A Logspace Constructive Proof of L=SL

2025-11-14

AI Summary

This paper resolves Kolokolova’s long-standing open problem: formalizing a proof of Reingold’s theorem (SL = L) within the bounded arithmetic theory VL, thereby determining whether VL equals VSL. Departing from the original algebraic proof—which relies on spectral properties of graphs—the authors adopt the combinatorial approach of Rozenman and Vadhan, augmented by the Buss–Kabanets–Kolokolova–Koucký characterization of graph expansion. This yields the first constructive, purely combinatorial proof of SL = L formalizable in VL. The result directly establishes VL = VSL, demonstrating logical equivalence between symmetric logspace and deterministic logspace within bounded arithmetic. Consequently, it significantly strengthens the proof-theoretic foundations of logspace computability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Social networks and social learning

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-Koucký [Bus+20] that allow VL to reason about graph expansion in combinatorial terms.

Problem

• Formalizing Reingold's Theorem in bounded arithmetic VL
• Proving VL equals VSL through logspace reasoning
• Avoiding eigenvalue reasoning by using combinatorial graph expansion

Innovation

• Formalized Reingold's Theorem in bounded arithmetic VL
• Avoided eigenvalue reasoning using combinatorial graph expansion
• Established VL equals VSL through logspace constructive proof