一句话：证明 iso-递归与 equi-递归类型的 STLC 在语义表达力上等价，且均与项级递归的 STLC 等价。
递归类型扩展了简单类型 λ 演算（STLC），使其支持发散计算并编码递归数据类型，其形式分为 iso-recursive 与 equi-recursive。两者对类型推断的影响已有研究，但语义表达力孰强孰弱仍不清楚。本文研究含 iso-recursive 与 equi-recursive 类型的 STLC 的语义表达力，证明二者表达力相同，并且都与仅含项级递归的 STLC 等价。作者在三门语言之间定义三个规范编译器，并以 full abstraction 表述这些等价性；语言选择使其能同时在简单类型接口和递归类型接口上考察交互。三个证明均依赖一种带类型的 approximate backtranslation 技术。本文聚焦简单类型设定，但作者认为结果可推广到 System F 等更强类型系统。
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.
@article{paperbot1198,
title = {On the semantic expressiveness of recursive types},
author = {Marco Patrignani and Eric Mark Martin and Dominique Devriese},
journal = {Proceedings of the ACM on Programming Languages},
volume = {5},
number = {POPL},
year = {2021},
doi = {10.1145/3434302}
}