Quantale Modules, with Applications to Logic and Image Processing

We propose a categorical and algebraic study of quantale modules. The results and constructions presented are also applied to abstract algebraic logic and to image processing tasks.Comment: 150 pages, 17 figures, 3 tables, Doctoral dissertation, Univ Salern

A neural implementation of multi-adjoint logic programs via sf-homogenization

A generalization of the homogenization process needed for the neural im- plementation of multi-adjoint logic programming (a unifying theory to deal with uncertainty, imprecise data or incomplete information) is presented here. The idea is to allow to represent a more general family of adjoint pairs, but maintaining the advantage of the existing implementation recently introduced in [6]. The soundness of the transformation is proved and its complexity is analysed. In addition, the corresponding generalization of the neural-like implementation of the fixed point semantics of multi-adjoint is presented

A General Framework for Representing, Reasoning and Querying with Annotated Semantic Web Data

We describe a generic framework for representing and reasoning with annotated Semantic Web data, a task becoming more important with the recent increased amount of inconsistent and non-reliable meta-data on the web. We formalise the annotated language, the corresponding deductive system and address the query answering problem. Previous contributions on specific RDF annotation domains are encompassed by our unified reasoning formalism as we show by instantiating it on (i) temporal, (ii) fuzzy, and (iii) provenance annotations. Moreover, we provide a generic method for combining multiple annotation domains allowing to represent, e.g. temporally-annotated fuzzy RDF. Furthermore, we address the development of a query language -- AnQL -- that is inspired by SPARQL, including several features of SPARQL 1.1 (subqueries, aggregates, assignment, solution modifiers) along with the formal definitions of their semantics

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

Mathematics in Software Reliability and Quality Assurance

This monograph concerns the mathematical aspects of software reliability and quality assurance and consists of 11 technical papers in this emerging area. Included are the latest research results related to formal methods and design, automatic software testing, software verification and validation, coalgebra theory, automata theory, hybrid system and software reliability modeling and assessment

Approximation Theory and Related Applications

In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics

Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences

Mathematical fuzzy logic (MFL) specifically targets many-valued logic and has significantly contributed to the logical foundations of fuzzy set theory (FST). It explores the computational and philosophical rationale behind the uncertainty due to imprecision in the backdrop of traditional mathematical logic. Since uncertainty is present in almost every real-world application, it is essential to develop novel approaches and tools for efficient processing. This book is the collection of the publications in the Special Issue “Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences”, which aims to cover theoretical and practical aspects of MFL and FST. Specifically, this book addresses several problems, such as:- Industrial optimization problems- Multi-criteria decision-making- Financial forecasting problems- Image processing- Educational data mining- Explainable artificial intelligence, etc

