A model-independent theory of computational complexity: from patience to precision and beyond - Blakey, E (2010)
The field of computational complexity theory--which chiefly aims to quantify the difficulty encountered when performing calculations--is, in the case of conventional computers, correctly practised and well understood (some important and fundamental open questions notwithstanding); however, such understanding is, we argue, lacking when unconventional paradigms are considered. As an illustration, we present here an analogue computer that performs the task of natural-number factorization using only polynomial time and space; the system's true, exponential complexity, which arises from requirements concerning precision, is overlooked by a traditional, `time-and-space' approach to complexity theory. Hence, we formulate the thesis that unconventional computers warrant unconventional complexity analysis; the crucial omission from traditional analysis, we suggest, is consideration of relevant resources, these being not only time and space, but also precision, energy, etc. In the presence of this multitude of resources, however, the task of comparing computers' efficiency (formerly a case merely of comparing time complexity) becomes difficult. We resolve this by introducing a notion of overall complexity, though this transpires to be incompatible with an unrestricted formulation of resource; accordingly, we define normality of resource, and stipulate that considered resources be normal, so as to rectify certain undesirable complexity behaviour. Our concept of overall complexity induces corresponding complexity classes, and we prove theorems concerning, for example, the inclusions therebetween. Our notions of resource, overall complexity, normality, etc. form a model-independent framework of computational complexity theory, which allows: insightful complexity analysis of unconventional computers; comparison of large, model-heterogeneous sets of computers, and correspondingly improved bounds upon the complexity of problems; assessment of novel, unconventional systems against existing, Turing-machine benchmarks; increased confidence in the difficulty of problems; etc. We apply notions of the framework to existing disputes in the literature, and consider in the context of the framework various fundamental questions concerning the nature of computation.