Lê Thành Dũng Nguyên - Simply typed convertibility is TOWER-complete even for safe lambda-terms

lmcs:11344 - Logical Methods in Computer Science, September 5, 2024, Volume 20, Issue 3 - https://doi.org/10.46298/lmcs-20(3:21)2024
Simply typed convertibility is TOWER-complete even for safe lambda-termsArticle

Authors: Lê Thành Dũng Nguyên ORCID

    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.


    Volume: Volume 20, Issue 3
    Published on: September 5, 2024
    Accepted on: June 24, 2024
    Submitted on: May 23, 2023
    Keywords: Computer Science - Logic in Computer Science,Computer Science - Programming Languages

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