Computational and computer complexity of analogic cellular wave computers

The CNN Universal Machine is generalized as the latest step in computational architectures: a Universal Machine on Flows. Computational complexity and computer complexity issues are studied in different architectural settings. Three mathematical machines are considered: the universal machine on integers (UMZ), the universal machine on reals (UMR) and the universal machine on flows (UMF). The three machines induce different kinds of computational difficulties: combinatorial, algebraic, and dynamic, respectively. After a broader overview on computational complexity issues, it is shown, following the reasoning related the UMR, that in many cases the size is not the most important parameter related to computational complexity. Emerging new computing and computer architectures as well as their physical implementation suggest a new look on computational and computer complexities. The new analog-and-logic (analogic) cellular array computer paradigm, based on the CNN Universal Machine, and its physical implementation in CMOS and optical technologies, proves experimentally the relevance of the role of accuracy and problem parameter in computational complexity. We introduce also the rigorous definition of computational complexity for UMF and prove an NP class of problems. It is also shown that choosing the spatial temporal elementary instructions, as well as taking into account the area and power dissipation, these choices inherently influence computational complexity and computer complexity, respectively. Comments related to relevance to biology of the UMF are presented in relation to complexity theory. It is shown that algorithms using spatial-temporal continuous elementary instructions (alpha-recursive functions) represent not only a new world in computing, but also, a more general type of logic inference.