Solution sets for equations over free groups are EDT0L languages

Ciobanu, L; Diekert, V; Elder, M

Solution sets for equations over free groups are EDT0L languages

Ciobanu, L Diekert, V Elder, M

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Publication Type:: Conference Proceeding
Citation:: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2015, 9135 pp. 134 - 145
Issue Date:: 2015-01-01

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Field	Value	Language
dc.contributor.author	Ciobanu, L	en_US
dc.contributor.author	Diekert, V	en_US
dc.contributor.author	Elder, M https://orcid.org/0000-0002-2438-3945	en_US
dc.date.issued	2015-01-01	en_US
dc.identifier.citation	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2015, 9135 pp. 134 - 145	en_US
dc.identifier.isbn	9783662476659	en_US
dc.identifier.issn	0302-9743	en_US
dc.identifier.uri	http://hdl.handle.net/10453/101378
dc.description.abstract	© Springer-Verlag Berlin Heidelberg 2015. We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been open for some years [9,10], apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products ⋆1≤i≤sFi, where each Fi is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by [8], which is a very restricted case. Our general result became possible due to the recent recompression technique of Jeż. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper [5]. For example, we improve the complexity from quadratic nondeterministic space in [5] to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE(n log n). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.	en_US
dc.relation.ispartof	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)	en_US
dc.relation.isbasedon	10.1007/978-3-662-47666-6_11	en_US
dc.subject.classification	Artificial Intelligence & Image Processing	en_US
dc.title	Solution sets for equations over free groups are EDT0L languages	en_US
dc.type	Conference Proceeding
utslib.citation.volume	9135	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Science
pubs.organisational-group	/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences
utslib.copyright.status	closed_access
pubs.publication-status	Published	en_US
pubs.volume	9135	en_US

Abstract:

© Springer-Verlag Berlin Heidelberg 2015. We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been open for some years [9,10], apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products ⋆1≤i≤sFi, where each Fi is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by [8], which is a very restricted case. Our general result became possible due to the recent recompression technique of Jeż. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper [5]. For example, we improve the complexity from quadratic nondeterministic space in [5] to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE(n log n). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.

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http://hdl.handle.net/10453/97307