On the semantic expressiveness of recursive types

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.

Polarized Subtyping

“…Our direct interpretation of isorecursive types and translation into an equirecursive setting furthers numerous works either comparing or relating both formulations [65,71,72]. In particular, Abadi and Fiore [1] and more recently Patrigniani et al [61] prove that terms in one equirecursive setting can be typed in the other (and vice versa) with varying approaches. The former treats type equality inductively and is focused on syntactic considerations.…”

“…As we discussed in Section 7, we can also directly interpret isorecurisve typestypes that are isomorphic, embodied by fold and unfold operators, but not equal to their expansions-in order to obtain a formulation for isorecursive semantic typing, and therefore semantic subtyping, within our equirecursive setting. Previous work has studied the relation between these two formulations from a syntactic perspective [1], via type assignment with positive recursive typing [71], and, more recently, in relation to semantic expressiveness [61]. For our needs, we demonstrate a semantic translation from the iso-to equi-recursive settings that showcases no significant differences between these formulations.…”

$\pi_{\mathbf{RA}}$: A $\pi\text{-calculus}$ for Verifying Protocols that Use Remote Attestation

“…10: Diagram of the proof of equivalence reflection of FAC. This diagram is adapted from the diagram in [29].…”