Word Problems Solvable in Logspace

Lipton, Richard J.; Zalcstein, Yechezkel

doi:10.1145/322017.322031

Cited by 119 publications

(69 citation statements)

References 8 publications

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“…Therefore, in order to prove Theorem E it remains only to find a generating set for SL 1 .Z/ with respect to which one can solve the word problem in polynomial time (cf. [28]). It is well known that SL 1 .Z/ is generated by fe i;j j i; j 2 N; i ¤ j g, where e i;j is the elementary matrix with .i; j / entry equal to 1.…”

Section: Lemma 74mentioning

confidence: 99%

On the difficulty of presenting finitely presentable groups

Bridson¹,

Wilton²

2011

Groups Geom. Dyn.

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Abstract. We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of its finitely presentable subgroups.

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Section: Lemma 74mentioning

confidence: 99%

On the difficulty of presenting finitely presentable groups

Bridson¹,

Wilton²

2011

Groups Geom. Dyn.

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“…Together with the well-known result that linear groups -in particular F 2 -have word problem in LOGSPACE [35,48], this leads to a LOGSPACE algorithm of the word problem. Moreover, in view of [12], Theorem A shows that the word problem of GBS groups is in the complexity class C = NC 1 (for a definition see [12]).…”

Section: Introductionmentioning

confidence: 86%

“…It was obtained by Lipton and Zalcstein [35] for fields of characteristic 0 and by Simon [48] for other fields.…”

Section: Complexitymentioning

confidence: 95%

A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups

Weiß¹

2016

Algebra and Computer Science

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Baumslag-Solitar groups were introduced in 1962 by Baumslag and Solitar as examples for finitely presented non-Hopfian two-generator groups. Since then, they served as examples for a wide range of purposes. As Baumslag-Solitar groups are HNN extensions, there is a natural generalization in terms of graph of groups.Concerning algorithmic aspects of generalized Baumslag-Solitar groups, several decidability results are known. Indeed, a straightforward application of standard algorithms leads to a polynomial time solution of the word problem (the question whether some word over the generators represents the identity of the group). The conjugacy problem (the question whether two given words represent conjugate group elements) is more complicated; still decidability has been established by Anshel and Stebe for ordinary Baumslag-Solitar groups and for generalized Baumslag-Solitar groups independently by Lockhart and Beeker. However, up to now, no precise complexity estimates have been given.In this work, we give a LOGSPACE algorithm for both problems. More precisely, we describe a uniform TC 0 many-one reduction of the word problem to the word problem of the free group. Then we refine the known techniques for the conjugacy problem and show it is AC 0 -Turing-reducible to the word problem of the free group.Finally, we consider uniform versions (where also the graph of groups is part of the input) of both word and conjugacy problem: while the word problem still is solvable in LOGSPACE, the conjugacy problem becomes EXPSPACE-complete.

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“…These groups and semigroups are algorithmically "tame": the word problem there is decidable in polynomial time and even log-space [34]. The Dehn function is a well-known indicator of complexity of the word problems: the smaller the Dehn function the easier the word problem.…”

Section: The Problem and Previous Approaches For A Solutionmentioning

confidence: 99%

“…In the case of finitely presented linear groups it is well known that the word problem can be solved in deterministic polynomial time [34,61]. This applies to most finitely presented groups (where "most" means "with overwhelming probability" in one of several probabilistic models): recent results of Agol [1] and Ollivier and Wise [45] together with the older result of Olshanskii [46] imply that most finitely presented groups are linear (even over Z).…”

Section: The "Yes" and "No" Parts Of The Mckinsey Algorithmmentioning

confidence: 99%