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Series Parallel Digraphs with Loops

Graphs Encoded by Regular Expression

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Abstract

In the conversion of finite automata to regular expressions, an exponential blowup in size can generally not be avoided. This is due to graph-structural properties of automata which cannot be directly encoded by regular expressions and cause the blowup combinatorially. In order to identify these structures, we generalize the class of arc-series-parallel digraphs beyond the acyclic case. The resulting digraphs are shown to be reversibly encoded by linear-sized regular expressions. Also, a characterization of this new class by a set of seven forbidden substructures is given. Automata that require expressions of superlinear size must contain some of these substructures.

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Acknowledgements

The author would like to express his gratitude to the anonymous referees, whose comments helped to improve this article considerably.

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  1. FB IV—Informatik, Universität Trier, 54286, Trier, Germany

    Stefan Gulan

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  1. Stefan Gulan
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Correspondence to Stefan Gulan.

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Gulan, S. Series Parallel Digraphs with Loops. Theory Comput Syst 53, 126–158 (2013). https://doi.org/10.1007/s00224-012-9409-0

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