Word Problem Languages for Free Inverse Monoids
Conference paper
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Abstract
This paper considers the word problem for free inverse monoids of finite rank from a language theory perspective. It is shown that no free inverse monoid has context-free word problem; that the word problem of the free inverse monoid of rank 1 is both 2-context-free (an intersection of two context-free languages) and ET0L; that the co-word problem of the free inverse monoid of rank 1 is context-free; and that the word problem of a free inverse monoid of rank greater than 1 is not poly-context-free.
Keywords
Word problems Co-word problems Inverse monoids ET0L languages Stack automata Poly-context-free languagesReferences
- 1.Aho, A.: Nested stack automata. J. ACM 16(3), 383–406 (1969)MathSciNetCrossRefGoogle Scholar
- 2.Boone, W.W.: The word problem. Ann. Math. 2(70), 207–265 (1959)CrossRefGoogle Scholar
- 3.Brough, T.: Groups with poly-context-free word problem. Groups Complex. Cryptol. 6(1), 9–29 (2014)MathSciNetCrossRefGoogle Scholar
- 4.Brough, T.: Inverse semigroups with rational word problem are finite, Unpublished note. arxiv:1311.3955 (2013)
- 5.Ciobanu, L., Diekert, V., Elder, M.: Solution sets for equations over free groups are EDT0L languages. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 134–145. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47666-6_11CrossRefzbMATHGoogle Scholar
- 6.Ciobanu, L., Elder, M., Ferov, M.: Applications of L systems to group theory. Int. J. Algebra Comput. 28, 309–329 (2018)MathSciNetCrossRefGoogle Scholar
- 7.Duncan, A., Gilman, R.H.: Word hyperbolic semigroups. Math. Proc. Camb. Philos. Soc. 136, 513–524 (2004)MathSciNetCrossRefGoogle Scholar
- 8.Gilbert, N.D., Noonan Heale, R.: The idempotent problem for an inverse monoid. Int. J. Algebra Comput. 21, 1170–1194 (2011)MathSciNetCrossRefGoogle Scholar
- 9.Gilman, R.H., Shapiro, M.: On groups whose word problem is solved by a nested stack automaton. arxiv:math/9812028
- 10.Ginsburg, S., Greibach, S.A., Harrison, M.A.: One-way stack automata. J. ACM 14(2), 389–418 (1967)MathSciNetCrossRefGoogle Scholar
- 11.Greibach, S.: Checking automata and one-way stack languages. J. Compt. Syst. Sci. 3, 196–217 (1969)MathSciNetCrossRefGoogle Scholar
- 12.Hoffman, M., Holt, D.F., Owens, M.D., Thomas, R.M.: Semigroups with a context-free word problem. In: Proceedings of the 16th International Conference on Developments in Language Theory, DLT 2012, pp. 97–108 (2012)Google Scholar
- 13.Holt, D.F., Owens, M.D., Thomas, R.M.: Groups and semigroups with a one-counter word problem. J. Aust. Math. Soc. 85, 197–209 (2005)MathSciNetCrossRefGoogle Scholar
- 14.Holt, D.F., Röver, C.E., Rees, S.E., Thomas, R.M.: Groups with a context-free co-word problem. J. Lond. Math. Soc. 71, 643–657 (2005)MathSciNetCrossRefGoogle Scholar
- 15.Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Boston (1979)zbMATHGoogle Scholar
- 16.Kambites, M.: Anisimov’s Theorem for inverse semigroups. Int. J. Algebra Comput. 25, 41–49 (2015)MathSciNetCrossRefGoogle Scholar
- 17.Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)CrossRefGoogle Scholar
- 18.Lohrey, M., Ondrusch, N.: Inverse monoids: decidability and complexity of algebraic questions. Inf. Comput. 205(8), 1212–1234 (2007)MathSciNetCrossRefGoogle Scholar
- 19.Lisovik, L.P., Red’ko, V.N.: Regular events in semigroups. Problemy Kibernetiki 37, 155–184 (1980)MathSciNetzbMATHGoogle Scholar
- 20.Munn, W.D.: Free inverse semigroups. Proc. Lond. Math. Soc. s3–29(3), 385–404 (1974)MathSciNetCrossRefGoogle Scholar
- 21.Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Am. Math. Soc. Transl. Ser. 2(9), 1–122 (1958)zbMATHGoogle Scholar
- 22.Pfeiffer, M.J.: Adventures in applying iteration lemmas, Ph.D. thesis, University of St Andrews (2013)Google Scholar
- 23.Post, E.: Recursive unsolvability of a problem of Thue. J. Symb. Log. 12(1), 1–11 (1947)MathSciNetCrossRefGoogle Scholar
- 24.Rozenberg, G., Salomaa, A.: The Book of L. Springer, Heidelberg (1986). https://doi.org/10.1007/978-3-642-95486-3CrossRefzbMATHGoogle Scholar
- 25.van Leeuwen, J.: Variations of a new machine model. In: Conference Record 17th Annual IEEE Symposium on Foundations of Computer Science, pp. 228–235 (1976)Google Scholar
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