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46
Finite Presentations of Infinite Structures: Automata and Interpretations
 Theory of Computing Systems
, 2002
"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations. ..."
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Cited by 41 (3 self)
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We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations.
PrefixRecognisable Graphs and Monadic SecondOrder Logic
, 2001
"... We present several characterisations of the class of prefixrecognisable ..."
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Cited by 20 (1 self)
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We present several characterisations of the class of prefixrecognisable
On the rational subset problem for groups
 Journal of Algebra
"... We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensi ..."
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Cited by 13 (9 self)
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We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for Gautomaton subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership. © 2006 Elsevier Inc. All rights reserved.
Growth and ergodicity of contextfree languages
 Trans. Amer. Math. Soc
, 2002
"... Abstract. A language L over a finite alphabet Σ is called growthsensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every ergodic unambiguous, nonlinear contextfree language is growthsensitive. “Ergodic ” ..."
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Cited by 11 (7 self)
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Abstract. A language L over a finite alphabet Σ is called growthsensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every ergodic unambiguous, nonlinear contextfree language is growthsensitive. “Ergodic ” means for a contextfree grammar and language that its dependency digraph is strongly connected. The same result as above holds for the larger class of essentially ergodic contextfree languages, and if growth is considered with respect to the ambiguity degrees, then the assumption of unambiguity may be dropped. The methods combine a construction of grammars for 2block languages with a generating function technique regarding systems of algebraic equations. 1. Introduction and
Multiplicative measures on free groups
 INT. J. ALGEBRA COMP
, 2002
"... 1 How one can measure subsets in the free group? 1.1 Motivation The present paper is motivated by needs of practical computations in finitely presented groups. In particular, we wish to develop tools which can be used in the analysis of the “practical ” complexity of algorithmic problems for discret ..."
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Cited by 10 (3 self)
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1 How one can measure subsets in the free group? 1.1 Motivation The present paper is motivated by needs of practical computations in finitely presented groups. In particular, we wish to develop tools which can be used in the analysis of the “practical ” complexity of algorithmic problems for discrete infinite groups, as well as in the analysis of the behaviour of heuristic (e.g. genetic) algorithms for infinite groups [22, 23]. In most computerbased computations in finitely presented groups G = F/R the elements are represented as freely reduced words in the free group F, with procedures for comparing their images in the factor group G = F/R. Therefore the ambient algebraic structure in all our considerations is the free group F = F(X) on a finite set X = {x1,...,xm}. We identify F with the set of all freely reduced words in the alphabet X ∪X −1, with the multiplication given by concatenation of words with the subsequent free reduction. The most natural and convenient way to generate pseudorandom elements
Counting paths in graphs
 Ensignment Math
, 1999
"... Abstract. We give a simple combinatorial proof of a formula that extends a result by Grigorchuk [Gri78a, Gri78b] (rediscovered by Cohen [Coh82]) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of ‘bumps ’ on paths in a graph: in a ..."
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Cited by 8 (2 self)
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Abstract. We give a simple combinatorial proof of a formula that extends a result by Grigorchuk [Gri78a, Gri78b] (rediscovered by Cohen [Coh82]) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of ‘bumps ’ on paths in a graph: in a dregular (not necessarily transitive) nonoriented graph let the series G(t) count all paths between two fixed points weighted by their length tlength, and F(u, t) count the same paths, weighted as unumber of bumpstlength. Then one has
Axiomatising Treeinterpretable Structures
 IN PROC. 19TH INT. SYMP. ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, LNCS 2285, 2002
, 2001
"... We introduce the class of treeinterpretable structures which generalises the notion of a prefixrecognisable graph to arbitrary relational structures. We prove that every treeinterpretable structure is finitely axiomatisable in guarded secondorder logic with cardinality quantifiers. ..."
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Cited by 8 (0 self)
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We introduce the class of treeinterpretable structures which generalises the notion of a prefixrecognisable graph to arbitrary relational structures. We prove that every treeinterpretable structure is finitely axiomatisable in guarded secondorder logic with cardinality quantifiers.
Groups and simple languages
 Trans. Amer. Math. Soc
, 1983
"... Abstract. With any finitely generated group presentation, one can associate a formal language (called the reduced word problem) consisting of those words on the generators and their inverses which are equal to the identity but which have no proper prefix equal to the identity. We show that the reduc ..."
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Cited by 6 (0 self)
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Abstract. With any finitely generated group presentation, one can associate a formal language (called the reduced word problem) consisting of those words on the generators and their inverses which are equal to the identity but which have no proper prefix equal to the identity. We show that the reduced word problem is a simple language if and only if each vertex of the presentation's Cayley diagram has only a finite number of simple closed paths passing through it. Furthermore, if the reduced word problem is simple, then the group is a free product of a free group of finite rank and a finite number of finite groups. Let tr = ( X; R) be a finitely generated (f.g.) presentation of a group G, and let 2 = X U X'x be the set of generators and their inverses. Define the word problem of 77, denoted by WP(7r), to be the set of all words on 2 which are equal to the identity element of G. Let the reduced word problem of 77, denoted by WP0(7r), be the subset of WP(7t) consisting of those words having no proper prefix equal to the identity. The general problem arises of determining the relationship between the properties of the group G and those of the formal languages WP(7r) and WP0(w).
Parallel algorithms for group word problems. Doctoral Dissertation
, 1993
"... quality and form for publication on microfilm: ..."
On groups whose word problem is solved by a nested stack automaton. arXiv:math.GR/9812028
, 1998
"... Abstract. Accessible groups whose word problems are accepted by a deterministic nested stack automaton with limited erasing are virtually free. 1. Introduction. During the past several years combinatorial group theory has received an infusion of ideas both from topology and from the theory of formal ..."
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Cited by 5 (0 self)
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Abstract. Accessible groups whose word problems are accepted by a deterministic nested stack automaton with limited erasing are virtually free. 1. Introduction. During the past several years combinatorial group theory has received an infusion of ideas both from topology and from the theory of formal languages. The resulting interplay between groups, the geometry of their Cayley diagrams, and associated formal languages has led to several developments including