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The word problem of ℤ n is a multiple context-free language

The word problem of ℤ n is a multiple context-free language AbstractThe word problem of a group G=〈Σ〉{G=\langle\Sigma\rangle}can be defined as the set of formal words in Σ*{\Sigma^{*}}that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of ℤ2{\mathbb{Z}^{2}}is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of ℤn{\mathbb{Z}^{n}}is a multiple context-free language for any n. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

The word problem of ℤ n is a multiple context-free language

Ho, Meng-Che
Groups Complexity Cryptology , Volume 10 (1): 7 – May 1, 2018
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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2018-0003
Publisher site
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Abstract

AbstractThe word problem of a group G=〈Σ〉{G=\langle\Sigma\rangle}can be defined as the set of formal words in Σ*{\Sigma^{*}}that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of ℤ2{\mathbb{Z}^{2}}is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of ℤn{\mathbb{Z}^{n}}is a multiple context-free language for any n.

Journal

Groups Complexity Cryptology – de Gruyter

Published: May 1, 2018

References

  • Three models for the description of language