ACM Computing Review CR137206: "Associative Digital Network Theory", ---
An associative algebra approach to logic, arithmetic and state machines.
(prof. Harvey Cohn, CUNY, aug 2009) -- Springer: 'out-of-print' 2009.
-- (c) N.F.Benschop 2010, "The Associative Structure of State Machines"
Publisher abc.nl (EBM), EUR 19.50
As a historical fact, mathematics developed from applications
-- in rational mechanics and number theory -- for which
commutative algebra was most natural. If the basic applications
were from network theory - Turing machines - the associative
algebra (ab)c = a(bc) would have been more natural, with Boolean
algebra aa = a, and commutative algebra ab = ba, as special cases.
Benschop develops this thesis in an idiosyncratic fashion,
reinforced by a long career of practical experience. This book
may well be an important historical document, also useful for
seminars, even if it is not presented primarily for class usage.
There are profuse illustrations in classic number theory, as
well as claims that the outlook sheds new light on classic
problems such as those of Fermat and Goldbach, interpreted as
machines. As unlikely as it is that this may be practical, it
makes for an interesting book.
PS: After re-acquiring the copyright from Springer in 2010, the
booktitle is now: "The Associative Structure of State Machines
--An associative algebra approach to logic, arithmetic and
automata", printed by abc.nl (American Book Center, Amsterdam,
Febr.2011) with an extended Ch.9 (elementary proof Goldbach).
[Ch.8 Fermat] http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html (p169-184)
[Ch.9 Goldbach] http://arxiv.org/abs/math.GM/0103091
====== Review 2:
Zentralblatt MATH, Vol.1169, 2009 (c 2010 FIZ Karlsruhe io-port 05500994)
Nico F. Benschop: "Associative digital network theory.
--- An associative algebra approach to logic, arithmetic and state machines".
Springer, Dordrecht (ISBN 978-1-4020-9828-4/hbk; 978-1-4020-9865-9/ebook).
180 p. EUR 96.25 (2009): 'out-of-print' 2009.
-- (c) N.F.Benschop 2010, "The Associative Structure of State Machines"
Publisher abc.nl (EBM, search for 'Benschop'), EUR 19.50
The book presents new ways for modeling digital networks (state
machines, sequential and combinational logic). It contains
applications for known principles of discrete mathematics.
The book has three parts. The first part presents state
machines and some algebraic ways to model them. Basically,
network composition is reduced to five basic types of state
machines. The second part is about Boolean logic. It introduces
the concept of spectrum, and some applications and algorithms
that are using it. It also presents symmetric Boolean functions
and some of their properties. An algorithm for symmetric logic
synthesis is provided. The last part of the book is about
residue arithmetic with two extremal types of prime related
moduli: m_k=p^k resp. m_k= \prod{first k primes}. A balanced focus on
closure- and generative properties of residues and carry is obtained
in the residue-and-carry method of arithmetic analysis.
The book also presents new ideas on the finite additive number
theory and a binary log-arithmetic microprocessor. This book
can be very useful for students and professors and also for
researchers interested in the digital network theory. It covers
a lot of fields, ranging from electrical engineering to computer
science and applied mathematics. --- prof. Eleonor Ciurea
(Univ. Brasov, Zentralblatt MATH, Vol. 1169, 2009)
Keywords: boolean functions; state machines;
sequential logic; combinatorial logic. \\
doi:10.1007/978-1-4020-9865-9 –