The Complexity of Local Stratification

The class of locally stratified logic programs is shown to be Π11-complete by the construction of a reducibility of the class of infinitely branching nondeterministic finite register machines.nondeterministic finite register machines

Designing Dependencies

Given a binary recursively enumerable relation R, one or more logic programs over a language L can be constructed and interconnected to produce a dependency relation D on selected predicates within the Herbrand base BL of L isomorphic to R. D can be, optionally, a positive, negative or mixed dependency relation. The construction is applied to representing any effective game of the type introduced by Gurevich and Harrington, which they used to prove Rabin\u27s decision method for S2S, as the dependency relation of a logic program. We allow games over an infinite alphabet of possible moves. We use this representation to reveal a common underlying reason, having to do with the shape of a program\u27s dependency relation, for the complexity of several logic program properties

The Expressiveness of Locally Stratified Programs

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be computed 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 obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of locally stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleene\u27s hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets

Four Lessons in Versatility or How Query Languages Adapt to the Web

Exposing not only human-centered information, but machine-processable data on the Web is one of the commonalities of recent Web trends. It has enabled a new kind of applications and businesses where the data is used in ways not foreseen by the data providers. Yet this exposition has fractured the Web into islands of data, each in different Web formats: Some providers choose XML, others RDF, again others JSON or OWL, for their data, even in similar domains. This fracturing stifles innovation as application builders have to cope not only with one Web stack (e.g., XML technology) but with several ones, each of considerable complexity. With Xcerpt we have developed a rule- and pattern based query language that aims to give shield application builders from much of this complexity: In a single query language XML and RDF data can be accessed, processed, combined, and re-published. Though the need for combined access to XML and RDF data has been recognized in previous work (including the W3C’s GRDDL), our approach differs in four main aspects: (1) We provide a single language (rather than two separate or embedded languages), thus minimizing the conceptual overhead of dealing with disparate data formats. (2) Both the declarative (logic-based) and the operational semantics are unified in that they apply for querying XML and RDF in the same way. (3) We show that the resulting query language can be implemented reusing traditional database technology, if desirable. Nevertheless, we also give a unified evaluation approach based on interval labelings of graphs that is at least as fast as existing approaches for tree-shaped XML data, yet provides linear time and space querying also for many RDF graphs. We believe that Web query languages are the right tool for declarative data access in Web applications and that Xcerpt is a significant step towards a more convenient, yet highly efficient data access in a “Web of Data”

