CL^{NP}_{2-round}: Catalytic Logspace with NP Queries

• CL^{NP}_{2-round} is a complexity class where deterministic logspace machines use a catalytic tape and two rounds of NP oracle queries to solve decision and search problems.
• It efficiently simulates randomized and interactive proof classes such as BPP, MA, and ZPP^{NP[1]} using techniques like the isolation lemma and Nisan-Wigderson pseudorandom generators.
• The class occupies a unique niche between P^{NP} and ZPP^{NP}, offering insights into space-bounded derandomization and the limitations imposed by catalytic resource constraints.

CL^{NP}_{2-round} is a complexity class within the framework of catalytic logspace computation, capturing decision and search problems that can be solved by machines using deterministic logspace augmented with a catalytic tape whose contents must be restored at termination, and with access to two rounds of nondeterministic polynomial-time (NP) oracle queries. This class is strictly larger than the standard class CL (catalytic logspace) and is characterized by its ability to efficiently simulate certain probabilistic and interactive proof classes within a catalytic setting. Recent work demonstrates that CL^{NP}_{2-round} contains central classes such as BPP, MA, ZPP^{NP[1]}, and the search variant of SAT, highlighting its nuanced position in the landscape between P^{NP} and ZPP^{NP} (Arvind et al., 10 Dec 2025).

1. Catalytic Logspace Computation: Definitions and Context

Catalytic logspace computation models augment the classical logspace Turing machine with an additional read/write catalytic tape. A computation is valid only if the catalytic tape is restored to its original configuration on all computation paths, enforcing a form of reversibility. The model was introduced by Buhrman et al. (STOC 2014) to investigate space-bounded computation in the presence of auxiliary resources that cannot be altered globally.

In the catalytic logspace setting:

• The work tape is of O(log n) space.
• The catalytic tape can be of arbitrary initial content with strict restoration requirements.
• The complexity class CL consists of languages decidable in deterministic catalytic logspace.

CL^{NP}_{2-round} generalizes this setting to allow interaction with NP oracles—notably in two "rounds" (i.e., the algorithm can make a polynomial number of adaptive queries to an NP oracle, possibly in two stages) while operating under catalytic constraints. This positions CL^{NP}_{2-round} as a nontrivial extension of CL, with additional power derived from limited nondeterminism.

2. Containment Results and Relationships to Standard Classes

Key containment results for CL^{NP}_{2-round} establish its relevance:

• SearchSAT ∈ CL^{NP}_{2-round}: Using the isolation lemma, the search version of SAT (finding a satisfying assignment if one exists) can be solved in CL^{NP}_{2-round}. This contrasts with deterministic logspace, where such search-to-decision reductions are not efficiently feasible.
• BPP, MA, ZPP^{NP[1]} ⊆ CL^{NP}_{2-round}: The class simulates these probabilistic and Arthur-Merlin classes through techniques including the compress-or-random method and the Nisan-Wigderson pseudorandom generator construction (Arvind et al., 10 Dec 2025).

These containments show that CL^{NP}_{2-round} is powerful enough to encompass most known randomized logspace and low-depth interactive proof classes, indicating that catalytic logspace with limited nondeterministic advice can efficiently derandomize or simulate randomness in a provable way. The comparison with ZPP^{NP} is particularly striking, as it is argued that CL^{NP}_{2-round} "resembles ZPP^NP more than P^{NP}" (Arvind et al., 10 Dec 2025), hinting at finer gradations in complexity arising from catalytic resource restrictions.

3. Isolation, Derandomization, and Pseudorandomness in CL^{NP}_{2-round}

The inclusion of SearchSAT in CL^{NP}_{2-round} fundamentally uses the isolation lemma, which ensures with high probability that a unique minimum-weight solution exists in a suitably weighted combinatorial structure. The ability to perform this isolation under catalytic constraints, and verify or construct witnesses accordingly, is central to the simulation of nondeterministic search within the class.

Other containments rely on the Nisan-Wigderson pseudorandom generator to simulate BPP, MA, and ZPP^{NP[1]}, demonstrating that pseudorandomness constructions can be integrated into the catalytic framework, provided that the restoration requirement is respected. The compress-or-random technique enables, for example, the reuse of catalytic bits for randomness extraction, trading off workspace and catalytic space to enable simulations otherwise not viable in strict logspace.

4. Structural Consequences and Implications

The structural properties revealed by the containments above suggest that CL^{NP}_{2-round} is distinguished more by its ability to efficiently simulate randomized computations than to capture the full power of P^{NP}. This supports the observation that CL-based classes more closely resemble ZPP (zero-error probabilistic polynomial time) and its relatives than P.

This position is significant because it hints at an underlying constraint in catalytic computation: while such machines are powerful relative to deterministic logspace, their restoration requirements prevent them from fully leveraging unbounded nondeterminism or capturing all of P^{NP}. Instead, their structure aligns with "probabilistic, but reversible" computation.

5. Comparison with Related Catalytic and Space-Bounded Classes

The study of CL^{NP}_{2-round} is motivated by parallel developments in catalytic logspace (CL), unambiguous catalytic classes (e.g., CUTISP), and standard space-bounded complexity classes such as NL, LogCFL, and ZPP^{NP}. A summary of relevant classes and inclusions:

Class	Computational Model	Known Containments
CL	Deterministic catalytic logspace	planar matching search, weighted arborescence
CL^{NP}_{2-round}	Catalytic logspace with two rounds of NP-oracle access	SearchSAT, BPP, MA, ZPP^{NP[1]}
CUTISP	Unambiguous, catalytic logspace with bounded workspace and catalytic tape	contains NL, LogCFL (Arvind et al., 10 Dec 2025)
ZPP^NP	Zero-error probabilistic poly-time with NP oracle	BPP, MA, ZPP^{NP[1]}

The close alignment of CL^{NP}_{2-round} with ZPP^NP—rather than P^NP—reflects both the limitations and strengths of catalytic space-bounded computation with controlled nondeterminism.

6. Broader Impact and Open Questions

The explicit characterization of CL^{NP}_{2-round} and its relationship to classical complexity classes refines the taxonomy of logspace and catalytic computation. Several natural problems in search, randomization, derandomization, and unambiguity coalesce within CL^{NP}_{2-round}. The evidence that "CL is more like ZPP than P" (Arvind et al., 10 Dec 2025) has implications for derandomization strategies, the feasibility of logspace kernelization, and the potential for catalytic resources to efficiently simulate probabilistic computation.

Open questions include:

• The exact boundaries between CL, CL^{NP}_{2-round}, ZPP^NP, and P^NP.
• Whether more general classes, e.g., CL^{NP}_{k-round} for k > 2, subsume all of P^NP or are constrained by further resource limitations.
• The potential for catalytic techniques to yield new algebraic and combinatorial algorithms falling within or near CL^{NP}_{2-round}.