Groups, the Theory of ends, and context-free languages

Muller, David E.; Schupp, Paul E.

doi:10.1016/0022-0000(83)90003-x

Cited by 222 publications

(220 citation statements)

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“…A celebrated theorem of Muller and Schupp [24] (which relies on Dunwoody's Accessibility Theorem [14]) states that, given a finite generating set for a group G, the set of words representing the identity in G is a context-free language if and only if G contains a free subgroup of finite index. We need a special case of (the easier direction of) this result: Proof.…”

Section: Grammarsmentioning

confidence: 99%

Combings of groups and the grammar of reparameterization

Bridson

2003

Commentarii Mathematici Helvetici

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Abstract.A new construction of combings is used to distinguish between several previously indistinguishable classes of groups associated to the theory of automatic groups and non-positive curvature in group theory. We construct synchronously bounded combings for a class of groups that are neither bicombable nor automatic. The linguistic complexity of these combings is analysed: in many cases the language of words in the combing is an indexed language.

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Section: Grammarsmentioning

confidence: 99%

Combings of groups and the grammar of reparameterization

Bridson

2003

Commentarii Mathematici Helvetici

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“…In Section 5 we study Cayley-graphs of automatic monoids. The Cayley-graph of a finitely generated monoid M with respect to a finite generating set Γ is a Γ-labeled directed graph with node set M and an a-labeled edge from a node x to a node y if y = xa in M. Cayley-graphs of groups are a fundamental tool in combinatorial group theory [32] and serve as a link to other fields like topology, graph theory, and automata theory, see, e.g., [34,35]. Results on the geometric structure of Cayley-graphs of automatic monoids can be found in [42,43].…”

Section: Introductionmentioning

confidence: 99%

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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“…Context-free graphs were introduced in the seminal papers [110,111,112] of Muller and Schupp. There are several equivalent definitions.…”

Section: From Context-free Graphs To Prefix-recognizable Structuresmentioning

confidence: 99%

“…This is independent of the choice of S. Moreover, a group is context-free if and only if its Cayley graph for some (and hence all) sets S of semigroup generators is a context-free graph. Finally, a finitely generated group is context-free if and only if it is virtually free, that is, if it has a free subgroup of finite index [111]. 2 Muller and Schupp have further shown that context-free graphs have a decidable MSO-theory.…”

Section: From Context-free Graphs To Prefix-recognizable Structuresmentioning

confidence: 99%

Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

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The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin's Theorem and the ImmermanVardi Theorem (see [78, Chapter 3] for a survey).However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones.There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic. Indeed, one of the central issues in algorithmic model theory is the effective evaluation of logical formulae, from a suitable logic such as, for instance, first-order logic (FO), monadic second-order logic (MSO), or a fixed point logic like LFP or the modal µ-calculus. But on computable structures, only the quantifier-free formulae generally admit effective evaluation, and already the existential fragment of first-order logic is undecidable, for instance on the computable structure (N, +, · ).This leads us to the central requirement that for a suitable logic L (depending on the intended application) the model-checking problem for the class C of finitely presented structures should be algorithmically solvable. At the very 1

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Groups, the Theory of ends, and context-free languages

Cited by 222 publications

References 3 publications

Combings of groups and the grammar of reparameterization

Combings of groups and the grammar of reparameterization

Decidability and Complexity in Automatic Monoids

Automata-based presentations of infinite structures

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