We consider a computational model which is known as set automata.

The set automata are one-way finite automata with additional storage—the set. There are two kinds of set automata—deterministic (DSA's) and nondeterministic (NSA's). The model was introduced by Kutrib, Malcher, Wendlandt in 2014. It was shown that DSA-recognizable languages look similar to DCFL's and NSA-recognizable languages look similar to CFL's.

In this paper, we extend this similarity and prove that languages recognizable by NSA's form a rational cone, as do CFL's.

The main topic of this paper is computational complexity: we prove that

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languages recognizable by DSA's belong to P and there are P-complete languages among them;

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languages recognizable by NSA's are in NP and there are NP-complete languages among them;

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the word membership problem is P-complete for DSA's without ε-loops and PSPACE-complete for general DSA's;

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the emptiness problem is PSPACE-complete for DSA's.

我们考虑一个被称为集合自动机的计算模型。

集合自动机是具有附加存储的单向有限自动机——集合。有两种集合自动机——确定性（DSA）和非确定性（NSA）。该模型由 Kutrib、Malcher、Wendlandt 于 2014 年引入。结果表明，DSA 可识别语言与 DCFL 相似，NSA 可识别语言与 CFL 相似。

在本文中，我们扩展了这种相似性并证明了 NSA 可识别的语言形成了一个有理锥，CFL 也是如此。

本文的主要主题是计算复杂度：我们证明

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DSA可识别的语言属于P，其中有P-完全语言；

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NSA 可识别的语言是NP，其中有NP 完全语言；

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词隶属度问题对于没有ε -loops 的DSA来说是P -complete，对于一般 DSA 来说是PSPACE -complete ；

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空性问题是DSA 的PSPACE -complete。