  This work proposes tractable bisimulations for the higher-order pi-calculus
with session primitives (HOpi) and offers a complete study of the expressivity
of its most significant subcalculi. First we develop three typed bisimulations,
which are shown to coincide with contextual equivalence. These
characterisations demonstrate that observing as inputs only a specific finite
set of higher-order values (which inhabit session types) suffices to reason
about HOp} processes. Next, we identify HO, a minimal, second-order subcalculus
of HOpi in which higher-order applications/abstractions, name-passing, and
recursion are absent. We show that HO can encode HOpi extended with
higher-order applications and abstractions and that a first-order session
pi-calculus can encode HOpi. Both encodings are fully abstract. We also prove
that the session pi-calculus with passing of shared names cannot be encoded
into HOpi without shared names. We show that HOpi, HO, and pi are equally
expressive; the expressivity of HO enables effective reasoning about typed
equivalences for higher-order processes.
