Automation and Remote Control, Vol. 65, No. 6, 2004, pp. 842–856. Translated from Avtomatika i Telemekhanika, No. 6, 2004, pp. 4–21. Original Russian Text Copyright c  2004 by Astola, Egiazarian, M. Stankovi´ c, R. Stankovi´ c. ARITHMETIC LOGIC Fibonacci Arithmetic Expressions 1 J. T. Astola ∗ , K. Egiazarian ∗ , M. Stankovi´ c ∗∗ , and R. S. Stankovi´ c ∗∗ * Tampere International Center for Signal Processing, Tampere University of Technology, Tampere, Finland ** University of Niˇ s, Dept. of Computer Science, Faculty of Electronics, Niˇ s, Serbia Received December 16, 2003 Abstract—In this paper, we extend the arithmetic (AR) expressions for functions on finite dyadic groups to functions used in Fibonacci interconnection topologies. We have introduced the Fibonacci-Arithmetic (FibAR) expressions for representation of these functions. We discussed the optimization of FibARs with respect to the number of non-zero coefficients through the Fixed-Polarity FibARs defined by using different polarities for the Fibonacci variables. In this way, we provide a base to extend the application of ARs and related powerful CAD design tools for switching functions to functions in Fibonacci interconnection topologies. 1. INTRODUCTION Arithmetic expressions (ARs) for switching functions were used in the description of logic net- works from the beginning of development of this area [1–4], see also [5, 6]. Although ARs for switching functions, are hybrid expressions in the sense that the coefficients are integers for the logic-valued functions, there is apparent a renewed interest in application of ARs for the following reasons: (1) ARs are useful in parallelization of algorithms for calculations with switching functions [5,7,8], (2) ARs can represent the multi-output switching functions by a single expression which cannot be done with bit-level expressions [9,10,12], (3) ARs belong to the class of word-level expressions for switching functions in form of which some word-level decision diagrams represent the switching functions [13–15], (4) ARs can be used to estimate the error probability in logic networks [5]. Boolean interconnection topologies are extensively used in systems design, and logic design, see for example [6,21]. However, in some applications, they express some inconveniences originating in their inherent features, as for example, restrictions to the power of two in the number of nodes or inputs, etc. For this reason, the Fibonacci interconnection topologies are offered as an alternative [22–26]. With this motivation, in this paper we extended the ARs to functions defined in a number of points equal to a generalized Fibonacci p-number. 2. NOTATIONS The Boolean interconnection topologies are used in the design of systems whose input and output signals are described by functions defined in 2 n points, where n ∈ N 0 = N ∪ 0, N is the set of natural numbers. Each point is represented by an n-tuple x =(x 1 ,...,x n ), x i ∈{0, 1}, determined as the binary representation for x, i.e., x = n ∑ i=1 2 i x i . We denote the set of such n-tuples by C n 2 and 1 This work was supported by the Academy of Finland, project no. 44876 and EXSITE, project no. 51520. 0005-1179/04/6506-0842 c  2004 MAIK “Nauka/Interperiodica”