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Improved Bounds on the Space Complexity of Circuit Evaluation

Author Yakov Shalunov

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LIPIcs.CCC.2026.3.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Communication complexity
Keywords
  • circuit value problem CVP
  • space complexity
  • tree evaluation problem

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Abstract

Williams (STOC 2025) recently proved that time-t multitape Turing machines can be simulated using O(√{t log t}) space using the Cook-Mertz (STOC 2024) tree evaluation procedure. As Williams notes, applying this result to fast algorithms for the circuit value problem implies an O(√s ⋅ polylog s) space algorithm for evaluating circuits with s gates.
In this work, we provide a direct reduction from circuit value to tree evaluation without passing through Turing machines, simultaneously improving the bound to O(√{s log s}) space and providing a proof with fewer layers of abstraction.
This result can be thought of as a "sibling" result to Williams' for circuit complexity instead of time; in particular, using the fact that time-t Turing machines have size O(t log t) circuits, we can recover a slightly weakened version of Williams' result, simulating time-t machines in space O(√t log t).

Cite As Get BibTex

Yakov Shalunov. Improved Bounds on the Space Complexity of Circuit Evaluation. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.3

Author Details

Yakov Shalunov
  • Department of Computer Science, University of Chicago, IL, USA

Acknowledgements

I would like to thank William Hoza for suggesting this research direction, helping review this article, and general guidance. I am also grateful to Alexander Razborov for feedback on presentation.

Supplementary Materials

References

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