Universal Algebra Algebraic Logic and Databases

Author: B. Plotkin
Publisher: Springer Science & Business Media
ISBN: 9789401108201
Release Date: 2012-12-06
Genre: Mathematics

Modern algebra, which not long ago seemed to be a science divorced from real life, now has numerous applications. Many fine algebraic structures are endowed with meaningful contents. Now and then practice suggests new and unexpected structures enriching algebra. This does not mean that algebra has become merely a tool for applications. Quite the contrary, it significantly benefits from the new connections. The present book is devoted to some algebraic aspects of the theory of databases. It consists of three parts. The first part contains information about universal algebra, algebraic logic is the subject of the second part, and the third one deals with databases. The algebraic material of the flI'St two parts serves the common purpose of applying algebra to databases. The book is intended for use by mathematicians, and mainly by algebraists, who realize the necessity to unite theory and practice. It is also addressed to programmers, engineers and all potential users of mathematics who want to construct their models with the help of algebra and logic. Nowadays, the majority of professional mathematicians work in close cooperation with representatives of applied sciences and even industrial technology. It is neces sary to develop an ability to see mathematics in different particular situations. One of the tasks of this book is to promote the acquisition of such skills.

Cylindric like Algebras and Algebraic Logic

Author: Hajnal Andreka
Publisher: Springer Science & Business Media
ISBN: 9783642350252
Release Date: 2014-01-27
Genre: Mathematics

Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.​

Formal Theories of Information

Author: Giovanni Sommaruga
Publisher: Springer Science & Business Media
ISBN: 9783642006586
Release Date: 2009-04-22
Genre: Computers

This book presents the scientific outcome of a joint effort of the computer science departments of the universities of Berne, Fribourg and Neuchâtel. Within an initiative devoted to "Information and Knowledge", these research groups collaborated over several years on issues of logic, probability, inference, and deduction. The goal of this volume is to examine whether there is any common ground between the different approaches to the concept of information. The structure of this book could be represented by a circular model, with an innermost syntactical circle, comprising statistical and algorithmic approaches; a second, larger circle, the semantical one, in which "meaning" enters the stage; and finally an outermost circle, the pragmatic one, casting light on real-life logical reasoning. These articles are complemented by two philosophical contributions exploring the wide conceptual field as well as taking stock of the articles on the various formal theories of information.

Handbook of Algebra

Author: M. Hazewinkel
Publisher: Elsevier
ISBN: 0080532969
Release Date: 2000-04-06
Genre: Mathematics

Handbook of Algebra

Universal Algebra for Computer Scientists

Author: Wolfgang Wechler
Publisher: Springer Science & Business Media
ISBN: 9783642767715
Release Date: 2012-12-06
Genre: Computers

A new model-theoretic approach to universal algebra is offered in this book. Written for computer scientists, it presents a systematic development of the methods and results of universal algebra that are useful in a variety of applications in computer science. The notation is simple and the concepts are clearly presented. The book concerns the algebraic characterization of axiomatic classes of algebras (equational, implicational, and universal Horn classes) by closure operators generalizing the famous Birkhoff Variety Theorem, and the algebraic characterization of the related theories. The book also presents a thorough study of term rewriting systems. Besides basic notions, the Knuth-Bendix completion procedure and termination proof methods are considered. A third main topic is that of fixpoint techniques and complete ordered algebras. Algebraic specifications of abstract data types and algebraic semantics of recursive program schemes are treated as applications. The book is self-contained and suitable both as a textbook for graduate courses and as a reference for researchers.

Graph Structure and Monadic Second Order Logic

Author: Bruno Courcelle
Publisher: Cambridge University Press
ISBN: 9781139644006
Release Date: 2012-06-14
Genre: Mathematics

The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The authors not only provide a thorough description of the theory, but also detail its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory.

Lectures in General Algebra

Author: A. G. Kurosh
Publisher: Elsevier
ISBN: 9781483149578
Release Date: 2014-07-10
Genre: Mathematics

Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. The book examines the parallelism between the theory of groups and the theory of rings; such examinations show the convenience of constructing a single theory from the results of group experiments and ring experiments which are known to follow simple corollaries. The text also presents algebraic structures that are not of binary nature. From this parallelism arise other concepts, such as that of the lattices, complete lattices, and modular lattices. The book then proves the Schmidt-Ore theorem, and also describes linear algebra, as well as the Birkhoff-Witt theorem on Lie algebras. The text also addresses ordered groups, the Archimedean groups and rings, and Albert's theorem on normed algebras. This book can prove useful for algebra students and for professors of algebra and advanced mathematicians.

Advanced ukasiewicz calculus and MV algebras

Author: D. Mundici
Publisher: Springer Science & Business Media
ISBN: 9400708408
Release Date: 2011-06-22
Genre: Mathematics

This is a continuation of Vol. 7 of Trends in Logic. It wil cover the wealth of recent developments of Lukasiewicz Logic and their algebras (Chang MV-algebras), with particular reference to (de Finetti) coherent evaluation of continuously valued events, (Renyi) conditionals for such events, related algorithms.

Relational Methods in Computer Science

Author: Chris Brink
Publisher: Springer Science & Business Media
ISBN: 3211829717
Release Date: 1997-04-23
Genre: Computers

The calculus of relations has been an important component of the development of logic and algebra since the middle of the nineteenth century, when Augustus De Morgan observed that since a horse is an animal we should be able to infer that the head of a horse is the head of an animal. For this, Aristotelian syllogistic does not suffice: We require relational reasoning. George Boole, in his Mathematical Analysis of Logic of 1847, initiated the treatment of logic as part of mathematics, specifically as part of algebra. Quite the opposite conviction was put forward early this century by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910 - 1913): that mathematics was essentially grounded in logic. Logic thus developed in two streams. On the one hand algebraic logic, in which the calculus of relations played a particularly prominent part, was taken up from Boole by Charles Sanders Peirce, who wished to do for the "calculus of relatives" what Boole had done for the calculus of sets. Peirce's work was in turn taken up by Schroder in his Algebra und Logik der Relative of 1895 (the third part of a massive work on the algebra of logic). Schroder's work, however, lay dormant for more than 40 years, until revived by Alfred Tarski in his seminal paper "On the calculus of binary relations" of 1941 (actually his presidential address to the Association for Symbolic Logic).

Simple Relation Algebras

Author: Steven Givant
Publisher: Springer
ISBN: 9783319676968
Release Date: 2018-01-09
Genre: Mathematics

This monograph details several different methods for constructing simple relation algebras, many of which are new with this book. By drawing these seemingly different methods together, all are shown to be aspects of one general approach, for which several applications are given. These tools for constructing and analyzing relation algebras are of particular interest to mathematicians working in logic, algebraic logic, or universal algebra, but will also appeal to philosophers and theoretical computer scientists working in fields that use mathematics. The book is written with a broad audience in mind and features a careful, pedagogical approach; an appendix contains the requisite background material in relation algebras. Over 400 exercises provide ample opportunities to engage with the material, making this a monograph equally appropriate for use in a special topics course or for independent study. Readers interested in pursuing an extended background study of relation algebras will find a comprehensive treatment in author Steven Givant’s textbook, Introduction to Relation Algebras (Springer, 2017).

Universal Algebra

Author: George Grätzer
Publisher: Springer Science & Business Media
ISBN: 0387774874
Release Date: 2008-12-15
Genre: Mathematics

Universal Algebra has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well selected additional bibliography of over 1250 papers and books which makes this an indispensable new edition for students, faculty, and workers in the field.

Handbook of Logic and Language

Author: Johan F.A.K. van Benthem
Publisher: Elsevier
ISBN: 0444537279
Release Date: 2010-12-17
Genre: Mathematics

The logical study of language is becoming more interdisciplinary, playing a role in fields such as computer science, artificial intelligence, cognitive science and game theory. This new edition, written by the leading experts in the field, presents an overview of the latest developments at the interface of logic and linguistics as well as a historical perspective. It is divided into three parts covering Frameworks, General Topics and Descriptive Themes. Completely revised and updated - includes over 25% new material Discusses the interface between logic and language Many of the authors are creators or active developers of the theories