String-Rewriting Systems

Book, Ronald V.; Otto, Friedrich

doi:10.1007/978-1-4613-9771-7_3

Cited by 97 publications

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“…More details and references concerning the material in this section can be found in [7]. In the following, let Γ be always a finite alphabet of symbols.…”

Section: Monoids and Word Problemsmentioning

confidence: 99%

“…s → R t and s → R u) there exists w ∈ Γ * with t * → R w and u * → R w. If R is terminating, then by Newman's lemma R is confluent if and only if R is locally confluent. Using critical pairs [7] which result from overlapping left-hand sides of R, local confluence is decidable for finite terminating semi-Thue systems. The system R is length-reducing if |s| > |t| for all (s, t) ∈ R, where |w| is the length of a word w. The system R is called length-lexicographic if there exists a linear order on the alphabet Γ such that for every rule (s, t) ∈ R either |s| > |t| or (|s| = |t| and there are u, v, w ∈ Γ * and a, b ∈ Γ such that s = uav, t = ubw, and a b).…”

Section: Introductionmentioning

confidence: 99%

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Decidability and Complexity in Automatic Monoids

Lohrey

2004

Developments in Language Theory

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Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [8].Keywords: automatic monoids, hyperbolic groups, word problems, Cayley-graphs, complexity, decidability IntroductionAutomatic groups attracted a lot of attention in combinatorial group theory during the last 15 years, see e.g. the textbook [15]. Roughly speaking, a finitely generated group G, generated by the finite set Γ, is automatic, if the elements of G can be represented by words from a regular language over Γ, and the multiplication with a generator can be recognized by a synchronized 2-tape automaton. This concept easily yields a quadratic time algorithm for the word problem of an automatic group.It is straight forward to extend the definition of an automatic group to the monoid case; this leads to the class of automatic monoids, see e.g. [10,18,22,36]. In the present paper, we study the complexity and decidability of basic algorithmic questions in automatic monoids. In Section 4 we consider the complexity of the word problem for automatic monoids. Analogously to the group case, it is easy to show that for every automatic monoid the word problem can be solved in quadratic time. Here, we prove that there exists a fixed automatic monoid with a P-complete word problem. Thus, unless P = NC, where NC is the class of all problems that can be solved in polylogarithmic time using a polynomial amount of hardware, there exist * This work was partly done while the author was at RWTH Aachen, Germany. 1 automatic monoids for which the word problem cannot be efficiently parallelized. Whether there exists an automatic group with a P-complete word problem was asked for the first time by Cai [8]. This problem remains open.An important subclass of the class of automatic groups is the class of hyperbolic groups, which are defined via a geometric hyperbolicity condition on the Cayleygraph [16]. In [8], Cai has shown that for every hyperbolic group the word problem belongs to the parallel complexity class NC 2 . Cai also asked, whether the upper bound of NC 2 can be improved. Using known results from formal language theory, we show in Section 4 that the word problem for every hyperbolic group belongs to the complexity class LOGCFL ⊆ NC 2 . LOGCFL is the class of all problems that are logspace reducible to a context-free language [44]. We also present a class of automatic monoids, namely monoids that can be presented by finite, terminating, confluent, and left-basic semi-Thue systems [41], for which the complexity of the word problem captures the class LOGDCFL (the logspace closure of the deterministic contex...

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“…More details and references concerning the material in this section can be found in [7]. In the following, let Γ be always a finite alphabet of symbols.…”

Section: Monoids and Word Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Decidability and Complexity in Automatic Monoids

Lohrey

2004

Developments in Language Theory

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“…A rewriting system S is confluent if and only if x * ⇐⇒ S y implies ∃ w : x * =⇒ S w and y * =⇒ S w (see [9,29]). Thus, if S is confluent and teminating, then in every class of Σ * /S there is exactly one element to which no rule of S can be applied.…”

Section: Preliminariesmentioning

confidence: 99%

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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“…First of all we have to give some definitions and results concerning rewriting systems (see e.g. [5]). …”

Section: Rewriting System For Idempotent Semigroupsmentioning

confidence: 99%