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Elements of finite order for finite monadic Church-Rosser Thue systems. (English) Zbl 0583.20054
Let T be a Thue system over S. The relation $$\leftrightarrow_ T$$ is defined for $$u,v\in S^*$$ by $$u\leftrightarrow_ Tv$$ iff $$\exists x,y\in S^*$$, ($$\ell,r)\in T:$$ $$(u=x\ell y$$ and $$v=xry)$$ or $$(u=xry$$ and $$v=x\ell y)$$. The reflexive and transitive closure of $$\leftrightarrow_ T$$ is denoted by $$\leftrightarrow^*_ T$$. A Thue system T over S allows nontrivial elements of finite order if there exist $$u\in S^*$$ and integers $$n\geq 0$$ and $$k\geq 1$$ such that $$u\leftrightarrow^*_ T\ell$$ and $$u^{n+k}\leftrightarrow^*_ Tu^ n$$, where $$\ell$$ is the empty word. The following decision problem is shown to be decidable: for a finite monadic Church-Rosser Thue system T over S, does T allow nontrivial elements of finite order?
Reviewer: J.Grzymala-Busse

##### MSC:
 20M35 Semigroups in automata theory, linguistics, etc. 68Q45 Formal languages and automata 20M05 Free semigroups, generators and relations, word problems
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##### References:
 [1] Jürgen Avenhaus, Ronald V. Book, and Craig C. Squier, On expressing commutativity by finite Church-Rosser presentations: a note on commutative monoids, RAIRO Inform. Théor. 18 (1984), no. 1, 47 – 52 (English, with French summary). · Zbl 0542.20038 [2] J. Avenhaus, K. Madlener and F. Otto, Groups presented by finite two-monadic Church-Rosser Thue systems, Technical Report 110/84, Department of Computer Science, University of Kaiserslautern, 1984. · Zbl 0604.20034 [3] J. Berstel, Congruences plus que parfaites et langages algébriques, Séminaire d’Informatique Théorique, Institut de Programmation 1976-1977, pp. 123-147. [4] Ronald V. Book, Confluent and other types of Thue systems, J. Assoc. Comput. Mach. 29 (1982), no. 1, 171 – 182. · Zbl 0478.68032 · doi:10.1145/322290.322301 · doi.org [5] Ronald V. Book, When is a monoid a group? The Church-Rosser case is tractable, Theoret. Comput. Sci. 18 (1982), no. 3, 325 – 331. · Zbl 0489.68021 · doi:10.1016/0304-3975(82)90072-X · doi.org [6] Ronald V. Book, Decidable sentences of Church-Rosser congruences, Theoret. Comput. Sci. 24 (1983), no. 3, 301 – 312. · Zbl 0525.68015 · doi:10.1016/0304-3975(83)90005-1 · doi.org [7] Ronald V. Book, Matthias Jantzen, and Celia Wrathall, Monadic Thue systems, Theoret. Comput. Sci. 19 (1982), no. 3, 231 – 251. · Zbl 0488.03020 · doi:10.1016/0304-3975(82)90036-6 · doi.org [8] Ronald V. Book and Friedrich Otto, Cancellation rules and extended word problems, Inform. Process. Lett. 20 (1985), no. 1, 5 – 11. · Zbl 0561.68030 · doi:10.1016/0020-0190(85)90122-X · doi.org [9] Y. Cochet and M. Nivat, Une généralisation des ensembles de Dyck, Israel J. Math. 9 (1971), 389 – 395 (French). · Zbl 0215.56005 · doi:10.1007/BF02771689 · doi.org [10] Michael A. Harrison, Introduction to formal language theory, Addison-Wesley Publishing Co., Reading, Mass., 1978. · Zbl 0411.68058 [11] John E. Hopcroft and Jeffrey D. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley Publishing Co., Reading, Mass., 1979. Addison-Wesley Series in Computer Science. · Zbl 0426.68001 [12] Gérard Lallement, On monoids presented by a single relation, J. Algebra 32 (1974), 370 – 388. · Zbl 0307.20034 · doi:10.1016/0021-8693(74)90146-X · doi.org [13] Gérard Lallement, Semigroups and combinatorial applications, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Pure and Applied Mathematics; A Wiley-Interscience Publication. · Zbl 0421.20025 [14] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023 [15] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. · Zbl 0362.20023 [16] A. Markov, The impossibility of algorithms for the recognition of certain properties of associative systems, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 953 – 956 (Russian). [17] Andrzej Mostowski, On direct products of theories, J. Symbolic Logic 17 (1952), 1 – 31. · Zbl 0047.00704 · doi:10.2307/2267454 · doi.org [18] David E. Muller and Paul E. Schupp, Groups, the theory of ends, and context-free languages, J. Comput. System Sci. 26 (1983), no. 3, 295 – 310. · Zbl 0537.20011 · doi:10.1016/0022-0000(83)90003-X · doi.org [19] Maurice Nivat, Congruences parfaites et quasi-parfaites, Séminaire P. Dubreil, 25e année (1971/72), Algèbre, Fasc. 1, Exp. No. 7, Secrétariat Mathématique, Paris, 1973, pp. 9 (French). · Zbl 0338.02018 [20] Friedrich Otto, Conjugacy in monoids with a special Church-Rosser presentation is decidable, Semigroup Forum 29 (1984), no. 1-2, 223 – 240. , https://doi.org/10.1007/BF02573327 Paliath Narendran and Friedrich Otto, Complexity results on the conjugacy problem for monoids, Theoret. Comput. Sci. 35 (1985), no. 2-3, 227 – 243. , https://doi.org/10.1016/0304-3975(85)90016-7 Paliath Narendran, Friedrich Otto, and Karl Winklmann, The uniform conjugacy problem for finite Church-Rosser Thue systems is NP-complete, Inform. and Control 63 (1984), no. 1-2, 58 – 66. · Zbl 0592.03025 · doi:10.1016/S0019-9958(84)80041-8 · doi.org
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