Compressed Word Problems for Inverse Monoids

  • Markus Lohrey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6907)


The compressed word problem for a finitely generated monoid M asks whether two given compressed words over the generators of M represent the same element of M. For string compression, straight-line programs, i.e., context-free grammars that generate a single string, are used in this paper. It is shown that the compressed word problem for a free inverse monoid of finite rank at least two is complete for \(\Pi^p_2\) (second universal level of the polynomial time hierarchy). Moreover, it is shown that there exists a fixed finite idempotent presentation (i.e., a finite set of relations involving idempotents of a free inverse monoid), for which the corresponding quotient monoid has a PSPACE-complete compressed word problem. The ordinary uncompressed word problem for such a quotient can be solved in logspace [10]. Finally, a PSPACE-algorithm that checks whether a given element of a free inverse monoid belongs to a given rational subset is presented. This problem is also shown to be PSPACE-complete (even for a fixed finitely generated submonoid instead of a variable rational subset).


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  1. 1.
    Berman, P., Karpinski, M., Larmore, L.L., Plandowski, W., Rytter, W.: On the complexity of pattern matching for highly compressed two-dimensional texts. J. Comput. Syst. Sci. 65(2), 332–350 (2002)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Diekert, V., Lohrey, M., Miller, A.: Partially commutative inverse monoids. Semigroup Forum 77(2), 196–226 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1999)Google Scholar
  4. 4.
    Lifshits, Y.: Processing compressed texts: A tractability border. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 228–240. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Lipton, R.J., Zalcstein, Y.: Word problems solvable in logspace. J. Assoc. Comput. Mach. 24(3), 522–526 (1977)MathSciNetMATHGoogle Scholar
  6. 6.
    Lohrey, M.: On the parallel complexity of tree automata. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 201–215. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Lohrey, M.: Word problems and membership problems on compressed words. SIAM J. Comput. 35(5), 1210–1240 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Lohrey, M.: Leaf languages and string compression. Inf. Comput. 209(6), 951–965 (2011)MATHCrossRefGoogle Scholar
  9. 9.
    Lohrey, M.: Compressed word problems for inverse monoids.,
  10. 10.
    Lohrey, M., Ondrusch, N.: Inverse monoids: decidability and complexity of algebraic questions. Inf. Comput. 205(8), 1212–1234 (2007)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lohrey, M., Schleimer, S.: Efficient computation in groups via compression. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 249–258. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Lohrey, M., Steinberg, B.: Tilings and submonoids of metabelian groups. Theory Comput. Syst. 48(2), 411–427 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Macdonald, J.: Compressed words and automorphisms in fully residually free groups. Internat. J. Algebra Comput. 20(3), 343–355 (2010)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Margolis, S., Meakin, J.: Inverse monoids, trees, and context-free languages. Trans. Amer. Math. Soc. 335(1), 259–276 (1993)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Margolis, S., Meakin, J., Sapir, M.: Algorithmic problems in groups, semigroups and inverse semigroups. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups, pp. 147–214. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  16. 16.
    Munn, W.: Free inverse semigroups. Proc.London Math. Soc. 30, 385–404 (1974)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Plandowski, W.: Testing equivalence of morphisms on context-free languages. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 460–470. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  18. 18.
    Plandowski, W., Rytter, W.: Complexity of language recognition problems for compressed words. In: Karhumäki, J., Maurer, H.A., Paun, G., Rozenberg, G. (eds.) Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa, pp. 262–272. Springer, Heidelberg (1999)Google Scholar
  19. 19.
    Rozenblat, B.V.: Diophantine theories of free inverse semigroups. Sib. Math. J. 26, 860–865 (1985); English translationMathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Schleimer, S.: Polynomial-time word problems. Comment. Math. Helv. 83, 741–765 (2008)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Silva, P.V.: Rational languages and inverse monoid presentations. Internat. J. Algebra Comput. 2, 187–207 (1992)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Stephen, J.: Presentations of inverse monoids. J. Pure Appl. Algebra 63, 81–112 (1990)MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag GmbH Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Lohrey
    • 1
  1. 1.Institut für InformatikUniversität LeipzigGermany

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