Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

This paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms

On modal expansions of t-norm based logics with rational constants

[eng] According to Zadeh, the term “fuzzy logic” has two different meanings: wide and narrow. In a narrow sense it is a logical system which aims a formalization of approximate reasoning, and so it can be considered an extension of many-valued logic. However, Zadeh also says that the agenda of fuzzy logic is quite different from that of traditional many-valued logic, as it addresses concepts like linguistic variable, fuzzy if-then rule, linguistic quantifiers etc. Hájek, in the preface of his foundational book Metamathematics of Fuzzy Logic, agrees with Zadeh’s distinction, but stressing that formal calculi of many-valued logics are the kernel of the so-called Basic Fuzzy logic (BL), having continuous triangular norms (t-norm) and their residua as semantics for the conjunction and implication respectively, and of its most prominent extensions, namely Lukasiewicz, Gödel and Product fuzzy logics. Taking advantage of the fact that a t-norm has residuum if, and only if, it is left-continuous, the logic of the left-continuous t-norms, called MTL, was soon after introduced. On the other hand, classical modal logic is an active field of mathematical logic, originally introduced at the beginning of the XXth century for philosophical purposes, that more recently has shown to be very successful in many other areas, specially in computer science. That are the most well-known semantics for classical modal logics. Modal expansions of non-classical logics, in particular of many-valued logics, have also been studied in the literature. In this thesis we focus on the study of some modal logics over MTL, using natural generalizations of the classical Kripke relational structures where propositions at possible words can be many-valued, but keeping classical accessibility relations. In more detail, the main goal of this thesis has been to study modal expansions of the logic of a left-continuous t-norm, defined over the language of MTL expanded with rational truth-constants and the Monteiro-Baaz Delta-operator, whose intended (standard) semantics is given by Kripke models with crisp accessibility relations and taking the unit real interval [0, 1] as set of truth-values. To get complete axiomatizations, already known techniques based on the canonical model construction are uses, but this requires to ensure that the underlying (propositional) fuzzy logic is strongly standard complete. This constraint leads us to consider axiomatic systems with infinitary inference rules, already at the propositional level. A second goal of the thesis has been to also develop and automated reasoning software tool to solve satisfiability and logical consequence problems for some of the fuzzy logic modal logics considered. This dissertation is structured in four parts. After a gentle introduction, Part I contains the needed preliminaries for the thesis be as self-contained as possible. Most of the theoretical results are developed in Parts II and III. Part II focuses on solving some problems concerning the strong standard completeness of underlying non-modal expansions. We first present and axiomatic system for the non-nodal propositional logic of a left-continuous t-norm who makes use of a unique infinitary inference rule, the “density rule”, that solves several problems pointed out in the literature. We further expand this axiomatic system in order to also characterize arbitrary operations over [0, 1] satisfying certain regularity conditions. However, since this axiomatic system turn out to be not well-behaved for the modal expansion, we search for alternative axiomatizations with some particular kind of inference rules (that will be called conjunctive). Unfortunately, this kind of axiomatization does not necessarily exist for all left-continuous t-norms (in particular, it does not exist for the Gödel logic case), but we identify a wide class of t-norms for which it works. This “well-behaved” t-norms include all ordinal sums of Lukasiewiczand Product t-norms. Part III focuses on the modal expansion of the logics presented before. We propose axiomatic systems (which are, as expected, modal expansions of the ones given in the previous part) respectively strongly complete with respect to local and global Kripke semantics defined over frames with crisp accessibility relations and worlds evaluated over a “well-behaved” left-continuous t-norm. We also study some properties and extensions of these logics and also show how to use it for axiomatizing the possibilistic logic over the very same t-norm. Later on, we characterize the algebraic companion of these modal logics, provide some algebraic completeness results and study the relation between their Kripke and algebraic semantics. Finally, Part IV of the thesis is devoted to a software application, mNiB-LoS, who uses Satisfability Modulo Theories in order to build an automated reasoning system to reason over modal logics evaluated over BL algebras. The acronym of this applications stands for a modal Nice BL-logics Solver. The use of BL logics along this part is motivated by the fact that continuous t-norms can be represented as ordinal sums of three particular t-norms: Gödel, Lukasiewicz and Product ones. It is then possible to show that these t-norms have alternative characterizations that, although equivalent from the point of view of the logic, have strong differences for what concerns the design, implementation and efficiency of the application. For practical reasons, the modal structures included in the solver are limited to the finite ones (with no bound on the cardinality)

Logics of formal inconsistency arising from systems of fuzzy logic

This article proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this article we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.The authors have been partially supported by the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Coniglio was also supported by FAPESP (Thematic Project LogCons 2010/51038-0), and by a research grant from CNPq (PQ 305237/2011-0). Esteva and Godo also acknowledge partial support by the MINECO project TIN2012-39348-C02-01Peer Reviewe

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

The approximation of left-continuous t-norms

Abstract A discrete t-norm is a binary operation on a finite subset of the real unit interval fulfilling the same algebraic conditions as t-norms. We show that any left-continuous t-norm can, in a natural sense, be approximated by a discrete t-norm with an arbitrary precision

From Quantum Metalanguage to the Logic of Qubits

The main aim of this thesis is to look for a logical deductive calculus (we will adopt sequent calculus, originally introduced in Gentzen, 1935), which could describe quantum information and its properties. More precisely, we intended to describe in logical terms the formation of the qubit (the unit of quantum information) which is a particular linear superposition of the two classical bits 0 and 1. To do so, we had to introduce the new connective "quantum superposition", in the logic of one qubit, Lq, as the classical conjunction cannot describe this quantum link.Comment: 138 pages, PhD thesis in Mathematic