Subtyping in concurrency has been extensively studied since early 1990s as
one of the most interesting issues in type theory. The correctness of subtyping relations has
been usually provided as the soundness for type safety. The converse direction, the com-
pleteness, has been largely ignored in spite of its usefulness to de ne the largest subtyping
relation ensuring type safety. This paper formalises preciseness (i.e. both soundness and
completeness) of subtyping for mobile processes and studies it for the synchronous and the
asynchronous session calculi. We rst prove that the well-known session subtyping, the
branching-selection subtyping, is sound and complete for the synchronous calculus. Next
we show that in the asynchronous calculus, this subtyping is incomplete for type-safety:
that is, there exist session types
T
and
S
such that
T
can safely be considered as a subtype
of
S
, but
T
6
S
is not derivable by the subtyping. We then propose an asynchronous sub-
typing system which is sound and complete for the asynchronous calculus. The method
gives a general guidance to design rigorous channel-based subtypings respecting desired
safety properties. Both the synchronous and the asynchronous calculus are rst consid-
ered with linear channels only, and then they are extended with session initialisations and
communications of expressions (including shared channels).