Unconventional Computers and Unconventional Complexity Measures

Abstract

Unconventional computers—for example those exploiting chemical, analogue or quantum phenomena in order to compute, as opposed to those employing the standard, digital-computer approach of electronically implementing discrete-value logic gates—are widely studied both theoretically and experimentally. One notable motivation driving this study is the desire efficiently to solve classically difficult problems—we recall for example a chemical-computer approach to the \(\mathsf {{NP}}\)-complete Travelling Salesperson Problem—, with computational complexity theory providing the criteria for judging this efficiency. However, care must be taken: conventional (Turing-machine-style) complexity analyses are in many cases inappropriate for assessing unconventional computers; new, non-standard computational resources, with correspondingly new complexity measures, may well be consumed during unconventional computation, and yet are overlooked by conventional analyses. Accordingly, we discuss in this chapter various resources beyond merely the conventional time and space, advocating such resources’ consideration during analysis of the complexity of unconventional computers (and, more fundamentally, we discuss various interpretations of the term ‘resource’ itself). We hope that this acts as a useful starting point for practitioners of unconventional computing and computational complexity.