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).


Polynomial Time Word Problem Inverse Semigroup Tree Automaton Free Monoid 
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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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