On the Semantic Expressiveness of Iso- and Equi-Recursive Types

TL;DR

This paper demonstrates that iso-recursive and equi-recursive types in simply-typed lambda calculus are equally expressive, both matching the expressiveness of term-level recursion through full abstraction proofs.

Contribution

It provides the first semantic comparison showing iso- and equi-recursive types are equally expressive, extending understanding beyond type inference implications.

Findings

Iso- and equi-recursive types are equally expressive.

Both formulations are as expressive as term-level recursion.

Results likely extend to more complex type systems like System F.

Abstract

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.