Recognizability of the support of recognizable series over the semiring of the integers is undecidable

Authors:
D. Kirsten;K. Quaas
Affiliations:
Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany;Institut für Informatik, Universität Leipzig, 04009 Leipzig, Germany
Venue:
Information Processing Letters
Year:
2011

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On the supports of recognizable series over a field and a single letter alphabet

Information Processing Letters

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Abstract

A recognizable series over the semiring of the integers is a function that maps each word over an alphabet to an integer. The support of such a series consists of all those words which are not mapped to zero. It is long known that there are recognizable series whose support is not a recognizable, but a context-free language. However, the problem of deciding whether the support of a given recognizable series is recognizable was never considered. Here we show that this problem is undecidable. The proof relies on an encoding of an instance of Post@?s correspondence problem.