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Groups presented by certain classes of finite length-reducing string-rewriting systems [chapter]

Klaus Madlener, Friedrich Otto
1987 Lecture Notes in Computer Science  
This corollary is very important in that it provides ways of proving that a certain group cannot be presented by a finite length-reducing and confluent string-rewriting system.  ...  So we ask which classes of groups can be presented by which classes of string-rewriting systems.  ... 
doi:10.1007/3-540-17220-3_12 fatcat:lciybdavdjckdpqztukuiq7efa

About the descriptive power of certain classes of finite string-rewriting systems

Klaus Madlener, Friedrich Otto
1989 Theoretical Computer Science  
Therefore, it is of interest to determine the descriptive power of these classes, i.e., to find algebraic characterizations for those classes of monoids that can be presented by certain types of finite  ...  Finite string-rewriting systems can be used to present monoids and groups. In general, these finite presentations do not give much information about the monoids and groups presented.  ...  Groups presented by finite weight-reducing string-rewriting systems Let R be a finite, length-reducing, and confluent string-rewriting system on 2.  ... 
doi:10.1016/0304-3975(89)90002-9 fatcat:onterhtqvncyxk34zrcszlhmli

Page 1649 of Mathematical Reviews Vol. , Issue 94c [page]

1994 Mathematical Reviews  
Summary: “For a finite weakly confluent monadic string-rewriting system R presenting a group the set of valid linear sentences is decidable.  ...  Here we show that this is no longer true in general if R is a finite weakly confluent monadic string-rewriting system that does not present a group.  ... 

Page 2401 of Mathematical Reviews Vol. , Issue 90D [page]

1990 Mathematical Reviews  
length-reducing and complete string-rewriting systems can present only context-free groups [K.  ...  Finally, it is shown that no finite length-reducing and complete string rewriting system can present the group G;, which is an easy consequence of the result by Muller and Schupp and the fact that finite  ... 

On weakly confluent monadic string-rewriting systems

K. Madlener, P. Narendran, F. Otto, L. Zhang
1993 Theoretical Computer Science  
It is investigated as to how far the various decidability results for finite, monadic, and confluent string-rewriting systems can be carried over to the class of finite monadic string-rewriting systems  ...  On the other hand, for finite, monadic, and weakly confluent systems that present groups, the validation problem for linear sentences is decidable.  ...  On the other hand, they have all been solved for certain restricted classes of finite string-rewriting systems.  ... 
doi:10.1016/0304-3975(93)90213-d fatcat:ba7rvt2rb5fepptcymu4ql6ln4

Thue systems as rewriting systems

Ronald V. Book
1987 Journal of symbolic computation  
This paper is a survey of recent results on Thue systems, where the systems are viewed as rewriting systems on strings over a finite alphabet.  ...  Since Thue systems may also be considered as presentations of monoids, the language of monoids (and groups) is used to describe some of these properties.  ...  problem that cannot be presented by any finite complete rewriting system.  ... 
doi:10.1016/s0747-7171(87)80021-4 fatcat:pqx6qbrwmngcdfsj6mabwfrxce

Contributions of Ronald V. Book to the theory of string-rewriting systems

Robert McNaughton
1998 Theoretical Computer Science  
Part of our appreciation of the impact that he had on the field of rewriting systems was what these students and post-dots went on to do after they left Santa Barbara.  ...  He was, in effect, the leader of a group that included all or most of these.  ...  Thue systems [15] and semi-Thue systems These are the two basic abstract concepts used in the study of string rewriting, and are presented here briefly.  ... 
doi:10.1016/s0304-3975(98)00053-x fatcat:wgpelyippzbfbo2fkyqzw3yb4y

When is an extension of a specification consistent? Decidable and undecidable cases

Friedrich Otto
1991 Journal of symbolic computation  
Question: Is (E l , El) a consistent extension of (:Z, E)? For the special ease of string-rewriting systems we present decidable and undeeidable eases of this problem.  ...  ) that are presented by (E.  ...  The following problem is decidable: Instance: A finite string-rewriting system R1 on an alphabet £1 such that the monoid • /~i presented by (£1; RI) is a group, and a finite length-reducing string. rewriting  ... 
doi:10.1016/s0747-7171(08)80150-2 fatcat:nkdhm6qvg5b4raop56t755nzpu

