ULGss: A Strategy to construct a Library of Universal Logic Gates for $N$-variable Boolean Logic beyond NAND and NOR (2308.11395v1)

Abstract: In literature, NAND and NOR are two logic gates that display functional completeness, hence regarded as Universal gates. So, the present effort is focused on exploring a library of universal gates in binary that are still unexplored in literature along with a broad and systematic approach to classify the logic connectives. The study shows that the number of Universal Gates in any logic system grows exponentially with the number of input variables $N$. It is revealed that there are $56$ Universal gates in binary for $N=3$. It is shown that the ratio of the count of Universal gates to the total number of Logic gates is $\approx $ $\frac{1}{4}$ or 0.25. Adding constants $0,1$ allow for the creation of $4$ additional (for $N=2$) and $169$ additional Universal Gates (for $N=3$). In this article, the mathematical and logical underpinnings of the concept of universal logic gates are presented, along with a search strategy $ULG_{SS}$ exploring multiple paths leading to their identification. A fast-track approach has been introduced that uses the hexadecimal representation of a logic gate to quickly ascertain its attribute.