Simply typed convertibility is TOWER-complete even for safe lambda-terms

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 p...

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Bibliographic Details
Published in:Logical methods in computer science 2024-09, Vol.20, Issue 3 (3)
Main Author: Nguyên, Lê Thành Dũng
Format: Article
Language:English
Subjects:
Computer Science
computer science - logic in computer science
computer science - programming languages
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Summary: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.
ISSN:1860-5974
1860-5974
DOI:10.46298/lmcs-20(3:21)2024