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This textbook presents a survey of research on boolean functions, circuits, parallel computation models, function algebras, and proof systems. Its main aim is to elucidate the structure of "fast" parallel computation. The complexity of parallel computation is emphasized through a variety of techniques ranging from finite combinatorics, probability theory and finite group theory to finite model theory and proof theory. Nonuniform computation models are studied in the form of boolean circuits; uniform ones in a variety of forms. Steps in the investigation of non-deterministic polynomial time are surveyed as is the complexity of various proof systems. The book will benefit advanced undergraduates and graduate students as well as researchers in the field of complexity theory. ... Read more

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The importance of equational axioms emerged initially with the axiomatic approach to Boolean algebras, groups, and rings, and later in lattices. This unique research monograph systematically presents minimal equational axiom-systems for various lattice-related algebras, regardless of whether they are given in terms of join and meet or other types of operations such as ternary operations. Each of the axiom-systems is coded in a handy way so that it is easy to follow the natural connection among the various axioms and to understand how to combine them to form new axiom systems.

A new topic in this book is the characterization of Boolean algebras within the class of all uniquely complemented lattices. Here, the celebrated problem of E V Huntington is addressed, which -- according to G Gratzer, a leading expert in modern lattice theory -- is one of the two problems that shaped a century of research in lattice theory. Among other things, it is shown that there are infinitely many non-modular lattice identities that force a uniquely complemented lattice to be Boolean, thus providing several new axiom systems for Boolean algebras within the class of all uniquely complemented lattices. Finally, a few related lines of research are sketched, in the form of appendices, including one by Dr Willian McCune of the University of New Mexico, on applications of modern theorem-proving to the equational theory of lattices. ... Read more

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Logic forms the basis of mathematics and is a fundamental part of any mathematics course. This book provides students with a clear and accessible introduction to this important subject, using the concept of model as the main focus and covering a wide area of logic. The chapters of the book cover propositional calculus, boolean algebras, predicate calculus and completelness theorems with answeres to all of the exercises and the end of the volume. This is an ideal introduction to mathematics and logic for the advanced undergraduate student. ... Read more

Customer Reviews (1)

OK but Hard
You'll find this very hard unless you are a competent math major at one of the better universities. Similar to Elliot Mendelson's text, but not quite as good. Good chapter on Boolean algebra as a
piece of pure math; Halmos and Givant is gentler, though.

Interesting topic covered: the resolution so dear to the AI crowd. Unlike most mathematicians, Cori and Lascar have time for
the way computer scientists think. At the same time, this book does not cover tableau methods (see Smullyan), natural deduction, Genzen's ideas, and so on. For pure logic at the advanced undergrad level, you're better off with Bostock.

Haven't seen Part II, so cannot comment on the treatment of set theory. This is something Mendelson and Machover already do well. ... Read more

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This collection of papers presents a series of in-depth examinations of a variety of advanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the collection is the common thread created by the fundamental language, concepts, models, and tools provided by Boolean theory. Many readers will be surprised to discover the countless links between seemingly remote topics discussed in various chapters of the book. This text will help them draw on such connections to further their understanding of their own scientific discipline and to explore new avenues for research. ... Read more

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Process Algebra is a formal description technique for complex computer systems, especially those involving communicating, concurrently executing components. It is a subject that concurrently touches many topic areas of computer science and discrete math, including system design notations, logic, concurrency theory, specification and verification, operational semantics, algorithms, complexity theory, and, of course, algebra.

This Handbook documents the fate of process algebra since its inception in the late 1970's to the present. It is intended to serve as a reference source for researchers, students, and system designers and engineers interested in either the theory of process algebra or in learning what process algebra brings to the table as a formal system description and verification technique. The Handbook is divided into six parts spanning a total of 19 self-contained Chapters.

