On Kolmogorov´s superpositions: novel gates and circuits for nanoelectronics?

Based on explicit numerical constructions for Kolmogorov´s superpositions (KS) linear size circuits implementing arbitrary Boolean functions (BFs) are possible. Because classical Boolean as well as threshold logic (TL) implementations, require exponential size in the worst case, it follows that, size-optimal solutions for implementing arbitrary BFs should rely (at least partly) on KS-inspired gates (KGs). In this paper, we examine BFs of three inputs in detail and show that even the size given by KS can be reduced when Boolean gates (BGs) could be optimally combined with KGs (low precision analog gates). This shows that there is room for improving on the synthesis of BFs. Finally, we show that the size obtained when optimally combining BGs and KGs can be reduced even further if we are allowed to also use TL gates. Such systematic size reductions could help alleviate the challenging power consumption problem. They advocate for the design of KGs, as well as for the development of the theory, the algorithms, and the CAD tools that could take advantage of optimal combinations of different logic gates and design styles.