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 $\Z^n$ is accepted by a nondeterministic $m$-counter automaton if and only if $m \geq n$.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 ..