# Generalizing Turing machines: $\omega_1$-computation

## Russell Miller

### City University of New York

While Turing machines are well established as the principal framework for theoretical computation on the natural numbers, no machine model for computation on uncountable domains has achieved any similarly dominant status. We will give an overview of various methods that have been explored, including computable analysis, the Blum-Shub-Smale model of computation on the real numbers, local computability, α-recursion theory, and certain models of computation over ordinal time (one of which turns out to be equivalent to α-recursion). Each of these has strengths and weaknesses, for which reason no one of them has come to dominate the discussion. Much of the talk will be expository, describing these different notions of computation, but we will finish with a specific example from ω_{1}-recursive model theory which shows how phenomena that were unknown under Turing computation on ω can arise in the context of ω_{1}-computable model theory.

Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center. He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.