Switching circuits for arithmetic operations are usually developed from algorithms based on the polyadic representation of numbers (integers and fractions). A few years ago M. Valach [1] pointed out that it is possible to design very simple switching circuits for multiplication and addition of integers if we based the coding on the Algebra of residual Classes. In an other paper A. Svoboda and M. Valach [2] introduced the numerical System of Residual Classes (SRC) for integers. This system was extended by A. Svoboda to cover operations with fractions, their rounding off and approximations. The extended system was called Rational System of Residual Classes (RSRC) [3]. In the present paper we will discuss some applications of this system in the logical design of automatic computers.

The numerical system of residual classes in mathematical machines

It has been a vexing question in recent years whether concepts are fuzzy. In this paper several views on the fuzziness of concepts are pointed out to have stemmed from dubious concepts of fuzziness. The underlying notions of the roles feasibly played byprototype, set, andprobability in modeling concepts strongly suggest that the controversy originates from a vague relation between intuitive and mathematical ideas in the cognitive sciences. It is argued that the application of fuzzy sets cannot resolve this vagueness since they are one sided,viz., defined on sets. An alternative definition based on classes (in the sense of axiomatic set theory) is proposed.

Fuzziness of concepts and concepts of fuzziness

“Prototypes” and “fuzziness” are regarded in this paper as fundamental phenomena in the inherent logic of concepts whose relationship, however, has not been sufficiently clarified. Therefore, modifications are proposed in the definition of both. Prototypes are defined as the elements possessing maximal degree of membership in the given category such thatthis membership has maximal cognitive efficiency in representing theelement. A modified fuzzy set (m-fuzzy set) is defined on aclass (possibly self-contradictory collection) such that its core (the collection of elements with full membership) is aset (self-consistent collection) comprising the finite set of the prototypes. In this scheme of recursive representation the possibility of contradictions, which has been proven to be inevitable in the logic of concepts, islocalized. Finally a related model of the brain's pattern recognition mechanism is briefly summarized.

“Prototypes” and “fuzziness” in the logic of concepts

Residue arithmetic offers a new approach to the development of general-purpose digital computers which detect and correct their own arithmetic and data-transmission errors. This approach is based on special properties inherent in a suitably chosen redundant residue number system (RRNS). In this paper, RRNS algorithms are described which can detect any number of residues in error, and can detect and correct one residue in error, in a data word suitable for scientific or control computation. Moreover, it is shown that there are RRNS algorithms available to perform all arithmetic operations needed in ordinary general-purpose computation. To this end, a new class of residue arithmetic techniques based on the properties of "macro-coefficients" is introduced, which permits all required fractional operations to be carried out and leads to a convenient form of the error-detection-and-correction procedures. The implications of these developments for ultra-reliable computer design are discussed, and the RRNS approach to computer ultrareliability is compared with several others which have been suggested; binary information with a residue check digit (BRCD), linear codes, and logic-circuit-level redundancy techniques such as majority voting.

Self-checked computation using residue arithmetic

A class of arithmetic code defined in residue number system (RNS) is proposed as a mathematical model of neural encoding The formalism of RNS provides for a vectorial representation in which the digits in the code-words (the "coordinates") contain the information in a distributed manner These digits and the code-words are assumed to represent the states of the neurons and neuron-modules, respectively The involved mechanism appears physiologically possible After the number-theoretical background is summarized in a brief though self-contained way, mathematical assertions are drawn on the error detecting and correction capacities of the formalized modules It is concluded that RNS provides an appropriate model of neural encoding though drawing inference by the neurons (the evaluation of the encoded information) requires another formalism

G. Fuhrmann

Papers

Arithmetic codes resembling neural encoding