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Catalytic Computing and Register Programs Beyond Log-Depth

Authors Yaroslav Alekseev , Yuval Filmus , Ian Mertz , Alexander Smal , Antoine Vinciguerra

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ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Complexity classes
Keywords
  • catalytic computing
  • circuit classes
  • polynomial method

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Abstract

In a seminal work, Buhrman et al. (STOC 2014) defined the class CSPACE(s,c) of problems solvable in space s with an additional catalytic tape of size c, which is a tape whose initial content must be restored at the end of the computation. They showed that uniform TC¹ circuits are computable in catalytic logspace, i.e., CL = CSPACE(O(log{n}), 2^{O(log{n})}), thus giving strong evidence that catalytic space gives L strict additional power. Their study focuses on an arithmetic model called register programs, which has been a focal point in development since then.
Understanding CL remains a major open problem, as TC¹ remains the most powerful containment to date. In this work, we study the power of catalytic space and register programs to compute circuits of larger depth. Using register programs, we show that for every ε > 0, 
SAC² ⊆ CSPACE (O((log²n)/(log log n)), 2^{O(log^{1+ε} n)}) . 
On the other hand, we know that SAC² ⊆ TC² ⊆ CSPACE(O(log²{n}) , 2^{O(log{n})}). Our result thus shows an O(log log n) factor improvement on the free space needed to compute SAC², at the expense of a nearly-polynomial-sized catalytic tape.
We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing NC² ⊆ CL.

Cite As Get BibTex

Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra. Catalytic Computing and Register Programs Beyond Log-Depth. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.6

Author Details

Yaroslav Alekseev
  • Technion Israel Institute of Technology, Haifa, Israel
Yuval Filmus
  • Technion Israel Institute of Technology, Haifa, Israel
Ian Mertz
  • Charles University, Prague, Czech Republic
Alexander Smal
  • JetBrains Research, Paphos, Cyprus
Antoine Vinciguerra
  • Technion Israel Institute of Technology, Haifa, Israel

Funding

  • Alekseev, Yaroslav: Supported by ISF grant 507/24.
  • Filmus, Yuval: Supported by ISF grant 507/24.
  • Mertz, Ian: Supported by the Grant Agency of the Czech Republic under the grant agreement no. 24-10306S and by the Center for Foundations of Contemporary Computer Science (Charles Univ. project UNCE 24/SCI/008).
  • Vinciguerra, Antoine: Supported by ISF grant 507/24.

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