A Logspace Constructive Proof of L=SL

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 forsymmetric-logspace reasoning''. This resolves in the affirmative an old open question from Kolokolova Kol05. 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-Koucký [Bus+20] that allow VL to reason about graph expansion in combinatorial terms.