13 research outputs found
The word problem distinguishes counter languages
Counter automata are more powerful versions of finite-state automata where
addition and subtraction operations are permitted on a set of n integer
registers, called counters. We show that the word problem of is accepted
by a nondeterministic -counter automaton if and only if .Comment: 8 page
On groups and counter automata
We study finitely generated groups whose word problems are accepted by
counter automata. We show that a group has word problem accepted by a blind
n-counter automaton in the sense of Greibach if and only if it is virtually
free abelian of rank n; this result, which answers a question of Gilman, is in
a very precise sense an abelian analogue of the Muller-Schupp theorem. More
generally, if G is a virtually abelian group then every group with word problem
recognised by a G-automaton is virtually abelian with growth class bounded
above by the growth class of G. We consider also other types of counter
automata.Comment: 18 page
The word problem distinguishes counter languages
Counter automata are more powerful versions of finite state automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of Zn is accepted by a nondeterministic m-counter automaton if and only if m >= n
On groups that have normal forms computable in logspace
We consider the class of finitely generated groups which have a normal form
computable in logspace. We prove that the class of such groups is closed under
finite extensions, finite index subgroups, direct products, wreath products,
and also certain free products, and includes the solvable Baumslag-Solitar
groups, as well as non-residually finite (and hence non-linear) examples. We
define a group to be logspace embeddable if it embeds in a group with normal
forms computable in logspace. We prove that finitely generated nilpotent groups
are logspace embeddable. It follows that all groups of polynomial growth are
logspace embeddable.Comment: 24 pages, 1 figure. Minor corrections from previous versio
Algorithms for Polycyclic-by-finite Matrix Groups
Let R be the ring of integers or a number field. We present several algorithms for working with polycyclic-by-finite subgroups of GL(n; R). Let G be a subgroup of GL(n; R) given by a finite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-finite. For polycyclic-by-finite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable-by-finite, providing an alternative to the algorithm proposed by Beals ([Be1]) for this problem. Baumslag, Cannonito, Robinson and Segal prove that the problem of determining whether or not a finitely generated subgroup of GL(n; Z) is polycyclic-by-finite is decidable and that the problem of testing membership in a polycyclic-by-finite subgroup of GL(n; Z) is also decidable ([BCRS]). In this report we extend these results by describing algorithms which appear to be suitable for computer implementation. Exper..
Practical Algorithms for Polycyclic Matrix Groups
Many fundamental problems are undecidable for infinite matrix groups. Polycyclic matrix groups represent a large class of groups for which these same problems are known to be decidable. In this paper we describe a suite of new algorithms for studying polycyclic matrix groups --- algorithms for testing membership and for uncovering the polycyclic structure of the group. We also describe an algorithm for deciding whether or not a group is solvable, which, in the important context of subgroups of GL(n; Z), is equivalent to deciding whether or not a group is polycyclic. In contrast to previous algorithms, the algorithms in this paper are practical: experiments show that they are efficient enough to be useful in studying some reasonably complex examples using current technology. 1 Introduction The development of efficient matrix group algorithms is one of the most active areas of computational group theory, but until recently, most of that activity has focused on algorithms for studying ..