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Some Recent Developments in Space Complexity (Invited Talk)

Author R. Ryan Williams

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LIPIcs.MFCS.2026.3.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Turing machines
Keywords
  • time lower bound
  • space complexity
  • multitape Turing machine
  • P versus PSPACE
  • tree evaluation problem

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Abstract

Given a function, what is the minimal memory necessary to compute it? We will describe some old [John E. Hopcroft et al., 1977; Wolfgang J. Paul and Rüdiger Reischuk, 1981; Joseph Y. Halpern et al., 1986] and new algorithmic developments that give surprisingly low-space solutions to this question in many cases. We will survey the recent proof [Ryan Williams, 2026] that TIME[t] is contained in SPACE[√{t log t}] for the multitape Turing machine model, and the engine that makes the proof possible: the amazing Cook-Mertz algorithm [James Cook and Ian Mertz, 2024] for a problem called Tree Evaluation [Stephen A. Cook et al., 2012]. We will also briefly outline some new developments and generalizations that we've recently proved, pushing the low-space frontier beyond multitape Turing machines. The latter is joint work with Danil Sibgatullin (to appear).

Cite As Get BibTex

R. Ryan Williams. Some Recent Developments in Space Complexity (Invited Talk). In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.3

Author Details

R. Ryan Williams
  • Massachusetts Institute of Technology, CSAIL, Cambridge, MA, USA

Funding

Parts of this work were completed while R.W. was visiting the Institute for Advanced Study, Princeton, NJ. This material is based upon work supported by the National Science Foundation under grants DMS-2424441 (at IAS) and CCF-2420092 (at MIT).

References

  1. James Cook and Ian Mertz. Tree evaluation is in space O(log n ⋅ log log n). In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC), pages 1268-1278. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649664.
  2. Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam. Pebbles and branching programs for tree evaluation. ACM Trans. Comput. Theory, 3(2):4:1-4:43, 2012. URL: https://doi.org/10.1145/2077336.2077337.
  3. Joseph Y. Halpern, Michael C. Loui, Albert R. Meyer, and Daniel Weise. On time versus space III. Math. Syst. Theory, 19(1):13-28, 1986. URL: https://doi.org/10.1007/BF01704903.
  4. John E. Hopcroft, Wolfgang J. Paul, and Leslie G. Valiant. On time versus space. J. ACM, 24(2):332-337, 1977. Conference version in FOCS'75. URL: https://doi.org/10.1145/322003.322015.
  5. Wolfgang J. Paul and Rüdiger Reischuk. On time versus space II. J. Comput. Syst. Sci., 22(3):312-327, 1981. URL: https://doi.org/10.1016/0022-0000(81)90035-0.
  6. Ryan Williams. Simulating time with square-root space. J. ACM, 73(3):16:1-16:25, 2026. URL: https://doi.org/10.1145/3798104.
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