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Word problems of groups: Formal languages, characterizations and decidability

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posted on 20.07.2018, 08:39 by Sam A. M. Jones, Richard M. Thomas
Let G be a group and let φ:Σ ⁎ →G be a monoid homomorphism from the set of all strings Σ ⁎ over some finite alphabet Σ onto the group G. The set Σ is then called a generating set for G and the language {1}φ −1 ⊆Σ ⁎ is called the word problem of G with respect to the generating set Σ (via the homomorphism φ) and is denoted by W(G,Σ). We consider nine conditions that hold in each such language of the form W(G,Σ) and determine which combinations of these conditions are equivalent to the property of the language in question being the word problem of a group. We show that each of these nine conditions is decidable for the family of regular languages but that each is undecidable for the family of one-counter languages (the languages accepted by one-counter pushdown automata). We also show that the property of a language being the word problem of a group is undecidable for the family of one-counter languages but is decidable for the family of deterministic context-free languages (the languages accepted by deterministic pushdown automata).

History

Citation

Theoretical Computer Science, 2018.

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Informatics

Version

AM (Accepted Manuscript)

Published in

Theoretical Computer Science

Publisher

Elsevier

issn

0304-3975

Acceptance date

30/04/2018

Copyright date

2018

Available date

07/05/2019

Publisher version

https://www.sciencedirect.com/science/article/pii/S030439751830313X?via=ihub

Notes

The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.

Language

en

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