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.

The DARE platform has been designed to help research developers deliver user-facing applications and solutions over diverse underlying e-infrastructures, data and computational contexts. The platform is Cloud-ready, and relies on the exposure of APIs, which are suitable for raising the abstraction level and hiding complexity. At its core, the platform implements the cataloguing and execution of fine-grained and Python-based dispel4py workflows as services. Reflection is achieved via a logical knowledge base, comprising multiple internal catalogues, registries and semantics, while it supports persistent and pervasive data provenance. This paper presents design and implementation aspects of the DARE platform, as well as it provides directions for future development.

Copernicus is the European programme for monitoring the Earth.It consists of a set of systems that collect data from satellites and in-situ sensors, process this data and provide users with reliable and up-to-date information on a range of environmental and security issues.The data and information processed and disseminated puts Copernicus at the forefront of the big data paradigm, giving rise to all relevant challenges, the so-called 5 Vs: volume, velocity, variety, veracity and value.In this short paper, we discuss the challenges of extracting information and knowledge from huge archives of Copernicus data.We propose to achieve this by scale-out distributed deep learning techniques that run on very big clusters offering virtual machines and GPUs.We also discuss the challenges of achieving scalability in the management of the extreme volumes of information and knowledge extracted from Copernicus data.The envisioned scientific and technical work will be carried out in the context of the H2020 project ExtremeEarth which starts in January 2019.

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.

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.

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.

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.

We describe an Inductive Logic Programming (ILP) approach to learning descriptions in Description Logics (DL) under uncertainty. The approach is based on implementing many-valued DL proofs as propositionalizations of the elementary DL constructs and then providing this implementation as background predicates for ILP. The proposed methodology is tested on a many-valued variation of eastbound-trains and Iris, two well known and studied Machine Learning datasets.