Exploiting Symmetrization and D-Reducibility for Approximate Logic Synthesis

Abstract

Approximate synthesis is a recent trend in logic synthesis where one changes some outputs of a logic specification, within the error tolerance of a given application, to reduce the complexity of the final implementation. We attack the problem by exploiting the allowed flexibility in order to maximize the regularity of the specified Boolean functions. Specifically, we consider two types of regularity: symmetry and D-reducibility, and contribute two algorithms to find, respectively, a symmetric and a D-reducible approximation of a given target function f , within the given error rate threshold if possible. When targeting symmetry, we characterize and compute polynomially the closest symmetric approximation, i.e., the symmetric function obtained by injecting the minimum number of errors in the original incompletely specified Boolean function, with an unbounded number of errors; then, we discuss strategies to achieve partial symmetrization of the original specification while satisfying given error bounds. Finally, we present a polynomial heuristic algorithm to compute a D-reducible approximation of an incompletely specified target function, under a bit error metric. Experimental results on classical and new benchmarks confirm the effectiveness of the proposed approaches.

Developing an algorithm to minimize boolean functions for the visual-matrix form of the analytical method

“…The minimum function (15) contains 11 inverted variables. This is one inverted variable more than in (14).…”

“…Approximate synthesis is a recent trend of logical synthesis when some results of the logical specification change within the permissible error of the assigned application in order to reduce complexity and accelerate the implementation of the final digital component; it was considered in paper [14]. The task of synthesis is solved by using the permissible flexibility of technology to maximize the regularity of the assigned Boolean functions.…”

“…The considered literary sources [13][14][15][16][17][18][19] mainly report complex algorithms for minimizing Boolean functions based on the Quine-McClusky method, its modifications, and cubic technique. Compensation for the complexity of the search for the optimal function for such algorithms may be an approximate synthesis -the tendency of logical synthesis when some results of the logical specification change within the permissible optimality of the digital circuit being designed.…”

“…Thus, the complex Quine-McClusky algorithms, their modifications, cubic technique, software tools developed for them, covering the general procedure for minimizing logical functions [13][14][15][16][17][18][19], and the visual-matrix form of the analytical method, follow different approaches (principles of minimization). Therefore, they imply different prospects regarding the possibility of algorithmic minimization of logical functions.…”

“…Such components of the technology as «approximate synthesis», «heuristic algorithm», «adaptation to computer processing» are technological tools for current practice, which is changing. This is the reason to believe that the software and technological base, represented by the Quine-McClusky algorithms, their modifications, and cubic technique [13][14][15][16][17][18][19], is insufficient (in a state of change) for theoretical research on the optimal Boolean function minimization. That necessitates research into the visual-matrix form of the analytical method of minimizing logical functions.…”