Sebastiaan Terwijn (Radboud University) – Computability theory and combinatory algebra

Partial combinatory algebras (pcas) are abstract models of computation. They are one of the earliest type of model that emerged in the 1930's when mathematicians were trying to define what it means to be computable. Notions such as recursive functions (used by Gödel in the proof of his incompleteness theorems), Turing machines, combinatory algebra and lambda calculus each turned out to be useful in the development of the theory of computation in their own way. Pcas are also used in the study of constructive mathematics. They play a key role in the connections between constructive mathematics, proof theory, and computability, and also serve as a basis for models of constructive mathematics.

In this talk we will give a quick review of the basics of combinatory algebra, and discuss some of the key examples, such as Kleene's models $\mathcal{K}_1$ (which describes the classical setting for computation on the natural numbers), $\mathcal{K}_2$ (which does the same for the real numbers), and Scott's graph model. We then discuss recent results about embeddings and completions of pcas, and in particular the complexity of various embeddings. Depending on time, we will also discus the complexity of the isomorphism problem, extensionality, and ordinal analysis of pcas.