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：给定两个简单类型的λ项，它们是否β可转换？等价地，它们是否具有相同的范式？该问题众所周知是非初等的，但其精确复杂度——即TOWER完全——却鲜为人知。本短文的一个目标是推广这一事实。我们的原创贡献在于证明：当两个输入项属于Blum和Ong的安全λ演算（一个源自高阶递归方案研究的简单类型λ演算片段）时，该问题仍然是TOWER完全的。此前，这一安全β可转换性问题的已知最佳下界为PSPACE困难性。我们的证明通过从无星表达式等价问题归约来完成，借鉴了作者与Pradic在“类型λ演算中的隐式自动机”工作中的思路。这些结论同样适用于βη可转换性。