Semantics of Negation in Extensional Higher-Order Logic Programming

Θεωρούμε τις δύο υπάρχουσες εκτατικές προσεγγίσεις στη σημασιολογία των θετικών λογικών προγραμμάτων ανώτερης τάξης, προταθείσες από τον W. W. Wadge και τον M. Bezem αντίστοιχα. Η πρώτη προσέγγιση χρησιμοποιεί κλασικά εργαλεία από τη θεωρία πεδίων ενώ η δεύτερη στηρίζεται στις συντακτικές οντότητες που εμφανίζονται στο πρόγραμμα και βασίζεται στην επεξεργασία του βασικού αναπτύγματος του προγράμματος. Οι σχέσεις μεταξύ των δύο προσεγγίσεων δεν είχαν ως τώρα διερευνηθεί, ενώ μόνο η προσέγγιση του Wadge είχε επεκταθεί ώστε να εφαρμοστεί σε προγράμματα ανώτερης τάξης με άρνηση. Δείχνουμε ότι οι σημασιολογίες του Wadge και του Bezem συμπίπτουν για μία ευρεία και ενδιαφέρουσα κλάση προγραμμάτων, τα οποία δεν περιλαμβάνουν υπαρξιακά ποσοτικοποιημένες μεταβλητές στα σώματα των προτάσεων. Σημειώνουμε ότι έχουν επίσης ουσιαστικές διαφορές, οι οποίες γίνονται εμφανείς όταν επεκτείνουμε την θεωρούμενη γλώσσα ώστε να επιτρέπονται υπαρξιακές μεταβλητές. Επιπλέον, εστιάζουμε στη λιγότερο ανεπτυγμένη ερευνητική κατεύθυνση εκ των δύο, δηλαδή τη σημασιολογία του Bezem, και προσαρμόζουμε για πρώτη φορά την τεχνική του Bezem ώστε να ορίσουμε μία εκτατική σημασιολογία για λογικά προγράμματα ανώτερης τάξης με άρνηση. Για τον σκοπό αυτό, αξιοποιούμε την απειρότιμη προσέγγιση στην άρνηση-μέσω-αποτυχίας. Από την άλλη, δείχνουμε ότι o συνδυασμός της τεχνικής με τη σημασιολογία σταθερού μοντέλου ή με την καλώς θεμελιωμένη σημασιολογία, αποτυγχάνει να παράξει εκτατικές σημασιολογίες, στη γενική περίπτωση. Αναλύουμε τις αιτίες αυτής της αποτυχίας και ισχυριζόμαστε ότι μία τρίτιμη λογική δεν μπορεί να διαχωρίσει μεταξύ τους ορισμένα κατηγορήματα, τα οποία έχουν διαφορετική συμπεριφορά μέσα σε ένα πρόγραμμα, αλλά τυγχάνει να εμφανίζονται ως πανομοιότυπες τρίτιμες σχέσεις. Τέλος, ορίζουμε για πρώτη φορά τις έννοιες της στρωματοποίησης και της τοπικής στρωματοποίησης για λογικά προγράμματα ανώτερης τάξης με άρνηση. Αποδεικνύουμε ότι κάθε στρωματοποιημένο πρόγραμμα έχει ένα διακριτό εκτατικό μοντέλο, το οποίο μπορεί να κατασκευαστεί ισοδύναμα μέσω της καλώς θεμελιωμένης, της σταθερής ή της απειρότιμης σημασιολογίας. Επιπλέον, δείχνουμε ότι αυτό το μοντέλο δεν αποδίδει ποτέ την άγνωστη τιμή αληθείας. Τα αποτελέσματα αυτά αναδεικνύουν τη σπουδαιότητα και την καλή φύση των στρωματοποιημένων προγραμμάτων, που ήταν ως τώρα γνωστή μόνο στην περίπτωση των λογικών προγραμμάτων πρώτης τάξης.We consider the two existing extensional approaches to the semantics of positive higher-order logic programming, originally introduced by W. W. Wadge and M. Bezem respectively. The former approach uses classical domain-theoretic tools while the latter builds on a fixed-point construction defined on a syntactic instantiation of the source program. The relationships between these two approaches had not been investigated until now, while only Wadge's approach had been extended to apply to higher-order programs with negation. We show that Wadge's semantics and Bezem's semantics coincide for a broad and interesting class of programs, which do not include existentially quantified predicate variables in the bodies of clauses. We indicate that they also have profound differences, which surface when we extend our source language to allow existential predicate variables. In addition, we focus on the less developed research direction of the two, namely Bezem's semantics, and we adapt, for the first time, Bezem's technique to define an extensional semantics for higher-order logic programs with negation. For this purpose, we utilize the infinite-valued approach to negation-as-failure. On the other hand, we show that an adaptation of the technique under the well-founded or the stable model semantics does not in general lead to an extensional semantics. We analyse the reasons for this failure 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. As an application of our developments, we define for the first time the notions of stratification and local stratification for higher-order logic programs with negation. We prove that every stratified program has a distinguished extensional model which can be equivalently obtained through the well-founded, stable or infinite-valued model semantics. Furthermore, we show that this model does not assign the unknown truth value. These results affirm the importance and the well-behaved nature of stratified programs, which was, until now, only known for the first-order case

Data Integration on the (Semantic) Web with Rules and Rich Unification

For the last decade a multitude of new data formats for the World Wide Web have been developed, and a huge amount of heterogeneous semi-structured data is flourishing online. With the ever increasing number of documents on the Web, rules have been identified as the means of choice for reasoning about this data, transforming and integrating it. Query languages such as SPARQL and rule languages such as Xcerpt use compound queries that are matched or unified with semi-structured data. This notion of unification is different from the one that is known from logic programming engines in that it (i) provides constructs that allow queries to be incomplete in several ways (ii) in that variables may have different types, (iii) in that it results in sets of substitutions for the variables in the query instead of a single substitution and (iv) in that subsumption between queries is much harder to decide than in logic programming. This thesis abstracts from Xcerpt query term simulation, SPARQL graph pattern matching and XPath XML document matching, and shows that all of them can be considered as a form of rich unification. Given a set of mappings between substitution sets of different languages, this abstraction opens up the possibility for format-versatile querying, i.e. combination of queries in different formats, or transformation of one format into another format within a single rule. To show the superiority of this approach, this thesis introduces an extension of Xcerpt called Xcrdf, and describes use-cases for the combined querying and integration of RDF and XML data. With XML being the predominant Web format, and RDF the predominant Semantic Web format, Xcrdf extends Xcerpt by a set of RDF query terms and construct terms, including query primitives for RDF containers collections and reifications. Moreover, Xcrdf includes an RDF path query language called RPL that is more expressive than previously proposed polynomial-time RDF path query languages, but can still be evaluated in polynomial time combined complexity. Besides the introduction of this framework for data integration based on rich unification, this thesis extends the theoretical knowledge about Xcerpt in several ways: We show that Xcerpt simulation unification is decidable, and give complexity bounds for subsumption in several fragments of Xcerpt query terms. The proof is based on a set of subsumption monotone query term transformations, and is only feasible because of the injectivity requirement on subterms of Xcerpt queries. The proof gives rise to an algorithm for deciding Xcerpt query term simulation. Moreover, we give a semantics to locally and weakly stratified Xcerpt programs, but this semantics is applicable not only to Xcerpt, but to any rule language with rich unification, including multi-rule SPARQL programs. Finally, we show how Xcerpt grouping stratification can be reduced to Xcerpt negation stratification, thereby also introducing the notion of local grouping stratification and weak grouping stratification

The Complexity of Local Stratification

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.