Titlebook: Associative Digital Network Theory; An Associative Algeb Nico F. Benschop Book Apr 2009Latest edition Springer Science+Business Media B.V.

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Samik Basu,Diptikalyan Saha,Scott A. Smolkahen can be binary encoded readily, allowing binary digital implementation. Partial specification of state transitions, due to an incomplete state transition table, is implemented by filling the . by the ‘dc’ rule: “don’t care ⟶ don’t change”.

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Thread–Based Analysis of Sequence Diagramscode optimization based on a fast algorithm for Boolean symmetry analysis. Apart from the well known majority voting (triplication), Hamming- and Product codes are described, having an implementation overhead much less than for majority voting.

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Simple Semigroups and the Five Basic Machines,which is shown to be a ., that is: a semi-direct product . |>(.×.) of a left- and a right-copy semigroup, and a group. So in general a CR-machine is a composition of a .-, a .- and a . machine, which are three of the five basic types of state machines to be derived.

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General Network Decomposition of State Machines,hen can be binary encoded readily, allowing binary digital implementation. Partial specification of state transitions, due to an incomplete state transition table, is implemented by filling the . by the ‘dc’ rule: “don’t care ⟶ don’t change”.

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Fault Tolerant Logic with Error Correcting Codes,code optimization based on a fast algorithm for Boolean symmetry analysis. Apart from the well known majority voting (triplication), Hamming- and Product codes are described, having an implementation overhead much less than for majority voting.

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Introduction,rse historical order: sequential logic (state machines), combinational logic (Boolean algebra) and arithmetic. The respective disciplines: CS/EE/NT (computer science/electrical engineering digital circuit design/number theory) are merged under one heading: finite associative algebra (semigroups), which is the algebra of function composition.

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,Powersums ∑,, Represent Residues mod ,,, from Fermat to Waring,nalysis of divisors of 0 mod ... For large enough .≥.. (critical precision ..<. depends on .), all nonzero pairsums of core residues are shown to be distinct, upto commutation. The known . case. is related to this, and the set ..+.. mod .. of .-th power pairsums is shown to cover half of ... This yields main result:

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Book Apr 2009Latest editionrical engineering, computer science and applied mathematics departments, interested in new developments of modeling and designing digital networks (DN: state machines, sequential and combinational logic) in general, as a combined math/engineering discipline. As background an undergraduate level of m

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,Fermat’s Small Theorem Extended to ,,mod ,,,. are shown to have distinct .. mod .., and divisors . of .−1 (resp. .+1) with different primesets have distinct .. mod ... Moreover 2.≢2  mod .. for prime ., related to . primes (Wieferich in J. Reine Angew. Math. 136:293–302, .) and . case. for integers (Chap. 8). .: Some .|.±1 is semi primitive root of 1 mod .., with units group {−1, .}..