# The recursion hierarchy for PCF is strict

Let $\PCF_k$ denote the sublanguage of Plotkin’s $\PCF$ in which fixed point operators $Y_\sigma$ are admitted only for types $\sigma$ of level $\leq k$. We show that the languages $\PCF_k$ form a strict hierarchy, in the sense that for each $k$, there are closed programs of $\PCF_{k+1}$ that are not observationally equivalent to any programs of $\PCF_k$. This answers a question posed by Berger in 1999. Our proof makes substantial use of the theory of \emph{nested sequential procedures} (also called \emph{$\PCF$ B\”ohm trees}).

I plan to give the talk in two parts: first, a discussion of the result with a general overview of the method of proof, and then (for any who are interested) a closer look at some of the technical details of the proof.

This talk is part of the Theoretical computer science seminar series.