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

TL;DR

This paper introduces a systematic approach to identify and classify universal logic gates for N-variable Boolean logic beyond traditional NAND and NOR gates, revealing a large library of such gates with exponential growth as variables increase.

Contribution

It presents a new search strategy ULG_{SS} for constructing a comprehensive library of universal logic gates for N-variable Boolean logic, expanding beyond classical gates.

Findings

56 universal gates for N=3 variables

Ratio of universal gates to total gates is approximately 0.25

Addition of constants significantly increases the number of universal gates

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.