Catalytic Computing and Register Programs Beyond Log-Depth

2025-04-24

AI Summary

This work investigates the computational power of circuits deeper than logarithmic depth under the catalytic space model, focusing on the relationship between the class CL and circuit classes SAC² and NC². We introduce a novel register-program construction that simulates SAC² circuits using only O(log²n / log log n) free space and subpolynomial catalytic tape capacity—improving the free-space complexity by a factor of O(log log n). Furthermore, we present the first nontrivial matrix-power register program enabling efficient NC²-related computations. Our main contributions are: (1) a proof that SAC² ⊆ CSPACE(O(log²n / log log n), 2^O(log^{1+ε}n)) for any ε > 0; (2) significant progress toward the long-standing conjecture NC² ⊆ CL; and (3) a new technical framework for depth–space trade-offs in the catalytic space model.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $TC^1$ 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 $TC^1$ 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 $epsilon>0$, $SAC^2 subseteq CSPACEleft(Oleft(frac{log^2{n}}{loglog{n}} ight), 2^{O(log^{1+epsilon} n)} ight)$ This is an $O(log log n)$ factor improvement on the free space needed to compute $SAC^2$, which can be accomplished with near-polynomial catalytic space. We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing $NC^2 subseteq CL$.

Problem

• Investigates catalytic space power for deeper circuits
• Improves free space efficiency for SAC^2 computation
• Advances register programs for matrix powering towards NC^2

Innovation

• Catalytic logspace improves circuit depth computation
• Register programs reduce free space for SAC^2
• Matrix powering advances NC^2 inclusion in CL