Realizability Models for Sequential Computation

We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the deenitions of these new models. We give some precise deenitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of \sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Although the constructions of these various models have quite similar motivations, and the models themselves share many similar properties, some of the developments are so recent that they have not yet become generally known even among the handful of people working in the area. The purpose of this note is to bring together all these new models in one place, and to give a broad overview of the subject area and what it is trying to achieve. In comparison to most of the other work presented at the Pisa workshop, the material discussed here is fairly theoretical in avour, but I will suggest a few ways in which this work might have more practical repercussions. If any other workers whose background and interests are more practical than my own feel inclined to pursue any of these suggestions, I would be delighted to hear from them. I would like to thank all the people mentioned above for explaining their ideas to me, and also Thomas Streicher for suggesting that I give the talk on which this note is based.