Abstract

We introduce a formal definition of Wolfram’s notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process. 1 Computational Processes Degrees of unsolvability were introduced in two important papers by Post [21] and Kleene and Post [12]. The object of these papers was the study of the complexity of decision problems and in particular their relative complexity: how does a solution to one problem contribute to the solution of another, a notion that can be formalized in terms of Turing reducibility and Turing degrees. Post was particularly interested in the degrees of recursively enumerable (r.e.) degrees. The Turing degrees of r.e. sets together with Turing reducibility form a partial order and in fact an upper semi-lattice R. It is easy to see that R has least element /0, the degree of decidable sets, and a largest element /0 ′ , the degree of the halting set. Post asked whether there are any other r.e. degrees and embarked on a program to establish the existence of such an intermediate degree by constructing a suitable r.e. set. Post’s efforts produced a number of interesting ideas such as simple, hypersimple and hyperhypersimple sets but failed to produce

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