A short overview of Hidden Logic

In this paper we review a hidden (sorted) generalization of k-deductive systems - hidden k-logics. They encompass deductive systems as well as hidden equational logics and inequational logics. The special case of hidden equational logics has been used to specify and to verify properties in program development of behavioral systems within the dichotomy visible vs. hidden data. We recall one of the main applications of this work - the study of behavioral equivalence. Related results are obtained through combinatorial properties of the Leibniz congruence relation. In addition we obtain a few new developments concerning hidden equational logic, namely we present a new characterization of the behavioral consequences of a theory

Leibniz hierarchy

Mestrado em MatemáticaA Lógica Algébrica Abstracta estuda o processo pelo qual uma classe de álgebras pode ser associada a uma lógica. Nesta dissertação, analisamos este processo agrupando lógicas partilhando certas propriedades em classes. O conceito central neste estudo é a congruência de Leibniz que assume o papel desempenhado pela equivalência no processo tradicional de Lindenbaum- Tarski. Apresentamos uma hierarquia entre essas classes que é designada por hierarquia de Leibniz, caracterizando as lógicas de cada classe por propriedades meta-lógicas, por exemplo propriedades do operador de Leibniz. Estudamos também a recente abordagem comportamental que usa lógicas multigénero, lógica equacional comportamental e, consequentemente, uma versão comportamental do operador de Leibniz. Neste contexto, apresentamos alguns exemplos, aos quais aplicamos esta nova teoria, capturando alguns fenómenos de algebrização que não era possível formalizar com a abordagem standard. ABSTRACT: Abstract Algebraic logic studies the process by which a class of algebras can be associated with a logic. In this dissertation, we analyse this process by grouping logics sharing certain properties into classes. The central concept in this study is the Leibniz Congruence that assumes the role developed by the equivalence in the traditional Lindenbaum-Tarski process. We show a hierarchy between these classes, designated by Leibniz hierarchy, by characterizing logics in each class by meta-logical properties, for example properties of the Leibniz operator. We also study a recent behavioral approach which uses many-sorted logics, behavioral equational logic and, consequently, a behavioral version of the Leibniz operator. In this context, we provide some examples, to which we apply this new theory, capturing some phenomena of algebraization that are not possible to formalize using the standard approach

Behavioral algebraization of logics

We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL toward providing a meaningful algebraic counterpart also to logics with a many-sorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way toward a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several well-known characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach. © 2009 Springer Science+Business Media B.V.FCT via UIM