The organization is as follows. Part 1, consisting of four chapters, covers a broad swath of the basic theory of process algebra. Part 2 contains two chapters devoted to the sub-specialization of process algebra known as finite-state processes, while the three chapters of Part 3 look at infinite-state processes, value-passing processes and mobile processes in particular. Part 4, also three chapters in length, explores several extensions to process algebra including real-time, probability and priority. The four chapters of Part 5 examine non-interleaving process algebras, while Part 6's three chapters address process-algebra tools and applications.
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Customer Reviews (1)

Helpful overview
Process algebra is not only of interest in mathematics, theoretical computer science, and mathematical logic but is of enormous importance in applications. Most of these applications have been in business process modeling, models of concurrent processing, and computational biology. This book gives a fairly comprehensive overview of the main approaches to process algebra, but emphasizing theoretical foundations and not practical applications. One of the process algebras discussed in the book is the pi-calculus, this discussion appearing in Part 3 of the book by Joachim Parrow. This article was the only one read by this reviewer, so the commentary here will deal with it exclusively.

The pi-calculus has been the subject of an enormous amount of research in the last fifteen years, and has found its way into practical applications. It is introduced by the author as a mathematical model of processes with interconnections that change as they interact. The computational paradigm in the pi-calculus is fairly simple: a computation consists of the transfer of a communication link between processes. The process receiving the link can then utilize it to engage in interactions with other processes or agents. If one approaches the pi-calculus with a knowledge of the lambda calculus underlying functional computation, one will find that there are many parts of the pi-calculus that admit an interpretation in the lambda calculus.

There is much more to the pi-calculus however, and the author has given the reader a general discussion of its theoretical foundations. His approach is fairly clear, with many examples given, and he devotes separate sections to the discussion of some of the variants of the pi-calculus. This allows readers to skip these sections if they only want to learn concentrate on the pi-calculus as it was originally conceived (by the researcher Robin Milner).

One of the first things that the author clears up concerns the view that pi-calculus is merely a value-passing process algebra (with the values being the links). Noting that the things that are transferred do not have any internal structure may reinforce this belief, he says. To counter this skepticism he notes that that expressive power of the pi-calculus comes from its ability to migrate `local scopes.' As in other process algebras, one can define a private or `restricted' link between processes in the pi-calculus, but this link can be sent by the processes that share it to another process that can use it.The scope of an object must go along with the object when it is transferred between the processes, and this makes the pi-calculus more than just a value-passing process algebra he argues.

As in other process algebras some processes in the pi-calculus should be viewed as essentially the same if they have the same effective behavior. There are many notions of equivalence of processes in the pi-calculus, but the first one that is discussed by the author concerns the `structural congruence' of processes. This notion is a purely syntactical one, where two processes are structurally congruent if they are the same under a change of bound names (called alpha-conversion in the lambda calculus). Structural congruence also enforces the parallel processes as an Abelian group, and has various scope extension laws.

Another notion of behavioural equivalence that arises in the pi-calculus (as in many other process algebras) is that of `strong bisimilarity.' Two processes are bisimilar if they can be related by a bisimulation, i.e. if they can both perform the same actions and essentially mimic the transitions of each other. However bisimilarity is not a congruence since it is not preserved by the input prefix, and so this motivates the introduction of `strong congruence'. Two agents are strongly congruent if they are bisimilar for all substitutions.The author proves that strong congruence is the largest congruence in bisimilarity. Of modern recent interest is the notion of `barbed' congruence, which is discussed by the author in this article and which involves the inability of an observer to find out if an action is enabled on a given channel. This is a `barbed bisimulation' and two agents are `barbed congruent' if they are related by a barbed bisimulation for all contexts.

The operational semantics of the pi-calculus is given by a labeled transition system. The transitions are labeled by a collection of actions: the internal action, the free output actions, and the input actions, and no actions are possible for a restricted name as subject. But there is also a notion of `bound output' action that the author explains in some detail, where a local name can be transmitted and its scope extended to the recipient. ... Read more

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The book constitutes the joint refereed proceedings of the 10th International Conference on Relational Methods in Computer Science, RelMiCS 2008, and the 5th International Conference on Applications of Kleene Algebras, AKA 2008, held in Manchester, UK in April 2008.

