Chapter 4: A Computable Economist’s Perspective on Computational Complexity
K. Vela Velupillai* 4.1 Prologue There are many levels of complexity in problems, and corresponding boundaries between them. Turing computability is an outer boundary, . . . any theory that requires more power than that surely is irrelevant to any useful definition of human rationality. A slightly stricter boundary is posed by computational complexity, especially in its common ‘worst case’ form. We cannot expect people (and/or computers) to find exact solutions for large problems in computationally complex domains. This still leaves us far beyond what people and computers actually CAN do. The next boundary . . . is computational complexity for the ‘average case’ . . . . That begins to bring us closer to the realities of real-world and real-time computation. Finally, we get to the empirical boundary, measured by laboratory experiments on humans and by observation, of the level of complexity that humans actually can handle, with and without their computers, and – perhaps more important – what they actually do to solve problems that lie beyond this strict boundary even though they are within some of the broader limits. (Herbert Simon, letter to the author, 25 May 2000) As it happens, Herbert Simon’s lucid letter to me, delineating a rough ordering along the complexity scale for problem-solving by means of a machine model of computation, was dated 25 May 2000. The day before, on 24 May 2000, Arthur Jaffe, as the then President of the Clay Mathematical Institute, had announced the seven millennium problems,1 the solutions for which – in the form of a proof or a counterexample – would...
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