We consider the following decision problem: given two simply typed $\lambda$-terms, are they $\beta$-convertible? Equivalently, do they have the same normal form? It is famously non-elementary, but the precise complexity - namely TOWER-complete - is lesser known. One goal of this short paper is to popularize this fact. Our original contribution is to show that the problem stays TOWER-complete when the two input terms belong to Blum and Ong's safe $\lambda$-calculus, a fragment of the simply typed $\lambda$-calculus arising from the study of higher-order recursion schemes. Previously, the best known lower bound for this safe $\beta$-convertibility problem was PSPACE-hardness. Our proof proceeds by reduction from the star-free expression equivalence problem, taking inspiration from the author's work with Pradic on "implicit automata in typed $\lambda$-calculi". These results also hold for $\beta\eta$-convertibility.

翻译：我们考虑以下判定问题：给定两个简单类型λ项，它们是否β-可转换？等价地，它们是否具有相同的正规形式？该问题的非初等性是众所周知的，但其精确复杂度——即TOWER-完全性——则较少被了解。本文的一个目标是普及这一事实。我们的原创贡献在于证明：当两个输入项属于Blum与Ong提出的安全λ演算（源于高阶递归方案研究的简单类型λ演算片段）时，该问题仍保持TOWER-完全性。此前，该安全β-可转换性问题最著名的下界仅为PSPACE-困难。我们的证明通过从无星号正则表达式等价问题归约实现，其思路源于作者与Pradic在"类型化λ演算中的隐式自动机"中的工作。这些结论对βη-可转换性同样成立。