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Some Undecidability Results For Finitely Generated Thue Congruences On A TwoLetter Alphabet
 Fundamenta Informaticae
, 1996
"... Following the course set by A. Markov (1951), S. Adjan (1958), and M. Rabin (1958), C. ' O'D'unlaing (1983) has shown that certain properties of finitely generated Thue congruences are undecidable in general. Here we prove that many of these undecidability results remain valid even when only finitel ..."
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Cited by 3 (3 self)
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Following the course set by A. Markov (1951), S. Adjan (1958), and M. Rabin (1958), C. ' O'D'unlaing (1983) has shown that certain properties of finitely generated Thue congruences are undecidable in general. Here we prove that many of these undecidability results remain valid even when only finitely generated Thue congruences on a fixed twoletter alphabet \Sigma 2 are considered. In contrast to a construction given by P. Schupp (1976) which applies to groups only, we use a modified version of a technical lemma from A. Markov's original paper. Based on this technical result we can carry the result of A. SattlerKlein (1996), which says that certain Markov properties remain undecidable even when they are restricted to finitely generated Thue congruences that are decidable, over to the alphabet \Sigma 2 . 1 Introduction A stringrewriting system R on some alphabet \Sigma is a set of pairs of strings over \Sigma. It induces a congruence $ R on \Sigma , the Thue congruence generat...
Some Decision Problems Related To The Regularity Of Monoids
 Semigroups, Automata and Languages
, 1994
"... The problem of deciding whether a monoid that is given through a finite presentation is a regular monoid as well as two closely related decision problems are considered. Since these problems are undecidable in general, they are investigated for certain restricted classes of finite presentations. Som ..."
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Cited by 1 (1 self)
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The problem of deciding whether a monoid that is given through a finite presentation is a regular monoid as well as two closely related decision problems are considered. Since these problems are undecidable in general, they are investigated for certain restricted classes of finite presentations. Some new decidability results as well as some new undecidability results are obtained, which show that for two of the problems considered, only a small margin separates the decidable from the undecidable cases. 1 Introduction In the present paper we deal with the general problem of obtaining information on the algebraic structure of monoids from finite monoidpresentations of the form (\Sigma; R). Here \Sigma is a finite alphabet (set of generators), and R is a finite stringrewriting system on \Sigma (set of defining relations). The monoid MR that is given through the presentation (\Sigma; R) is the factor monoid \Sigma = $ R of the free monoid \Sigma generated by \Sigma modulo the...
Uniform decision problems for certain restricted classes of finite monoidpresentations  A survey on recent undecidability results
, 1997
"... It is wellknown that many decision problems are undecidable in general for the class of all finite monoidpresentations. Here we consider the question of which of these problems remain undecidable even for very restricted classes of finite monoidpresentations. The restrictions we are interested in ..."
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It is wellknown that many decision problems are undecidable in general for the class of all finite monoidpresentations. Here we consider the question of which of these problems remain undecidable even for very restricted classes of finite monoidpresentations. The restrictions we are interested in concern the number of generators admitted and the condition that each finite monoidpresentation considered has a decidable word problem. As we will see many of these problems remain undecidable even for these restricted classes of finite monoidpresentations. This follows from recent results on the behavior of the KnuthBendix completion procedure (SattlerKlein 1996) and a refined version of Markov's embedding lemma (Madlener and Otto 1997). 1 Introduction In recent years the effort to actually perform computations in algebraic structures has grown considerably. For example, the algorithmical aspects of the theory of polynomial rings are getting a lot of attention as a consequence of the...
A Survey on the Computational Power of Some Classes of Finite MonoidPresentations
, 1998
"... An overview is given on the decidability results that have been obtained for the various classes of finite monoidpresentations that involve stringrewriting systems which are noetherian and (weakly) confluent. Further, the KnuthBendix completion procedure and some of its extensions are described. ..."
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An overview is given on the decidability results that have been obtained for the various classes of finite monoidpresentations that involve stringrewriting systems which are noetherian and (weakly) confluent. Further, the KnuthBendix completion procedure and some of its extensions are described. Finally, a current research project supported by the Deutsche Forschungsgemeinschaft (DFG) is presented that aims at the development of a software system XSSR that realizes the various decidability results presented in the paper. 1 Introduction One of the forerunners of modern computational algebra is certainly its subfield of computational group theory, which today is being extended to computational semigroup theory. Not just results on the algebraic structure of a (semi) group are of interest any more, but emphasis is being placed on actually solving algorithmic problems for (semi) groups, that is, one is interested in effectively performing computations with the elements of a (semi) ...
ELEMENTS OF FINITE ORDER FOR FINITE MONADIC CHURCHROSSER TRUE SYSTEMS BY
, 1985
"... ABSTRACT. A Thue system T over ~ is said to allow nontrivial elements of finite order, if there exist a word u E ~ * and integers n;;. 0 and k;;. 1 such that u.... fA and u,,+k.... f u". Here the following decision problem is shown to be decidable: Instance. A finite, monadic, ChurchRosser Thue sys ..."
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ABSTRACT. A Thue system T over ~ is said to allow nontrivial elements of finite order, if there exist a word u E ~ * and integers n;;. 0 and k;;. 1 such that u.... fA and u,,+k.... f u". Here the following decision problem is shown to be decidable: Instance. A finite, monadic, ChurchRosser Thue system Tover~. Question. Does T allow nontrivial elements of finite order? By a result of Muller and Schupp this implies in particular that given a finite monadic ChurchRosser Thue system T it is decidable whether the monoid presented by T is a free group or not. Introduction. Thue systems are string rewriting systems often studied in computability theory, combinatorial (semi) group theory, and formal language theory. They are used to present monoids and groups [13, 15], and to specify languages as unions of congruence classes, leading to the class of "congruentiallanguages " [3, 9, 19]. Thue systems that satisfy the ChurchRosser property [3, 4, 9] are of special