ON A CLASS OF AUTOMATA GROUPS GENERALIZING LAMPLIGHTER GROUPS

RAIRO - Theoretical Informatics and Applications

We investigate the language classes recognized by group automata over matrix groups. For the case of 2 × 2 matrices, we prove that the corresponding group automata for rational matrix groups are more powerful than the corresponding group automata for integer matrix groups. Finite automata over some special matrix groups, such as the discrete Heisenberg group and the Baumslag-Solitar group are also examined. We also introduce the notion of time complexity for group automata and demonstrate some separations among related classes. The case of linear-time bounds is examined in detail throughout our repertory of matrix group automata.

Journal of Mathematical Sciences, 2009

Classification of groups generated by 3-state automata over a 2-letter alphabet started in [BGK + 06] is continued.

2016

In this paper we define Autocamina Group. In Camina group, conjugacy class of an element outside the commutator subgroup coincides with the coset of the commutator subgroup. Similarly we call a group an Autocamina group, if fusion class is the coset of autocommutator subgroup. We study the structure of Autocamina groups in this paper.

2012

One of the classic models of automata is finite automata, which determine whether a string belongs to a particular language or not. The string accepted by automata is said to be recognized by that automata. Another type of automata, so-called Watson-Crick automata, with two reading heads that work on double-stranded tapes using the complimentary relation. Finite automata over groups extend the possibilities of finite automata and allow studying the properties of groups using finite automata. In this paper, we consider finite automata over some Abelian groups ℤn and ℤn × ℤn. The relation of Cayley table to finite automata diagram is introduced in the paper. Some properties of groups ℤn and ℤn × ℤn in terms of automata are also presented in this paper.

arXiv: Combinatorics, 2017

We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and $L$ functions of Schreier graphs of few groups namely the Grigorchuk group, Gupta-Sidki $p$ group, Gupta-Fabrykowski group and BSV torsion-free group.

2008

This article contains most of the known results on the classification of groups generated by 3-state automata over a 2-letter alphabet, extending the previous papers 0704.3876 and math/0612178.

2014

The aim of this short note is to revisit some old results about groups of intermediate growth and groups of the lamplighter type and to show that the Lamplighter group L = Z 2 ≀ Z is a condensation group and has a minimal presentation by generators and relators. The condensation property is achieved by showing that L belongs to a Cantor subset of the space M 2 of marked 2-generated groups consisting mostly of groups of intermediate growth.

AIP Conference Proceedings

Recently, the relation of automata and groups has been studied. It was shown that properties of groups can be studied using state diagrams of modified automata and modified Watson-Crick automata. In this work, we investigate the relation of subgroups with the modified finite and Watson-Crick automata. We also establish the conditions for the recognition of subgroups by using the modified automata.

Discrete Applied Mathematics, 2001

Some results from Dassow and Mitrana (Internat. J. Comput. Algebra (2000)), Griebach (Theoret. Comput. Sci. 7 (1978) 311) and Ibarra et al. (Theoret. Comput. Sci. 2 (1976) 271) are generalized for ÿnite automata over arbitrary groups. The closure properties of these automata are poorer and the accepting power is smaller when abelian groups are considered. We prove that the addition of any abelian group to a ÿnite automaton is less powerful than the addition of the multiplicative group of rational numbers. Thus, each language accepted by a ÿnite automaton over an abelian group is actually a unordered vector language. Characterizations of the context-free and recursively enumerable languages classes are set up in the case of non-abelian groups. We investigate also deterministic ÿnite automata over groups, especially over abelian groups. ?

Natural Computing, 2017

For a finite group G and a finite set A, we study various algebraic aspects of cellular automata over the configuration space A G. In this situation, the set CA(G; A) of all cellular automata over A G is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units ICA(G; A) of CA(G; A). We obtain a decomposition of ICA(G; A) into a direct product of wreath products of groups that depends on the numbers α [H] of periodic configurations for conjugacy classes [H] of subgroups of G. We show how the numbers α [H] may be computed using the Möbius function of the subgroup lattice of G, and we use this to improve the lower bound recently found by Gao, Jackson and Seward on the number of aperiodic configurations of A G. Furthermore, we study generating sets of CA(G; A); in particular, we prove that CA(G; A) cannot be generated by cellular automata with small memory set, and, when all subgroups of G are normal, we determine the relative rank of ICA(G; A) on CA(G; A), i.e. the minimal size of a set V ⊆ CA(G; A) such that CA(G; A) = ICA(G; A) ∪ V .