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ON SEMIGROUPS WITH PSPACE-COMPLETE SUBPOWER MEMBERSHIP PROBLEM

Part of: Theory of computing Semigroups

Published online by Cambridge University Press:  30 May 2018

MARKUS STEINDL
MARKUS STEINDL*
Affiliation:
Institute for Algebra, Johannes Kepler University Linz, Altenberger St 69, 4040 Linz, Austria Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO 80309-0395, USA email markus.steindl@colorado.edu
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Abstract

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Fix a finite semigroup $S$ and let $a_{1},\ldots ,a_{k},b$ be tuples in a direct power $S^{n}$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_{1},\ldots ,a_{k}$. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the six-element Brandt monoid, the full transformation semigroup on three or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\geq 2$.

Keywords

subalgebras of powersmembership testcomputational complexityPSPACE-completeNP-complete

MSC classification

Primary: 20M99: None of the above, but in this section
Secondary: 68Q25: Analysis of algorithms and problem complexity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 1 , February 2019 , pp. 127 - 142
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author was supported by the Austrian Science Fund (FWF): P24285.

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