On semigroups with PSPACEcomplete subpower membership problem
Abstract
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 NPcomplete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NPcomplete, or PSPACEcomplete. This result yields various semigroups with PSPACEcomplete SMP including the $6$element Brandt monoid, the full transformation semigroup on $3$ or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\ge 2$.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 DOI:
 10.48550/arXiv.1604.01757
 arXiv:
 arXiv:1604.01757
 Bibcode:
 2016arXiv160401757S
 Keywords:

 Mathematics  Group Theory;
 Computer Science  Computational Complexity;
 20M99;
 68Q25
 EPrint:
 J. Aust. Math. Soc. 106 (2019) 127142