Towards a Church-Turing-Thesis for Infinitary Computations

We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines ($IAM$s) of this which will turn out to be equivalent in strength to the Ordinal Turing Machines defined by P. Koepke

Open questions about Ramsey-type statements in reverse mathematics

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page

Local stability of ergodic averages

The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining."Comment: Minor errors corrected. To appear in Transactions of the AM

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Infinite time decidable equivalence relation theory

We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely Delta_1^2 reductions, and hence to the infinite time computable reductions.Comment: 30 pages, 3 figure