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In the catalytic logspace (CL) model of (Buhrman et. al. STOC 2013), we are given a small work tape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. This model is of interest from a complexity-theoretic perspective as it gains surprising power over traditional space. However, many fundamental structural questions remain open. We substantially advance the understanding of the structure of CL, addressing several questions raised in prior work. Our main results are as follows. 1: We unconditionally derandomize catalytic logspace: CBPL = CL. 2: We show time and catalytic space bounds can be achieved separately if and only if they can be achieved simultaneously: any problem in both CL and P can be solved in polynomial time-bounded CL. 3: We characterize deterministic catalytic space by the intersection of randomness and time: CL is equivalent to polytime-bounded, zero-error randomized CL. Our results center around the compress--or--random framework. For the second result, we introduce a simple yet novel compress--or--compute algorithm which, for any catalytic tape, either compresses the tape or quickly and successfully computes the function at hand. For our first result, we further introduce a compress--or--compress--or--random algorithm that combines runtime compression with a second compress--or--random algorithm, building on recent work on distinguish-to-predict transformations and pseudorandom generators with small-space deterministic reconstruction.
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Collapsing Catalytic Classes
IEEE.org IEEE Xplore IEEE SA IEEE Spectrum More Sites Donate Cart Create Account Personal Sign In IEEE Xplore logo - Link to home MIT logo Access provided by: MIT Sign Out IEEE logo - Link to IEEE main site homepage Search by Content Type ADVANCED SEARCH Conferences >2025 IEEE 66th Annual Symposi... Collapsing Catalytic Classes PDF Michal Koucký; Ian Mertz; Edward Pyne; Sasha Sami All Authors Abstract Document Sections I. Introduction II. Catalytic Machines III. Main Technical theorem IV. Proof of Main Result Authors References Keywords Metrics Footnotes Abstract: A catalytic machine is a space-bounded Turing machine with additional access to a second, much larger work tape, with the caveat that this tape is full, and its contents must be preserved by the computation. Catalytic machines were defined by Buhrman et al. (STOC 2014), who, alongside many follow-up works, exhibited the power of catalytic space (CSPACE) and, in particular, catalytic logspace machines (CL) beyond that of traditional space-bounded machines. Several variants of CL have been proposed, including nondeterministic and co-non-deterministic catalytic computation by Buhrman et al. (STACS 2016) and randomized catalytic computation by Datta et al. (CSR 2020). These and other works proposed several questions, such as catalytic analogues of the theorems of Savitch and Immerman and Szelepcsényi. Catalytic computation was recently derandomized by Cook et al. (STOC 2025), but only in certain parameter regimes. We settle almost all questions regarding randomized and nondeterministic catalytic computation by giving an optimal reduction from catalytic space with additional resources to the corresponding non-catalytic space classes. With regards to non-determinism, our main result is that CL = CNL and with regards to randomness we show CL = CPrL where CPrL denotes randomized catalytic logspace where the accepting probability can be arbitrarily close to 1/2. We also have a number of near-optimal partial results for non-deterministic and randomized catalytic computation with less catalytic space. We show catalytic versions of Savitch’s theorem, Immerman-Szelepscényi, and the derandomization results of Nisan and Saks and Zhou, all of which are unconditional and hold for all parameter settings. Our results build on the compress-or-compute framework of Cook et al. (STOC 2025). Despite proving broader and stronger results, our framework is simpler and more modular.
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- Award ID(s):
- 2310818
- PAR ID:
- 10676568
- Publisher / Repository:
- 66th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2025)
- Date Published:
- Format(s):
- Medium: X
- Location:
- Sydney, Australia
- Sponsoring Org:
- National Science Foundation
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