27 research outputs found
TC^0 Circuits for Algorithmic Problems in Nilpotent Groups
Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC^0 - uniformly for all r-generated nilpotent groups of class at most c for fixed r and c.
Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform TC^0, while all the other problems we examine are shown to be TC^0-Turing reducible to the problem of computing greatest common divisors and expressing them as linear combinations
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
The Power Word Problem
In this work we introduce a new succinct variant of the word problem in a finitely generated group G, which we call the power word problem: the input word may contain powers p^x, where p is a finite word over generators of G and x is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G). The main result of the paper states that the power word problem for a finitely generated free group F is AC^0-Turing-reducible to the word problem for F. Moreover, the following hardness result is shown: For a wreath product G Wr Z, where G is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where G is abelian: then the power word problem is shown to be in TC^0