Statistical Complexity of Algorithms for Boolean Function Minimization

ABSTRACT

The first problem in a two level Boolean minimization is the determination of the prime k-cubes. This paper is concerned with the estimation of the statistical complexity of some well-known algorithms which solve this problem. Formulas are given for the average number of comparison operations among k-cubes occurring in Quine's method and in Mc-Cluskey's method; these quantities provide indications of the average execution time of computer programs based on the corresponding algorithms. Numerical values are given and commented on. Formulas are also obtained for the variance of the number of k-cubes and the variance of the number of cubes of a Boolean function; in fact the calculation of these quantities is strictly related to that of the average number of comparison operations among k-cubes. These variances give an idea of the probable error made by using the corresponding average values (obtained in a previous paper by the authors) to make forecasts. It turns out that this error is quite small.

REFERENCES

• 1: MILETO, F., AND PUTZOLU, G. Average values of quantities appearing ia Boolean function minimization. IEEE Trans. EC-I3, 2 (Apr. 1964), 87-92.
• 2: QuINE, W. V. The problem of simplifying truth functions. Amer. Math. Monthly 59, 8 (Oct. 1952), 521-531.
• 3: McCLusY, E. J. Jn. Miniraization of Boolean functions. Bell Sys. Tech. Y. 85, 6 (Nov. 1956), 1417-1444.
• 4: FELLER, W. An Introduction to Probability Theory and Its Applications, Vol. I. Wiley, New York, 1960.
• 5: DIg TROYE, N. C. Classification and minimization of switching functions. Philips Res. Reps. 14, 1959, 250-292.
• 6: ROTH, J. D. Algebraic topological methods for the synthesis of switching systems, I. Trans. Amer. Math. Soc. 88 (July, 1958), 301-326.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

General Terms:
Algorithms, Measurement, Performance, Theory