On sparseness and Turing reducibility over the reals

On Turing Reducibility

We show that the transitivity of pointwise Turing reducibility on the recursively enumerable sets of integers cannot be proven in P− + IΣ1, first order arithmetic with induction limited to Σ1 predicates. We produce a example of intransitivity in a nonstandard model of P+IΣ1 by a finite injury priority construction.

On the Turing Degrees of Divergence Bounded Computable Reals

The d-c.e. (difference of c.e.) and dbc (divergence bounded computable) reals are two important subclasses of ∆2-reals which have very interesting computability-theoretical as well as very nice analytical properties. Recently, Downey, Wu and Zheng [2] have shown by a double witness technique that not every ∆2-Turing degree contains a d-c.e. real. In this paper we show that the classes of Turing...

Infinite-Time Turing Machines and Borel Reducibility

In this document I will outline a couple of recent developments, due to Joel Hamkins, Philip Welch and myself, in the theory of infinite-time Turing machines. These results were obtained with the idea of extending the scope of the study of Borel equivalence relations, an area of descriptive set theory. I will introduce the most basic aspects of Borel equivalence relations, and show how infinite...

Turing cones and set theory of the reals

Computing over the Reals: Where Turing Meets Newton, Volume 51, Number 9

1024 NOTICES OF THE AMS VOLUME 51, NUMBER 9 The classical (Turing) theory of computation has been extraordinarily successful in providing the foundations and framework for theoretical computer science. Yet its dependence on 0s and 1s is fundamentally inadequate for providing such a foundation for modern scientific computation, in which most algorithms—with origins in Newton, Euler, Gauss, et al...