Reviews of Books and Papers in the Computer Field DONALD

This paper presents an empirical formula C=Ki(K2)ng h/(g+h) for the average cost C of a two-level combinational logic network (switching circuit) implementing a single Boolean function with specified numbers of independent variables n, "one" vertices g, and "zero" vertices h (or don't-care vertices d =2n-g -h). K, and Kp are constants that depend, generally, on the used technology and available types of moduli. Although the results presented in the paper are not quite satisfactory (limitations to one output and two levels, K, and K2 are presented only for two specific definitions of the cost, the average percentage error of the formula was found to exceed 15 percent), they represent a first step in an important new direction in switching theory-an experimental study of statistical properties of logic functions and switching circuits. When neglecting the possibility of multiple minimization, the formula can help significantly to get a fairly good cost evaluation of several different designs of a large scale digital system. It also throws a little light upon the following general question: "Using the same technology, is a switching circuit with -n inputs and ml outputs cheaper or more expensive (in average) than a switching circuit with n2 inputs and m2 outputs, where ni>n2 and ml <iM2?" The more precisely we can answer the question, the better we can evaluate different models of sequential switching circuits.' Incidentally, the paper reminds one of the results presented by Mileto and Putzolu some years ago.2"3 They derived, in a strictly combinatorial way, formulas for average number of k cubes, prime k cubes, and essential k cubes of a Boolean function specified by the same parameters as those which are used in the above-mentioned formula. One wonders if their method could be adopted for the problem stated by Kellerman. If not, further experimentation, similar to that described by Kellerman, should be done for multiple output switching circuits. An excellent tool for such an experimentation seems to be the Boolean Analyzer-a special logic unit attached with a general-purpose digital computer-suggested by Svoboda.4