# On computational complexity of Set Automata

## TL;DR

This paper analyzes the computational complexity of set automata, showing that deterministic set automata recognize languages in P, nondeterministic ones in NP, and both have PSPACE-complete problems, clarifying their computational boundaries.

## Contribution

It establishes the complexity classes of languages recognized by deterministic and nondeterministic set automata, including completeness results and problem complexities.

## Key findings

- DSA-recognized languages are in P, with some P-complete languages.
- NSA-recognized languages are in NP, with some NP-complete languages.
- Word membership is P-complete for DSA without epsilon-loops and PSPACE-complete for general DSA.

## Abstract

We consider a computational model which is known as set automata. The set automata are one-way finite automata with an additional storage---the set. There are two kinds of set automata---the deterministic and the nondeterministic ones. We denote them as DSA and NSA respectively. The model was introduced by M. Kutrib, A. Malcher, M. Wendlandt in 2014. It was shown that DSA-languages look similar to DCFL due to their closure properties and NSA-languages look similar to CFL due to their undecidability properties. In this paper we show that this similarity is natural: we prove that languages recognizable by NSA form a rational cone, so as CFL. The main topic of this paper is computational complexity: we prove that - languages recognizable by DSA belong to P and there are P-complete languages among them; - languages recognizable by NSA are in NP and there are NP-complete languages among them; - the word membership problem is P-complete for DSA without epsilon-loops and PSPACE-complete for general DSA; - the emptiness problem is in PSPACE for NSA and, moreover, it is PSPACE-complete for DSA.