Abstract. The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky and Weinberger on applications of computability to differential geometry. Yu and Ding showed that the relevant degree structure restricted to the c.e. reals has no greatest element, and asked for maximal elements. We answer this question for the case of c.e. sets. Using a doubly non-uniform argument we show that there are no maximal elements in the sw degrees of the c.e. sets. We note that the same holds for the Solovay degrees of c.e. sets. 1

Abstract. To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B>st A) if for every computable function f, for all but finitely many x, mB(x)> f(mA(x)). This settling-time ordering, which is a natural extension to an ordering of the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it, that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x)> f(mA(g(x))).

We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physics-like computation. These processes admit a natural classification into decidable, intermediate and complete, where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting. It is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process.

Abstract. This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut–off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.

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