A language is said to be in catalytic logspace if we can test membership using a deterministic logspace machine that has an additional read/write tape filled with arbitrary data whose contents have to be restored to their original value at the end of the computation. The model of catalytic computation was introduced by Buhrman et al [STOC2014]. As our first result, we obtain a catalytic logspace algorithm for computing a minimum weight witness to a search problem, with small weights, provided the algorithm is given oracle access for the corresponding weighted decision problem. In particular, our reduction yields CL algorithms for the search versions of the following three problems: planar perfect matching, planar exact perfect matching and weighted arborescences in weighted digraphs. Our second set of results concern the significantly larger class CL^{NP}_{2-round}. We show that CL^{NP}_{2-round} contains SearchSAT and the complexity classes BPP, MA and ZPP^{NP[1]}. While SearchSAT is shown to be in CL^{NP}_{2-round} using the isolation lemma, the other three containments, while based on the compress-or-random technique, use the Nisan-Wigderson [JCSS 1994] based pseudo-random generator. These containments show that CL^{NP}_{2-round} resembles ZPP^NP more than P^{NP}, providing some weak evidence that CL is more like ZPP than P. For our third set of results we turn to isolation well inside catalytic classes. We consider the unambiguous catalytic class CTISP[poly(n),logn,log^2n]^UL and show that it contains reachability and therefore NL. This is a catalytic version of the result of van Melkebeek & Prakriya [SIAM J. Comput. 2019]. Building on their result, we also show a tradeoff between the workspace of the oracle and the catalytic space of the base machine. Finally, we extend these catalytic upper bounds to LogCFL.

翻译：若存在一个确定性对数空间机器，该机器配备一个额外的读写磁带，磁带初始填充任意数据且计算结束时必须恢复其原始内容，且能用于测试成员资格，则称该语言属于催化对数空间。催化计算模型由Buhrman等人[STOC2014]引入。作为我们的第一个结果，我们获得了一个催化对数空间算法，用于计算搜索问题的最小权重见证（权重较小），前提是该算法被赋予对相应加权决策问题的预言机访问权限。特别地，我们的归约产生了以下三个问题搜索版本的CL算法：平面完美匹配、平面精确完美匹配以及加权有向图中的加权树形图。我们的第二组结果涉及显著更大的类CL^{NP}_{2-round}。我们证明了CL^{NP}_{2-round}包含SearchSAT以及复杂度类BPP、MA和ZPP^{NP[1]}。虽然SearchSAT通过隔离引理被证明属于CL^{NP}_{2-round}，但其他三个包含关系虽然基于压缩或随机技术，却使用了基于Nisan-Wigderson [JCSS 1994]的伪随机生成器。这些包含关系表明，CL^{NP}_{2-round}更类似于ZPP^NP而非P^{NP}，这为CL更类似于ZPP而非P提供了一些弱证据。对于我们的第三组结果，我们转向催化类内部的隔离问题。我们考虑明确催化类CTISP[poly(n),logn,log^2n]^UL，并证明它包含可达性问题，因此包含NL。这是van Melkebeek & Prakriya [SIAM J. Comput. 2019]结果的催化版本。基于他们的结果，我们还展示了预言机的工作空间与基础机器的催化空间之间的权衡。最后，我们将这些催化上界扩展到LogCFL。