  Recursive types extend the simply-typed lambda calculus (STLC) with the
additional expressive power to enable diverging computation and to encode
recursive data-types (e.g., lists). Two formulations of recursive types exist:
iso-recursive and equi-recursive. The relative advantages of iso- and
equi-recursion are well-studied when it comes to their impact on
type-inference. However, the relative semantic expressiveness of the two
formulations remains unclear so far. This paper studies the semantic
expressiveness of STLC with iso- and equi-recursive types, proving that these
formulations are equally expressive. In fact, we prove that they are both as
expressive as STLC with only term-level recursion. We phrase these
equi-expressiveness results in terms of full abstraction of three canonical
compilers between these three languages (STLC with iso-, with equi-recursive
types and with term-level recursion). Our choice of languages allows us to
study expressiveness when interacting over both a simply-typed and a
recursively-typed interface. The three proofs all rely on a typed version of a
proof technique called approximate backtranslation. Together, our results show
that there is no difference in semantic expressiveness between STLCs with iso-
and equi-recursive types. In this paper, we focus on a simply-typed setting but
we believe our results scale to more powerful type systems like System F.
