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arXiv · 2511.12011

A Logspace Constructive Proof of L=SL

Sam Buss·Anant Dhayal·Valentine Kabanets·Antonina Kolokolova·Sasank Mouli
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Abstract

We formalize the proof of Reingold's Theorem that SL=L [Rei05] in the theory of bounded arithmetic VL, which corresponds to ``logspace reasoning''. As a consequence, we get that VL=VSL, where VSL is the theory of bounded arithmetic for ``symmetric-logspace reasoning''. This resolves in the affirmative an old open question from Kolokolova [Kol05] (see also Cook-Nguyen [NC10]). Our proof relies on the Rozenman-Vadhan alternative proof of Reingold's Theorem ([RV05]). To formalize this proof in VL, we need to avoid reasoning about eigenvalues and eigenvectors (common in both original proofs of SL=L). We achieve this by using some results from Buss-Kabanets-Kolokolova-Kouck\'y [Bus+20] that allow VL to reason about graph expansion in combinatorial terms.

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Sam Buss, Anant Dhayal, Valentine Kabanets, Antonina Kolokolova, Sasank Mouli. 2025-11-15. A Logspace Constructive Proof of L=SL. https://arxiv.org/abs/2511.12011

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