Three-level logic minimization based on function regularities

Abstract

We exploit the "regularity" of Boolean functions with the purpose of decreasing the time for constructing minimal three-level expressions, in the sum of pseudoproducts (SPP) form recently developed. The regularity of a Boolean function f of n variables can be expressed by an autosymmetry degree k (with 0 /spl les/ k /spl les/ n). k = 0 means no regularity, that is we are not able to provide any advantage over standard synthesis. For k /spl ges/ 1 the function f is said to be autosymmetric, and a new function f/sub k/ depending on n - k variables only, called the restriction of f, is identified in time polynomial in the number of points of f. The relation between f and f/sub k/ is discussed in depth to show how a minimal SPP form for f can be build in linear time from a minimal SPP form for f/sub k/. The concept of autosymmetry is then extended to functions with don't care conditions, and the SPP minimization technique is duly extended to such functions. A large set of experimental results is presented, showing that 61% of the outputs for the functions in the classical ESPRESSO benchmark suite are autosymmetric. The minimization time for such functions is critically reduced, and cases otherwise intractable are solved. The quality of the corresponding circuits, measured with some well established cost functions, is also improved. Finally, we discuss the role and meaning of autosymmetric functions, and why a great amount of functions of practical interest fall in this class.