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Logical Aspects of CayleyGraphs: The Group Case
 TO APPEAR IN ANNALS OF PURE AND APPLIED LOGIC
"... We prove that a finitely generated group is contextfree whenever its Cayleygraph has a decidable monadic secondorder theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of contextfree groups and also proves a conjecture of Schupp. To derive this re ..."
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Cited by 5 (3 self)
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We prove that a finitely generated group is contextfree whenever its Cayleygraph has a decidable monadic secondorder theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of contextfree groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is contextfree whenever its monadic secondorder theory is decidable. Further, it is shown that the word problem of a finitely generated group is decidable if and only if the firstorder theory of its Cayleygraph is decidable.
Decidable Theories of Cayleygraphs
 PROCEEDINGS OF THE 20TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS 2003), BERLIN (GERMANY), NUMBER 2607 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2003
"... We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSOtheory if and only if it is contextfree. This implies that a group is contextfree if and only if its Cayleygraph has a decidable MSOtheory. On the other hand, the rstorder theory of the Cayl ..."
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Cited by 2 (2 self)
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We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSOtheory if and only if it is contextfree. This implies that a group is contextfree if and only if its Cayleygraph has a decidable MSOtheory. On the other hand, the rstorder theory of the Cayleygraph of a group is decidable if and only if the group has a decidable word problem. For Cayleygraphs of monoids we prove the following closure properties. The class of monoids whose Cayleygraphs have decidable MSOtheories is closed under free products. The class of monoids whose Cayleygraphs have decidable firstorder theories is closed under general graph products. For the latter result on firstorder theories we introduce a new unfolding construction, the factorized unfolding, that generalizes the treelike structures considered by Walukiewicz. We show and use that it preserves the decidability of the firstorder theory. Most of
Automatabased presentations of infinite structures
, 2009
"... The model theory of finite structures is intimately connected to various fields ..."
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The model theory of finite structures is intimately connected to various fields
Computational and logical aspects of
"... The present work constitutes the Habilitationsschrift of the author written at the University of Stuttgart. It contains a treatise of several computational and logical aspects of infinite monoids. The first chapter is devoted to the word problem for finitely generated monoids. In particular, the rel ..."
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The present work constitutes the Habilitationsschrift of the author written at the University of Stuttgart. It contains a treatise of several computational and logical aspects of infinite monoids. The first chapter is devoted to the word problem for finitely generated monoids. In particular, the relationship between the the computational complexity of the word problem and the syntactical properties of monoid presentations is investigated. The second chapter studies Cayleygraphs of finitely generated monoids under a logical point of view. Cayleygraphs of groups play an important role in combinatorial group theory. We will study firstorder and monadic secondorder theories of Cayleygraphs for both groups and monoids. The third chapter deals with word equations over monoids. Using the graph product operation, which generalizes both the free and the direct product, we generalize the seminal decidability results of Makanin on free monoids and groups to larger classes of monoids. Acknowledgments. It is a great pleasure for me to thank Professor V. Diekert for his enthusiastic and inspiring support over the last six years that I spent in his research group at the University of Stuttgart. Many ideas in this Habilitationsschrift have their origin in joint research efforts with Professor V. Diekert. I am grateful to Professor Volker Claus (University of Stuttgart), Professor Erich Gr&quot;adel (RWTH Aachen), and Professor G'eraud S'enizergues (Universit'e Bordeaux I Nouvelle) who were further referees of this Habilitationsschrift. I am also greatly indebted to Dr. Dietrich Kuske for a fruitful research collaboration during the last few years. His insights had substantial influence on the content of this work. Thanks to all members of the group Theoretische Grundlagen der Informatik at the University of Stuttgart for many fruitful discussions.