Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

Optimization and inference under fuzzy numerical constraints

Εκτεταμένη έρευνα έχει γίνει στους τομείς της Ικανοποίησης Περιορισμών με διακριτά (ακέραια) ή πραγματικά πεδία τιμών. Αυτή η έρευνα έχει οδηγήσει σε πολλαπλές σημασιολογικές περιγραφές, πλατφόρμες και συστήματα για την περιγραφή σχετικών προβλημάτων με επαρκείς βελτιστοποιήσεις. Παρά ταύτα, λόγω της ασαφούς φύσης πραγματικών προβλημάτων ή ελλιπούς μας γνώσης για αυτά, η σαφής μοντελοποίηση ενός προβλήματος ικανοποίησης περιορισμών δεν είναι πάντα ένα εύκολο ζήτημα ή ακόμα και η καλύτερη προσέγγιση. Επιπλέον, το πρόβλημα της μοντελοποίησης και επίλυσης ελλιπούς γνώσης είναι ακόμη δυσκολότερο. Επιπροσθέτως, πρακτικές απαιτήσεις μοντελοποίησης και μέθοδοι βελτιστοποίησης του χρόνου αναζήτησης απαιτούν συνήθως ειδικές πληροφορίες για το πεδίο εφαρμογής, καθιστώντας τη δημιουργία ενός γενικότερου πλαισίου βελτιστοποίησης ένα ιδιαίτερα δύσκολο πρόβλημα. Στα πλαίσια αυτής της εργασίας θα μελετήσουμε το πρόβλημα της μοντελοποίησης και αξιοποίησης σαφών, ελλιπών ή ασαφών περιορισμών, καθώς και πιθανές στρατηγικές βελτιστοποίησης. Καθώς τα παραδοσιακά προβλήματα ικανοποίησης περιορισμών λειτουργούν βάσει συγκεκριμένων και προκαθορισμένων κανόνων και σχέσεων, παρουσιάζει ενδιαφέρον η διερεύνηση στρατηγικών και βελτιστοποιήσεων που θα επιτρέπουν το συμπερασμό νέων ή/και αποδοτικότερων περιορισμών. Τέτοιοι επιπρόσθετοι κανόνες θα μπορούσαν να βελτιώσουν τη διαδικασία αναζήτησης μέσω της εφαρμογής αυστηρότερων περιορισμών και περιορισμού του χώρου αναζήτησης ή να προσφέρουν χρήσιμες πληροφορίες στον αναλυτή για τη φύση του προβλήματος που μοντελοποιεί.Extensive research has been done in the areas of Constraint Satisfaction with discrete/integer and real domain ranges. Multiple platforms and systems to deal with these kinds of domains have been developed and appropriately optimized. Nevertheless, due to the incomplete and possibly vague nature of real-life problems, modeling a crisp and adequately strict satisfaction problem may not always be easy or even appropriate. The problem of modeling incomplete knowledge or solving an incomplete/relaxed representation of a problem is a much harder issue to tackle. Additionally, practical modeling requirements and search optimizations require specific domain knowledge in order to be implemented, making the creation of a more generic optimization framework an even harder problem.In this thesis, we will study the problem of modeling and utilizing incomplete and fuzzy constraints, as well as possible optimization strategies. As constraint satisfaction problems usually contain hard-coded constraints based on specific problem and domain knowledge, we will investigate whether strategies and generic heuristics exist for inferring new constraint rules. Additional rules could optimize the search process by implementing stricter constraints and thus pruning the search space or even provide useful insight to the researcher concerning the nature of the investigated problem

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

Reasoning with Very Expressive Fuzzy Description Logics

It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). We illustrate why transitive role axioms are difficult to handle in the presence of fuzzy interpretations and how to handle them properly. Then we extend these results by adding role hierarchies and finally number restrictions. The main contributions of the paper are the decidability proof of the fuzzy DL languages fuzzy-SI and fuzzy-SHIN, as well as decision procedures for the knowledge base satisfiability problem of the fuzzy-SI and fuzzy-SHIN

Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems

Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

Managing Uncertainty and Vagueness in Semantic Web

Ο Σημασιολογικός Ιστός στοχεύει στην διεκπεραίωση εργασιών σε υπολογιστικά συστήματα χωρίς την ανθρώπινη παρέμβαση. Προκειμένου να επιτευχθεί ο στόχος αυτός, εισάγεται η έννοια της πληροφορίας που είναι επεξεργάσιμη από μηχανές. Στα περισσότερα προβλήματα, η έννοια της πληροφορίας είναι συνυφασμένη με την έννοια της αβεβαιότητας και της ασάφειας. Και οι δύο έννοιες περιγράφονται με την κοινή ονομασία ατελής πληροφορία. Δεδομένου ότι ο Σημασιολογικός Ιστός απαρτίζεται από ένα σύνολο τεχνολογιών και των θεωριών που τις διέπουν, οποιαδήποτε μέθοδος αναπαράστασης θα πρέπει να βρίσκεται σε συμφωνία με άλλες υπάρχουσες. Συγκεκριμένα, το θεωρητικό πλαίσιο πρέπει να εντάσσεται ομαλά στη θεωρία που εφαρμόζεται στο Σημασιολογικό Ιστό. Η δε υλοποίησή του, ιδανικό είναι, να υποστηριχθεί με χρήση μεθόδων του Σημασιολογικού Ιστού, στις οποίες κυριαρχεί εκείνη των οντολογιών. Στη διατριβή μας, ορίσαμε μία μέθοδο αναπαράστασης της αβεβαιότητας και της ασάφειας μέσω ενός ενιαίου πλαισίου. Το μοντέλο Dempster-Shafer χρησιμοποιήθηκε για την αναπαράσταση της αβεβαιότητας και το μοντέλο Ασαφούς Λογικής και Ασαφών Συνόλων για την αναπαράσταση της ασάφειας. Για το λόγο αυτό, ορίσαμε το θεωρητικό πλαίσιο, στοχεύοντας σε ένα συνδυασμό ALC Λογικών Περιγραφών (Description Logics) με το μοντέλο Dempster-Shafer. Κατά τη διάρκεια της έρευνάς μας υλοποιήσαμε μεταοντολογίες για την αναπαράσταση της αβεβαιότητας και της ασάφειας και στη συνέχεια μελετήσαμε την συμπεριφορά τους σε πραγματικές εφαρμογές.Semantic Web has been designed for processing tasks without human intervention. In this context, the term machine processable information has been introduced. In most Semantic Web tasks, we come across information incompleteness issues, aka uncertainty and vagueness. For this reason, a method that represents uncertainty and vagueness under a common framework has to be defined. Semantic Web technologies are defined through a Semantic Web Stack and are based on a clear formal foundation. Therefore, any representation scheme should be aligned with these technologies and be formally defined. As the concept of ontologies is significant in the Semantic Web for representing knowledge, any framework is desirable to be built upon it. In our work, we have defined an approach for representing uncertainty and vagueness under a common framework. Uncertainty is represented through Dempster-Shafer model, whereas vagueness has been represented through Fuzzy Logic and Fuzzy Sets. For this reason, we have defined our theoretical framework, aimed at a combination of the classical crisp DL ALC with a Dempster-Shafer module. As a next step, we added fuzziness to this model. Throughout our work, we have implemented metaontologies in order to represent uncertain and vague concepts and, next, we have tested our methodology in real-world applications

Foundations of fuzzy answer set programming

Answer set programming (ASP) is a declarative language that is tailored towards combinatorial search problems. Although ASP has been applied to many problems, such as planning, configuration and verification of software, and database repair, it is less suitable for describing continuous problems. In this thesis we therefore studied fuzzy answer set programming (FASP). FASP is a language that combines ASP with ideas from fuzzy logic -- a class of many-valued logics that are able to describe continuous problems. We study the following topics: 1. An important issue when modeling continuous optimization problems is how to cope with overconstrained problems. In many cases we can opt to allow imperfect solutions, i.e. solutions that do not satisfy all constraints, but are sufficiently acceptable. However, the question which one of these imperfect solutions is most suitable then arises. Current approaches to fuzzy answer set programming solve this problem by attaching weights to the rules of the program. However, it is often not clear how these weights should be chosen and moreover weights do not allow to order different solutions. We improve upon this technique by using aggregators, which eliminate the aforementioned problems. This allows a richer modeling language and bridges the gap between FASP and other techniques such as valued constraint satisfaction problems. 2. The wishes of users and implementers of a programming language are often in direct conflict with each other. Users prefer a rich language that is easy to model in, whereas implementers prefer a small language that is easy to implement. We reconcile these differences by identifying a core language for FASP, called core FASP (CFASP), that only consists of non-constraint rules with monotonically increasing functions and negators in the body. We show that CFASP is capable of simulating constraint rules, monotonically decreasing functions, aggregators, S-implicators and classical negation. Moreover we remark that the simulations of constraints and classical negation bear a great resemblance to their simulations in classical ASP, which provides further insight into the relationship between ASP and FASP. 3. As a first step towards the creation of an implementation method for FASP we research whether it is possible to translate a FASP program to a fuzzy SAT problem. We introduce the concept of the completion of a FASP program and show that for programs without loops the models of the completion coincide with the answer sets. Furthermore we show that if a program has loops, we can translate the program to a fuzzy SAT problem by generalizing the concept of loop formulas. We illustrate this on a continuous version of the k-center problem. Such a translation is important because it allows us to solve FASP programs by means of solvers for fuzzy SAT. Under the appropriate conditions it is for example possible to solve FASP programs by means of off-the-shelf solvers for mixed integer programming (MIP)