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

Substructurality and residuation in logic and algebra

A very and natural way of introducing a logic is by using a sequent calculus, or Gentzen system. These systems are determined by specifying a set of axioms and a set of rules. Axioms are then starting points from which we can derive new consequences by using the rules. Hilbert systems consist also on a set of axioms and a set of rules that are used to deduce consequences. The main difference is that, whereas the axioms in Hilbert systems are formulas, and the rules allow to deduce certain formulas from other sets of formulas, in the case of Gentzen systems the axioms are sequents and the rules indicate which sequents can be inferred from other sets of sequents. By a sequent we understand a pair hG, Si, where G and S are finite sequences of formulas. We denote the sequent hG, Si by G . S.1 The sequent G . S intends to formalize – at least in its origin – the concept “the conjunction of all the formulas of G implies the disjunction of all the formulas of S.” The notion of a sequent calculus was invented by G. Gentzen in order to give axiomatizations for Classical and Intuitionistic Propositional Logics. And the rules he gave in both cases can be grouped in different categories: because of its character, the Cut rule deserves a special category for itself; then we have the rules of introduction and elimination of each one of the connectives, both on the left and on the right – of the symbol . –; and finally a set of rules that do not involve any particular connective. These rules are necessary in Classical and Intuitionistic logics because in these logics 1Traditional notations for sequents are G ) S and G ` S, but since both the symbols ) and ` have many other meanings, we prefer to denote sequents by using the less overloaded symbol ., which can also be found in literature with this use. the order in which we are given the premises, or if we have them repeated, is irrelevant, and we do not loose consequences if we extend the set of hypotheses. But there are other logics that do not satisfy all these rules: for instance, relevance logics and linear logic. At first, these logics were studied separately, and different theories were developed for their investigation. But later on, researches arrived to the conclusion that all of them share a common feature, which became more apparent after the work of W. Blok and D. Pigozzi. It was discovered that (pointed) residuated lattices – or FL algebras – are the algebraic counterpart of substructural logics. In the XIX century, Boole noticed a close connection between “the laws of thought,” as he put it, and algebra. After him, other mathematicians put together all the pieces and described a sort of algebras, named Boole algebras after him, and shed light on the connection anticipated by Boole: Boole algebras are the “natural” semantics for Classical Propositional Logic. More connections were discovered between other logics and other sorts of algebras: for instance, Heyting algebras are the “natural” semantics for Intuitionistic Propositional Logic, and MV algebras for Łukasievicz Multivalued Logic. But it was not until 1989, when Blok and Pigozzi published their book Algebraizable Logics, that for the first time the connections between these logics and classes of algebras were finally described with absolute precision. According to their definitions, these classes of algebras are the equivalent algebraic sematics of the corresponding logics. That is, these classes of algebras are the algebraic counterparts of the corresponding logics. Their ideas paved the way to a new branch of mathematics called Abstract Algebraic Logic, which investigates the connections between logics and classes of algebras, and the so-called bridge theorems: that is, theorems that establish bridges between some property of one realm (logic or algebra) with another property of the other realm. The core of the connection between substructural logics and residuated lattices is that in all these logics, some theorem of the following form could always be proven. Thus, we could think that the metalogical symbol ’,’ is acting as a real connective. More precisely, we could introduce a new connective , called fusion, and impose the following rule. Given an algebraic model with a lattice reduct, it is usually the case that the meet and join operations serve as the interpretations of the conjunction and disjunction connectives. What should be then the interpretation of the fusion? Usually, the elements of the lattice are thought as different degrees of truth, and “a . b is provable” is interpreted as “for every assignment, the degree of truth of a is less than that of b.” Under this natural interpretation, the condition (1) becomes: That is, the fusion is interpreted as a residuated operation on the lattice. Being the algebraic semantics of substructural logics and containing many interesting subvarieties such as Heyting algebras, MV algebras, and lattice-ordered groups, to name a few, the variety of residuated lattices is of utmost importance to the studies of Logic and Algebra, hence our interest. In this dissertation we carry out some investigations on different problems concerning residuated lattices. In what follows we give a brief description of the contents and organization of this dissertation. Every chapter – except for the first one, which is devoted to setting the preliminaries – starts with an introduction in which the reader will find a lengthier explanation of the subject of the chapter, the way the material is organized, and references. We start by compiling in Chapter 1 all the essential well-known results about residuated lattices that we will need in the subsequent chapters. We present here the definitions of those concepts that are not specific to some particular chapter, but general. We define the variety of residuated lattices, and some of its more significant subvarieties. We also introduce nuclei, and nucleus retracts. As it is widely known, the lattice of normal convex subalgebras of a residuated lattice is