative arrays. Copyright © 1999 John Wiley & Sons, Ltd.

The purpose of this paper is to demonstrate that first-order functional programs can be transformed into intensional programs of nullary variables, in a semantics preserving way. On the foundational side, the goal of our study is to bring new insights and a better understanding of the nature of functional languages. From a practical point of view, our investigation provides a formal basis for the tagging mechanism that is used in the implementation of first-order functional languages on dataflow machines.