Computational Processes and Incompleteness

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

The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.Comment: 15 page

The Borel complexity of the class of models of first-order theories

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\boldsymbol\Pi_\omega^0$-complete set of models. We also give sharp conditions for theories to have a $\boldsymbol\Pi^0_n$-complete set of models. Finally, we determine the Turing degrees needed to witness the completeness

Point-Separable Classes of Simple Computable Planar Curves

In mathematics curves are typically defined as the images of continuous real functions (parametrizations) defined on a closed interval. They can also be defined as connected one-dimensional compact subsets of points. For simple curves of finite lengths, parametrizations can be further required to be injective or even length-normalized. All of these four approaches to curves are classically equivalent. In this paper we investigate four different versions of computable curves based on these four approaches. It turns out that they are all different, and hence, we get four different classes of computable curves. More interestingly, these four classes are even point-separable in the sense that the sets of points covered by computable curves of different versions are also different. However, if we consider only computable curves of computable lengths, then all four versions of computable curves become equivalent. This shows that the definition of computable curves is robust, at least for those of computable lengths. In addition, we show that the class of computable curves of computable lengths is point-separable from the other four classes of computable curves

On the jumps of degrees below an recursively enumerable degree

We consider the set of jumps below a Turing degree, given by JB(a) = {x(1) : x <= a}, with a focus on the problem: Which recursively enumerable (r.e.) degrees a are uniquely determined by JB(a)? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high(2) r.e. degree a is determined by JB(a), then R cannot have a nontrivial automorphism. We then defeat the strategy-at least in the form presented-by constructing pairs a(0), a(1) of distinct r.e. degrees such that JB(a(0)) = JB(a(1)) within any possible jump class {x : x' = c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity