Multilevel Logic Minimization Using K-map XOR Patterns

OMBINING entered variable (EV) subfunctions and C XOR patterns in a Karnaugh map (K-map) extraction process [3] is a special and powerful form of K-map multilevel function minimization. Used with two-level logic forms (AND and OR functions) this multilevel minimization approach leads to XOIUSOP, EQVPOS and hybrid forms that often represent a substantial reduction in the hardware not possible otherwise. XOWSOP and EQVROS forms are those connecting p-terms with XOR operators or s-terms with EQV operators, respectively. Hybrid forms are those containing a mixture of these. Conventional (1’s and 0’s) K-map methods involving XOR pattems have been used to obtain coefficient values for ReedMuller XOR expansions [4]. Such methods have been shown to provide minimum fixed-polarity solutions to such expansions. Sasao [2] has shown that the use of conventional K-maps together with Boolean manipulation can be used to yield a XOIUSOP minimum for some functions. Except for the work cited in [3] no other published work is known to exist on the subject of logic minimization using EV K-map XOR patterns. In the past the XOR gate (or EQV gate) has been viewed as a two-level device, meaning two units of path delay. Thus, its use with other gates, including other XOR or EQV gates, resulted in multiple units of path delay, hence the term multilevel. Also, XOR logic was known to be expensive, hardware-wise. But the emergence of CMOS IC technology has moved the XOR and EQV gates close to single level gates with respect to compactness and speed [3]. Furthermore, the arrival of FPGA’s and FPLD’s has made XOWSOP and EQV/POS based logic synthesis practical [2]. In this article it will be shown how simple “pencil-andpaper” methods can be used to extract gate-minimum multilevel logic designs not yet possible by any other method including the use of CAD techniques. The methods described in this article make possible multilevel IC designs that occupy much less real estate than would be possible for an equivalent