In this talk, I will discuss two problems on examining in which cases context-free processes and pushdown processes are the same. In particular, we depart from the well-known case of language equivalence and instead look at processes using process theory and more fine-grained equivalences, such as branching bisimulation and contrasimulation. We identify two difficulties when looking at process specifications: head-recursion and transparency. Here two new results are achieved: we prove that when excluding transparency, context-free processes and pushdown processes are equivalent up to at least divergence-insensitive variant of branching bisimilarity. When including transparency, we prove that they are equivalent up to at least contrasimulation.

This is a joint work with Jos Baeten and Zeno de Hoop.