On Computational Complexity of Set Automata

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 in [3] and [4]. 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, which is an extended version of the conference paper [10], 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 ε-loops and PSPACE-complete for general DSA; -the emptiness problem is in PSPACE for NSA and, moreover, it is PSPACE-complete for DSA.

Automata Equipped with Auxiliary Data Structures and Regular Realizability Problems

“…Most of the results of this section directly generalize the results of [13, Section 3] (see the full journal version [15]). In most cases, to get a generalized result, one can substitute SA protocols (see Example 8) with general protocols as defined in Definition 6.…”

“…Set supports the following operations: in(x) : S → S ∪ {x}, out(x) : S → S \ {x}, test(x) : x ? ∈ S. We define the protocol language SA-PROT consistently with [13,14], so the elements of alphabets below are individual symbols while they are words in [13,14].…”

“…So, our general approach comes from the generalization of the technique that was developed for SAs. One can also find in [15] more technically detailed proofs. Lemma 12.…”