isomorphic to its congruence lattice, and hence its importance. But it turns out that also the lattice of convex (not necessarily normal) subalgebras is of great significance, specially in the case of e-cyclic residuated lattices. Many of its properties depend on the fact that it is a pseudo-complemented lattice. Actually, it is a Heyting algebra. For instance, polars are special sets usually defined in terms of a certain notion of orthogonality; in the case of e-cyclic residuated lattices, polars are the pseudo-complements of the convex subalgebras. We end the chapter by briefly explaining the notions of semilinearity and projectability for residuated lattices. In the 1960’s, P. F. Conrad and other authors set in motion a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex `-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. We devote Chapters 2 and 3 to two different problems that can be framed within Conrad’s program for residuated lattices. More specifically, in Chapter 2 we revisit the Galatos-Tsinakis categorical equivalence between integral GMV algebras and negative cones of `-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Afterwards, we introduce the notion of Gödel GMV algebras, which are expansions of projectable integral GMV algebras by a binary term that realizes a positive Gödel implication in every such algebra. We see that Gödel GMV algebras and projectable integral GMV algebras are essentially the same thing. Analogously, Gödel negative cones are those Gödel GMV algebras whose residuated lattice reducts are negative cones of `-groups. Thus, we turn projectable integral GMV algebras and negative cones of projectable `-groups into varieties by including this implication in their signature. We prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications. We devote Chapter 3 to the study of certain kinds of completions of semilinear residuated lattices. We can find in the literature different notions of completions for residuated lattices, like for example Dedekind-McNeil completions, regular completions, complete ideal completions, . . . Very often it happens that for a certain algebra in a variety of residuated lattices, those completions exists but do not belong to the same variety. That is, varieties are not closed, in general, under the operations of taking these kinds of completions. But there are other notions of completions that might have better properties in this regard. Conrad and other authors proved the existence of lateral completions, projectable completions, and orthocompletions for representable `-groups, and moreover, that the varieties of representable `-groups are closed under these completions. Our goal in this chapter is to prove the existence of lateral completions, (strongly) projectable completions, and orthocompletions for semilinear e-cyclic residuated lattices, as they are a natural generalization of representable `-groups. We introduce all these concepts along the chapter, and prove first that every semilinear e-cyclic residuated lattice can be densely embedded into another residuated lattice which is latterly complete and strongly projectable. We obtain this lattice as a direct limit of a certain family of algebras obtained from the original lattice by taking quotients and products, so the direct limit stays in the same variety where the original algebra lives. Finally, we prove that for semilinear GMV algebras, we can find minimal dense extensions satisfying all the required properties. In Chapter 4 we study the failure of the Amalgamation Property on several varieties of residuated lattices. The Amalgamation Property is of particular interest in the study of residuated lattices due to its relation with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the Amalgamation Property. The variety of semilinear Abstract 5 residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the Amalgamation Property are semilinear. However, we prove that this is not the case, and in the process we establish that the same happens for the variety of semilinear cancellative residuated lattices, that is, it also lacks the Amalgamation Property. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ^ z)w xyw ^ xzw also fails the Amalgamation Property. In Chapter 5 we show how some well-known results of the theory of automata, in particular those related to regular languages, can be viewed within a wider framework. In order to do so, we introduce the concept of module over a residuated lattice, and show that modules over a fixed residuated lattice – that is, partially ordered sets acted upon by a residuated lattice – provide a suitable algebraic framework for extending the concept of a recognizable language as defined by Kleene. More specifically, we introduce the notion of a recognizable element of a residuated lattice by a finite module and provide a characterization of such an element in the spirit of Myhill’s characterization of recognizable languages. Further, we investigate the structure of the set of recognizagle elements of a residuated lattice, and also provide sufficient conditions for a recognizable element to be recognized by a Boolean module. We summarize in Chapter 6 the main results of this dissertation and propose some of the problems that still remain open. We end this dissertation with an appendix on directoids. These structures were introduced independently three times, and their aim is to study directed ordered sets from an algebraic perspective. The structures that we have studied in this dissertations have an underlying order, but moreover they have a lattice reduct. That is not always the case for directed ordered sets. Hence the importance of the study of directoids. We prove some properties of directoids and their expansions by additional and complemented directoids. Among other results, we provide a shorter proof of the direct decomposition theorem for bounded involute directoids. We present a description of central elements of complemented directoids. And finally we show that the variety of directoids, as well as its expansions mentioned above, all have the strong amalgamation property