# Blog Archives

# Topic Archive: computable analysis

# Weihrauch reducibility and Ramsey theorems

Weihrauch reducibility is a common tool in computable analysis for understanding and comparing the computational content of theorems. In recent years, variations of Weihrauch reducibility have been used to study Ramsey type theorems in the context of reverse mathematics where they give a finer analysis than implications in **RCA _{0}** and they allow comparisons of computably true principles. In this talk, we will give examples of recent results and techniques in this area.

# Generalizing Turing machines: $\omega_1$-computation

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.