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Globally deterministic CD-systems of stateless R-automata with window size 1

Benedek Nagy, Friedrich Otto

2013
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International Journal of Computer Mathematics
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It is known that cooperating distributed systems (CD-systems) of stateless deterministic restarting automata with window size 1 accept a class of semi-linear languages that properly includes all rational trace languages. Although the component automata of such a CD-system are all deterministic, the CD-system itself is not. Here we study CD-systems of stateless deterministic restarting automata with window size 1 that are themselves completely deterministic. In fact, we consider two such types
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... CD-systems, the strictly deterministic systems and the globally deterministic systems. International Journal of Computer Mathematics GlobalDetrevision1 On Globally Deterministic CD-Systems of Stateless R-Automata 3 as expressive as the CD-systems of stl-det-R(1)-automata considered in [11], as they do not accept all rational trace languages. This paper is structured as follows. In Section 2 we give the definition of the stl-det-R(1)-automaton and of the stl-det-local-CD-R(1)-system from [11], and we restate some of the main results on these systems. In Section 3 we define the strictly deterministic CD-systems of stl-det-R(1)-automata, and we show that they have a rather weak expressive power. In addition, we prove that the class of languages accepted by these systems is an anti-AFL that is not even closed under reversal; however, this language class is closed under complementation. Then in Section 4, we define the main notion of this paper, the globally deterministic CD-system of stldet-R(1)-automata (stl-det-global-CD-R(1)-system, for short). We show that these systems accept all regular languages, we present a normal form result for them, and we prove that they are not sufficiently expressive to accept all rational trace languages. Thus, they are strictly less expressive than the locally deterministic systems of [11] . Also we show that the class of languages accepted by the globally deterministic CD-systems of stl-det-R(1)-automata is closed under complementation, but that it is not closed under union, intersection with regular languages, product, Kleene star, reversal, or commutation. Thus, with respect to closure properties these systems are much weaker than the locally deterministic systems. Finally we turn to decision problems for stl-det-global-CD-R(1)-systems in Section 5. While the decidability of the membership, emptiness, and finiteness problems follows immediately from the corresponding results for stl-det-local-CD-R(1)-systems, the closure under complementation implies that also the universe problem is decidable for stl-det-global-CD-R(1)-systems. This is an important contrast to the situation for stl-det-local-CD-R(1)-systems, where the regularity, inclusion, and equivalence problems are shown to be undecidable by a reduction from the universe problem. Here we present a reduction from the Post Correspondence Problem to show that the inclusion problem is still undecidable for stl-det-global-CD-R(1)-systems. The paper closes with a short summary and some open problems in Section 6.

doi:10.1080/00207160.2012.688820
fatcat:mtwzfllhfjc3vjqo7uqg7jnuca