24 research outputs found
Recent advances in algorithmic problems for semigroups
In this article we survey recent progress in the algorithmic theory of matrix
semigroups. The main objective in this area of study is to construct algorithms
that decide various properties of finitely generated subsemigroups of an
infinite group , often represented as a matrix group. Such problems might
not be decidable in general. In fact, they gave rise to some of the earliest
undecidability results in algorithmic theory. However, the situation changes
when the group satisfies additional constraints. In this survey, we give an
overview of the decidability and the complexity of several algorithmic problems
in the cases where is a low-dimensional matrix group, or a group with
additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New
The Complexity of Knapsack Problems in Wreath Products
We prove new complexity results for computational problems in certain wreath
products of groups and (as an application) for free solvable group. For a
finitely generated group we study the so-called power word problem (does a
given expression , where are
words over the group generators and are binary encoded
integers, evaluate to the group identity?) and knapsack problem (does a given
equation , where are words
over the group generators and are variables, has a solution in
the natural numbers). We prove that the power word problem for wreath products
of the form with nilpotent and iterated wreath products
of free abelian groups belongs to . As an application of the
latter, the power word problem for free solvable groups is in .
On the other hand we show that for wreath products , where
is a so called uniformly strongly efficiently non-solvable group (which
form a large subclass of non-solvable groups), the power word problem is
-hard. For the knapsack problem we show
-completeness for iterated wreath products of free abelian groups
and hence free solvable groups. Moreover, the knapsack problem for every wreath
product , where is uniformly efficiently non-solvable, is
-hard
A Characterization of Wreath Products Where Knapsack Is Decidable
The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and
Ushakov. It is defined for each finitely generated group and takes as input
group elements and asks whether there are
with . We study the knapsack
problem for wreath products of groups and . Our main result is
a characterization of those wreath products for which the knapsack
problem is decidable. The characterization is in terms of decidability
properties of the indiviual factors and . To this end, we introduce two
decision problems, the intersection knapsack problem and its restriction, the
positive intersection knapsack problem. Moreover, we apply our main result to
, the discrete Heisenberg group, and to Baumslag-Solitar
groups for . First, we show that the knapsack
problem is undecidable for for any . This
implies that for and for infinite and virtually nilpotent groups ,
the knapsack problem for is decidable if and only if is virtually
abelian and solvability of systems of exponent equations is decidable for .
Second, we show that the knapsack problem is decidable for
if and only if solvability of systems of exponent
equations is decidable for