General Model Theoretic Semantics for Higher-Order Horn Logic Programming

We introduce model-theoretic semantics [6] for Higher-Order Horn logic programming language. One advantage of logic programs over conventional non-logic programs has been that the least fixpoint is equal to the least model, therefore it is associated to logical consequence and has a meaningful declarative interpretation. In simple theory of types [9] on which Higher-Order Horn logic programming language is based, domain is dependent on interpretation [10]. To define T p operator for a logic program P, we need a fixed domain without regard to interpretation which is usually taken to be a set of atomic propositions. We build a semantics where we can fix a domain while changing interpretations. We also develop a fixpoint semantics based on our model, and show that we can get the least fixpoint which is the least model. Using this fixpoint we prove the completeness of the interpreter of our language in [14]

Logic-Based Analogical Reasoning and Learning

Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper contributes to the foundations of artificial general intelligence by developing an abstract algebraic framework for logic-based analogical reasoning and learning in the setting of logic programming. The main idea is to define analogy in terms of modularity and to derive abstract forms of concrete programs from a `known' source domain which can then be instantiated in an `unknown' target domain to obtain analogous programs. To this end, we introduce algebraic operations for syntactic program composition and concatenation and illustrate, by giving numerous examples, that programs have nice decompositions. Moreover, we show how composition gives rise to a qualitative notion of syntactic program similarity. We then argue that reasoning and learning by analogy is the task of solving analogical proportions between logic programs. Interestingly, our work suggests a close relationship between modularity, generalization, and analogy which we believe should be explored further in the future. In a broader sense, this paper is a first step towards an algebraic and mainly syntactic theory of logic-based analogical reasoning and learning in knowledge representation and reasoning systems, with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity

Equivalence of two Fixed-Point Semantics for Definitional Higher-Order Logic Programs

Two distinct research approaches have been proposed for assigning a purely extensional semantics to higher-order logic programming. 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. In this paper we demonstrate that for a very broad class of programs, namely the class of definitional programs introduced by W. W. Wadge, the two approaches coincide (with respect to ground atoms that involve symbols of the program). On the other hand, we argue that if existential higher-order variables are allowed to appear in the bodies of program rules, the two approaches are in general different. The results of the paper contribute to a better understanding of the semantics of higher-order logic programming.Comment: In Proceedings FICS 2015, arXiv:1509.0282

Extensionality of simply typed logic programs

We set up a framework for the study of extensionality in the context of higher-order logic programming. For simply typed logic programs we propose a novel declarative semantics, consisting of a model class with a semi-computable initial model, and a notion of extensionality. We show that the initial model of a simply typed logic program, in case the program is extensional, collapses into a simple, set-theoretic representation. Given the undecidability of extensionality in general, we develop a decidable, syntactic criterion which is sufficient for extensionality. Some typical examples of higher-order logic programs are shown to be extensional

Higher-Order Chronolog

Η Chronolog είναι μια γλώσσα λογικού προγραμματισμού βασισμένη στη χρονική λογική και στη σημασιολογία πιθανών κόσμων. Η λογική χρόνου μας επιτρέπει να περιγράψουμε δυναμικές και εξαρτόμενες από το χρόνο ιδιότητες με ένα άμεσα τρόπο. Ορίζουμε μία επέκταση της Chronolog που μπορεί να χρησιμοποιηθεί για χρονικό λογικό προγραμματισμό υψηλότερης τάξης. Παρουσιάζουμε το συντακτικό και τη σημασιολογία της επέκτασής μας και χρησιμοποιούμε την γλώσσα μας για να περιγράψουμε κάποια προβλήματα.Chronolog is a logic programming language based on temporal logic and possible worlds semantics. Temporal logic allows us to describe dynamic and time-dependent properties in a direct way. We define an extension of Chronolog which can be used for higher- order temporal logic programming. We present the syntax and semantics of our extended Chronolog and use the language to describe some problems

The Expressive Power of Higher-order Datalog: An XSB Implementation

Η πτυχιακή εργασία ακολουθεί τα αποτελέσματα του μη δημοσιευμένου ακόμα άρθρου των, Α. Χαραλαμπίδη, Χ. Νομικού και Π. Ροντογιάννη με τίτλο “The Expressive Power of Higher-order Datalog”. Το άρθρο παρουσιάζει μία απόδειξη ισοδυναμίας σε εκφραστική ισχύ της Datalog υψηλής-τάξης, με τις χρονικά εκθετικά περιορισμένες μηχανές Turing. Με άλλα λόγια ότι η Datalog υψηλής-τάξης εκφράζει τις κλάσεις πολυπλοκότητας των προβλημάτων απόφασης EXP^(k)TIME. Η πτυχιακή εργασία αυτή, θα παρουσιάσει με λεπτομέρεια τα παραπάνω αποτελέσματα, καθώς και θα υποδείξει κάποια σφάλματα στα προγράμματα που αναγράφονται στο άρθρο, καθώς και θα προτείνει τρόπους επίλυσής τους. Επιπλέον θα παρατεθεί μία λειτουργική υλοποίηση των προγραμμάτων στο σύστημα XSB, με στόχο την τεκμηρίωση των παραπάνω αποτελεσμάτων.This thesis follows the results in the yet unpublished paper of A. Charalambidis, Ch. Nomikos and P. Rondogiannis namely “The Expressive Power of Higher-order Datalog”. That paper proposes a proof which shows that Higher-order Datalog is equivalent in computational power to exponentially time bounded Turing Machines. In other words that higher-order Datalog captures the complexity class of decision problems EXP^(k)TIME. This thesis will review the above result in detail while demonstrating and proposing solutions for the flaws in the programs written in that paper. In addition a working implementation of the programs in the XSB system will be provided which shows that the proposed results hold