Compressed Word Problems for Inverse Monoids
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 . 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).
Unable to display preview. Download preview PDF.
- 3.Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1999)Google Scholar
- 9.Lohrey, M.: Compressed word problems for inverse monoids., http://arxiv.org/abs/1106.1000
- 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
- 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