Advanced Computational Complexity Theory from an Elementary Standpoint

A computable economist's view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministric computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called 'Post's Program for Research for Higher Recursion Theory'. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix.