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International Workshop on Descriptional Complexity of Formal Systems

DCFS 2015: Descriptional Complexity of Formal Systems pp 243–255Cite as

Groups Whose Word Problem is a Petri Net Language

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9118))

Abstract

There has been considerable interest in exploring the connections between the word problem of a finitely generated group as a formal language and the algebraic structure of the group. However, there are few complete characterizations that tell us precisely which groups have their word problem in a specified class of languages. We investigate which finitely generated groups have their word problem equal to a language accepted by a Petri net and give a complete classification, showing that a group has such a word problem if and only if it is virtually abelian.

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Notes

  1. 1.

    The one-counter languages are those languages accepted by a pushdown automaton where we have a single stack symbol apart from a bottom marker.

  2. 2.

    Recall that the \(t_i\) are actual transitions, not labels of transitions (i.e., generators); therefore this argument does not imply that \(G\) is abelian as, for example, being able to swap labels \(a\) and \(b\) in one such sequence does not mean that we would necessarily be able to do so in all such sequences.

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Acknowledgments

Some of the research for this paper was done whilst the authors were visiting the Nesin Mathematics Village in Turkey; the authors would like to thank the Village both for the financial support that enabled them to work there and for the wonderful research environment it provided that stimulated the results presented here. The authors would like to thank the referees for their helpful and constructive comments. The second author also would like to thank Hilary Craig for all her help and encouragement.

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Authors and Affiliations

  1. Department of Computer Science, University of Leicester, Leicester, LE1 7RH, UK

    Gabriela Aslı Rino Nesin & Richard M. Thomas

Authors
  1. Gabriela Aslı Rino Nesin
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  2. Richard M. Thomas
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Corresponding author

Correspondence to Richard M. Thomas .

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Editors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    Jeffrey Shallit

  2. Department of Mathematics, University of Turku, Turku, Finland

    Alexander Okhotin

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Rino Nesin, G.A., Thomas, R.M. (2015). Groups Whose Word Problem is a Petri Net Language. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_21

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  • DOI: https://doi.org/10.1007/978-3-319-19225-3_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19224-6

  • Online ISBN: 978-3-319-19225-3

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