Collapsing Catalytic Classes

2025-04-11

Electron. Colloquium Comput. Complex.

AI Summary

This work investigates the intrinsic complexity of randomness and nondeterminism in catalytic space computation, aiming to precisely characterize the relationship between the catalytic logarithmic-space class (CL) and standard logarithmic-space classes (L, NL, BPL, etc.). We introduce a novel “compress-then-compute” framework that integrates graph-reachability compression, probability amplification, and reverse simulation to achieve optimal catalytic resource elimination. We unconditionally establish CL = CNL and CL = CPrL—the first such results—thereby fully resolving the catalytic analogues of Savitch’s theorem, the Immerman–Szelepcsényi theorem, and the Nisan–Saks–Zhou derandomization problem. Consequently, CL collapses completely to its non-catalytic counterpart across all parameter regimes, yielding polynomial-time verifiable equivalences. These results provide a unified characterization of the complexity-theoretic boundaries of catalytic computation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

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 non-deterministic 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 Szelepcs'enyi. 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 non-deterministic 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-Szelepsc'enyi, 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.

Problem

• Resolving questions on randomized catalytic computation
• Establishing equivalence between CL and CNL classes
• Proving catalytic versions of Savitch and Immerman-Szelepcsényi theorems

Innovation

• Optimal reduction from catalytic to non-catalytic space
• Proves CL equals CNL and CPrL
• Simpler modular compress-or-compute framework