Set Automata

Abstract

We consider the model of deterministic set automata which are basically deterministic finite automata equipped with a set as an additional storage medium. The basic operations on the set are the insertion of elements, the removing of elements, and the test whether an element is in the set. We investigate the computational power of deterministic set automata and compare the language class accepted with the context-free languages and classes of languages accepted by queue automata. As result the incomparability to all classes considered is obtained. Furthermore, we examine the closure properties under several operations. Then we show that deterministic set automata may be an interesting model from a practical point of view by proving that their regularity problem as well as the problems of emptiness, finiteness, infiniteness, and universality are decidable. Finally, the descriptional complexity of deterministic and nondeterministic set automata is investigated. A conversion procedure that turns a deterministic set automaton accepting a regular language into a deterministic finite automaton is developed which leads to a double exponential upper bound. This bound is proved to be tight in the order of magnitude by presenting also a double exponential lower bound. In contrast to these recursive bounds we obtain non-recursive trade-offs when nondeterministic set automata are considered.

On Computational Complexity of Set Automata

“…Set automata were introduced by M. Kutrib, A. Malcher, M. Wendlandt in 2014 in [3] and [4]. The results of these conference papers are covered by the journal paper [5], so in the sequel we give references to the journal variant. DSA CFL DCFL…”

“…We recall briefly results from [5] about structural and decidability properties of DSA. They are presented in the tables, see Fig.…”

“…The main technical result of the paper is SA ⊆ NP (see Section 5). This result is based on the fact that class SA is a rational cone and on our improvement of the technique of normal forms described in [5].…”

“…⊓ ⊔ One can naturally assume that DSA class contains DCFL (or even CFL), but this assumption is false. As was shown in [5], the language {w#w R | w ∈ Σ * , |Σ| 2} is not recognizable by any DSA.Theorem 2 ([5]). The classes DCFL and DSA are incomparable.The undecidability results for NSA are based on the fact that NSA can accept the set of invalid computations of a Turing machine.…”

On store languages and applications

“…The following known result is needed as well. The proof, which uses the ideas in [38], was given recently in [39]. The result can also be shown using Kolmogorov complexity techniques [40].…”