Some undecidability results concerning the property of preserving regularity

Friedrich Otto
1998 Theoretical Computer Science  
In addition, some undecidability results are presented that generalize results of Gilleron and Tison (1995) from term-rewriting systems to string-rewriting systems.  ...  A finite string-rewriting system R preserves regularity if and only if it preserves Z-regularity, where C is the alphabet containing exactly those letters that have occurrences in the rules of R.  ...  Acknowledgements The author wants to thank Klaus Madlener and Paliath Narendran for very fruitful discussions regarding the result presented in Section 6 and for pointing out the papers [15, 17] .  ... 
doi:10.1016/s0304-3975(98)00055-3 fatcat:ighxqdcmazhzbm4v7pxenpq7wu

On deciding the confluence of a finite string-rewriting system on a given congruence class

Friedrich Otto
1987 Journal of computer and system sciences (Print)  
In general it is undecidable whether or not a given finite string-rewriting system R is confluent on a given congruence class [w]~, even when only length-reducing systems are considered.  ...  A string-rewriting system R on alphabet C induces a congruence ++ 2 on the free monoid 27 generated by Z, and hence, the set M, of congruence classes modulo -2 is a monoid.  ...  Madlener for many fruitful discussions regarding the results presented in this paper.  ... 
doi:10.1016/0022-0000(87)90017-1 fatcat:uuahdoiqerd7lmkymitfrgngd4

Geodesic Rewriting Systems and Pregroups [chapter]

Volker Diekert, Andrew J. Duncan, Alexei G. Myasnikov
2010 Combinatorial and Geometric Group Theory  
We show that these are well-behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations.  ...  In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist.  ...  Finite convergent rewriting systems for certain classes of Coxeter groups have been found by Hermiller [28] (see also [19, 8] HNN-extensions Let G be any group with isomorphic subgroups A and B.  ... 
doi:10.1007/978-3-7643-9911-5_3 fatcat:2kt4ma2ztbfbzmkiiqiwff5tta

Infinite Convergent String-rewriting Systems and Cross-sections for Finitely Presented Monoids

1998 Journal of symbolic computation  
string-rewriting systems.  ...  A finitely presented monoid has a decidable word problem if and only if it can be presented by some left-recursive convergent string-rewriting system if and only if it has a recursive cross-section.  ...  This class of languages was defined by McNaughton et al. (1988) using length-reducing and confluent string-rewriting systems.  ... 
doi:10.1006/jsco.1998.0230 fatcat:66f3r5ntwvdz5p6extzc7wlt3e

Length-Based Attacks for Certain Group Based Encryption Rewriting Systems [article]

James Hughes, Allen Tannenbaum
2003 arXiv   pre-print
In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld  ...  The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack.  ...  Word and Conjugacy Problems for Finitely Presented Groups Let G =< s 1 , s 2 , · · · , s n : r 1 , · · · , r k > be a finitely presented group. Let U be the free monoid generated by s i and s −1 i .  ... 
arXiv:cs/0306032v1 fatcat:x6j7ch3o7vcjte7dunngqf6iiq

Page 1592 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
Further, we prove that for each finite monadic 68 COMPUTER SCIENCE 1592 string-rewriting system R such that R presents a group and R is confluent on [e]pr, the set of valid linear sentences is decidable  ...  Since the class of groups that can be presented in this way is exactly the class of finitely generated context-free groups, we see that the linear sentences can be used to solve uniformly a large class  ... 


2015 Bulletin of the Australian Mathematical Society  
We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman's  ...  We also prove that the finite cyclic groups of order at least three are the only finite groups admitting a presentation by more than one finite convergent monadic rewriting system (up to relabelling),  ...  Acknowledgement The author wishes to thank the University of Wollongong for its hospitality as this work was undertaken.  ... 
doi:10.1017/s0004972715000015 fatcat:nmry2rkfvjaubf7twovqizjtra
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