The 26 revised full papers presented together with 2 invited papers were carefully reviewed and selected from numerous submissions. The papers describe the calculus of relations and similar algebraic formalisms as methodological and conceptual tools with special focus on formal methods for software engineering, logics of programs and links to neighbouring disciplines. Their scope comprises relation algebra, fixpoint calculi, semiring theory, iteration algebras, process algebras and dynamic algebras. Applications include formal algebraic modeling, the semantics, analysis and development of programs, formal language theory and combinatorial optimization.

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This book presents a systematic investigation of the theory of those commutative, unital subalgebras (of bounded linear operators acting in a Banach space) which are closed for some given topology and are generated by a uniformly bounded Boolean algebra of projections. One of the main aims is to employ the methods of vector measures and integration as a unifying theme throughout. This yields proofs of several classical results which are quite different to the classical ones. This book is directed to both those wishing to learn this topic for the first time and to current experts in the field. ... Read more

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Many fundamental combinatorial problems, arising in suchdiverse fields as artificial intelligence, logic, graph theory, andlinear algebra, can be formulated as Boolean constraint satisfactionproblems (CSP). This book is devoted to the study of the complexity ofsuch problems. The authors' goal is to develop a framework forclassifying the complexity of Boolean CSP in a uniform way. In doingso, they bring out common themes underlying many concepts and resultsin both algorithms and complexity theory. The results and techniquespresented here show that Boolean CSP provide an excellent frameworkfor discovering and formally validating "global" inferences about thenature of computation.

This book presents a novel and compact form of a compendium thatclassifies an infinite number of problems by using a rule-basedapproach. This enables practitioners to determine whether or not agiven problem is known to be computationally intractable. It alsoprovides a complete classification of all problems that arise inrestricted versions of central complexity classes such as NP, NPO, NC,PSPACE, and #P. ... Read more

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Boolean function complexity has seen exciting advances in the past few years. It is a long established area of discrete mathematics that uses combinatorial and occasionally algebraic methods. Professor Paterson brings together papers from the 1990 Durham symposium on Boolean function complexity. The list of participants includes very well known figures in the field, and the topics covered will be significant to many mathematicians and computer scientists working in related areas. ... Read more

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Timing research in high performance VLSI systems has advancedat a steady pace over the last few years. Tools, however, especiallytheoretical mechanisms, lag behind. Much of the present timingresearch relies heavily on timing diagrams, which although intuitive,are inadequate for analysis of large designs with many parameters.Further, timing diagrams offer only approximations, not exactsolutions to many timing problems and provide little insight in thecases where temporal properties of a design interact intricately withthe design's logical functionalities. Timed Boolean Functions presents a methodology for timingresearch which facilitates analysis and design of circuits and systemsin a unified temporal and logical domain. The goal of the book is topresent the central idea of representing logical and timinginformation in a common structure, TBFs, and to present a canonicalform suitable for efficient manipulation. This methodology is thenapplied to practical applications to provide intuition and insightinto the subject so that these general methods can be adapted tospecific engineering problems and also to further the researchnecessary to enhance the understanding of the field. Timed Boolean Functions is written for professionals involved intiming research and digital designers who want to enhance theirunderstanding of the timing aspects of high speed circuits. Theprerequisites are a common background in logic design, computeralgorithms, combinatorial optimization and a certain degree ofmathematical sophistication. ... Read more

