936 research outputs found
Orbit decidability and the conjugacy problem for some extensions of groups
Given a short exact sequence of groups with certain conditions, , we prove that has solvable conjugacy problem if and only if
the corresponding action subgroup is orbit decidable. From
this, we deduce that the conjugacy problem is solvable, among others, for all
groups of the form , , , and with virtually solvable action
group . Also, we give an easy way of constructing
groups of the form and with
unsolvable conjugacy problem. On the way, we solve the twisted conjugacy
problem for virtually surface and virtually polycyclic groups, and give an
example of a group with solvable conjugacy problem but unsolvable twisted
conjugacy problem. As an application, an alternative solution to the conjugacy
problem in is given
Equations over free inverse monoids with idempotent variables
We introduce the notion of idempotent variables for studying equations in
inverse monoids.
It is proved that it is decidable in singly exponential time (DEXPTIME)
whether a system of equations in idempotent variables over a free inverse
monoid has a solution. The result is proved by a direct reduction to solve
language equations with one-sided concatenation and a known complexity result
by Baader and Narendran: Unification of concept terms in description logics,
2001. We also show that the problem becomes DEXPTIME hard , as soon as the
quotient group of the free inverse monoid has rank at least two.
Decidability for systems of typed equations over a free inverse monoid with
one irreducible variable and at least one unbalanced equation is proved with
the same complexity for the upper bound.
Our results improve known complexity bounds by Deis, Meakin, and Senizergues:
Equations in free inverse monoids, 2007.
Our results also apply to larger families of equations where no decidability
has been previously known.Comment: 28 pages. The conference version of this paper appeared in the
proceedings of 10th International Computer Science Symposium in Russia, CSR
2015, Listvyanka, Russia, July 13-17, 2015. Springer LNCS 9139, pp. 173-188
(2015
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