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 threelevel expressions, in the sum of pseudoproducts (SPP) form recently developed. The regularity of a Boolean function of variables can be expressed by an autosymmetry degree (with 0). = 0 means no regularity, that is we are not able to provide any advantage over standard synthesis. For 1 the function is said to be autosymmetric, and a new function depending on variables only, called the restriction of , is identified in time polynomial in the number of points of. The relation between and is discussed in depth to show how a minimal SPP form for can be build in linear time from a minimal SPP form for. 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.

Minimization of Boolean functions by combinatorial method

“…In [11], a discussion is presented on the role of the autosymmetry degree of variables in a Boolean function and why it deserves attention on minimizing a logical function. The regularity of the variables of a Boolean function can be expressed by the degree of autosymmetry, which in the end gives a new tool for effective minimization.…”

“…In contrast to [7][8][9][10][11][12][13][14][15][16], in this paper, the object of solving the problem of minimizing a Boolean function is a combinatorial block-diagram with repetition, which allows to concentrate the minimization principle within the truth table of a given function. The peculiarities of the combinatorial method consist in greater informativeness of the process of solving the problem in comparison with the algebraic method of minimizing the function, due to tabular organization and the introduction of figurative calculus apparatus.…”

“…No. Using Table 5, let's compose the PDNF of the given 5-bit function from the blocks for which the function receives the value of 1, that is, for the sets 1,4,5,7,9,11,12,13,16,17,18,22,28,31. …”

DRedSOP: Synthesis of a New Class of Regular Functions

“…Factorization of literal terms is a widely studied field in multi-level logic [4,18]. Finally note that D-reducibility and autosymmetry (described in [2]) are different regularities, since autosymmetric functions can be studied in a new space whose variables are EXOR combinations of the original ones, and D-reducible functions are studied in a projection space producing an expression where the EXOR gates are in AND with a SOP form. However, D-reducible functions have an interesting connection with autosymmetric functions through their Walsh transform [1].…”

Conjugate symmetry

“…Although we have confined our attention to a single application, our on-going work suggests very strongly that the concepts introduced here will be useful in other areas of design automation such as three-level function minimization [34][35][36][37] and other areas.…”