Typicality and the Turing degrees

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov’s 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. In this paper, we prove results in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree.

The typical Turing degree, with Barmpalias and Day, Proceedings of the London Mathematical Society (2014) 109 (1). pp. 1-39, pdf. 

Computable structures

A computable structure is given by a computable domain, and then a set of computable relations and functions defined on that domain. The study of computable structures, going back as far as the work of Frohlich and Shepherdson, Rabin, and Malcev is part of a long-term programme to understand the algorithmic content of mathematics.

In mathematics generally, the notion of isomorphism is used to determine structures which are essentially the same. Within the context of effective (algorithmic) mathematics, however, one is presented with the possibility that pairs of computable structures may exist which, while isomorphic, fail to have a computable isomorphism between them. Thus the notion of computable categoricity has become of central importance: a computable structure S is computably categorical if any two computable presentations A and B of S are computably isomorphic. In this paper, my co-authors Downey, Kach, Lempp, Montalban, Turetsky and I, answer one of the longstanding questions in computable structure theory, showing the class of computably categorical structures has no simple structural or syntactic characterisation.

The complexity of computable categoricity, with Downey, Kach, Lempp, Montalban, and Turetsky, Advances in Mathematics 268 (2015), 423–466, pdf.