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Chapters: Kleene Algebra, Introduction to Boolean Algebra, Canonical Form, Relation Algebra, Predicate Functor Logic, Higher-Dimensional Algebra, Residuated Boolean Algebra, Abstract Algebraic Logic, Action Algebra, Mv-Algebra, Monadic Boolean Algebra, Cylindric Algebra, Lindenbaum-tarski Algebra, Polyadic Algebra. Source: Wikipedia. Pages: 107. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Boolean algebra, developed in 1854 by George Boole in his book An Investigation of the Laws of Thought, is a variant of ordinary elementary algebra. Boolean algebra differs from ordinary algebra in three ways: in the values that variables may assume, which are of a logical instead of a numeric character, prototypically 0 and 1; in the operations applicable to those values; and in the properties of those operations, that is, the laws they obey. Applications include mathematical logic, digital logic, computer programming, set theory, and statistics. Whereas elementary algebra deals mainly with real numbers, Boolean algebra deals with the values 0 and 1. These can be thought of as two integers, or as the truth values false and true respectively. In either case they are called bits or binary digits, in contrast to the decimal digits 0 through 9. Boolean algebra also deals with other values on which Boolean operations can be defined, such as sets or sequences of bits. However, Boolean algebra is unlike many other systems of algebra in that it obeys exactly the same laws (equational properties), neither more nor fewer, no matter which of these other values are employed. Much of the subject can therefore be introduced without reference to any values besides 0 and 1. We therefore postpone to the section on Boolean algebras the treatment of these other values and focus on these two prototypical Boolean value...More: http://booksllc.net/?id=15410556 ... Read more

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Distributed by Elsevier Science on behalf of Science Press.

This book is mainly designed for graduate students who are interested in the theory of BCK and BCI-algebras. It introduces the general theoretical basis of BCI-algebras, omitting difficult proofs and abstract topics which are less necessary for beginners to learn. With abundant examples and exercises arranged after each section, it provides readers with easy-to-follow steps into this field.

* Specially designed for graduate students with emphasis on elementary knowledge in this field
* Organizes knowledge points systematically and highlights various arguments on vital topics to make them easy to be understand
* Gives many examples to clarify important notations and terminologies and abundant of classified exercises after each chapter for revision purposes
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This work represents a comprehensive account of all existing methods of statistical estimation for Boolean models. It summarizes the available theory concerning Boolean models in a form accessible for mathematically aware scientists working in stochastic geometry, image analysis and microscopy. Written in lecture note style, it has two levels of presentation - one orientated to applied mathematicians and the other for general mathematicians. ... Read more

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Since the early work of Montague, Boolean semantics and its subfield of generalized quantifier theory have become the model-theoretic foundation for the study of meaning in natural languages. This book uses this framework to develop a new semantic theory of central linguistic phenomena involving coordination, plurality, and scope. The proposed theory makes use of the standard Boolean interpretation of conjunction, a choice-function account of indefinites, and a novel semantics of plurals that is not based on the distributive/collective distinction. The key to unifying these mechanisms is a version of Montagovian semantics that is augmented by flexibility principles: semantic operations that have no counterpart in phonology.This is the first book to cover these areas in a way that is both linguistically comprehensive and formally explicit. On one hand, it addresses questions of primarily linguistic concern: the semantic functions of words like and and or in different languages, the interpretation of indefinites and their scope, and the semantic typology of noun phrases and predicates. On the other hand, it addresses formal questions that are motivated by the treatment of these linguistic problems: the use of Boolean algebras in linguistics, the proper formalization of choice functions within generalized quantifier theory, and the extension of this theory to the domain of plurality. While primarily intended for readers with a background in theoretical linguistics, the book will also be of interest to researchers and advanced students in logic, computational linguistics, philosophy of language, and artificial intelligence. ... Read more

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This book contains the proceedings of the NATO-Russia Advanced Study Institute (ASI) 'Boolean Functions in Cryptology and Information Security', which was held in Zvenigorod, Moscow region, Russia. These proceedings consist of three parts. The first part contains survey lectures on various areas of Boolean function theory that are of primary importance for cryptology. These lectures were delivered by leading researchers from many countries and contain both classic and recent results. The second part contains research papers written by graduate and postgraduate students of Lomonosov University, Moscow.The third part contains a list of open problems in Boolean function theory. The book includes lectures and papers concern the following areas: cryptographic properties of Boolean functions and mappings; algebraic and combinatorial constructions of Boolean functions and mappings with prescribed cryptographic properties; Boolean functions and mappings in cryptosynthesis; classification of Boolean functions; cryptanalysis of ciphers; and, efficient computations in finite fields.

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