Language classes associated with automata over matrix groups

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.

Journal of Symbolic Computation, 2005

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.

Journal of Mathematical Sciences, 2009

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

Mathematics and Statistics, 2024

This paper presents a comprehensive examination of the underlying structures and behaviours of finite group automata and group machines, delving into their intricate relationships and properties. Researchers have developed a comprehensive structure for finite group automata applicable to any finite group. Their work utilizes state complexity (SC) and accepting state complexity (ASC) as key metrics. They have also computed syntactic and quotient complexities (SNC and QC) specifically for cyclic groups. While the primary emphasis lies on cyclic groups, their versatile methodology establishes a foundation for broadening these complexity analyses to encompass other categories of finite groups. Building upon existing research on finite group automata, this study investigates the structures of group machines, group machine recognizers, and their properties, including strong connectivity, cyclicality, perfection, bideterminism, and permutation behaviours. Our analysis reveals the interconnected nature of group machines and group machine recognizers, shedding light on their distinct characteristics. A key finding of this research is that all group machines are finite group automata, although the converse is not always true. By differentiating these structures based on their properties, we can effectively handle machines according to their characteristics. This study contributes to the advancement of research in this field by providing a thorough understanding of the fundamental structures and behaviours of finite group automata and group machines, ultimately enriching the theoretical foundations of computational models.

Information and Computation, 1990

Springer eBooks, 2015

In the setting of symbolic dynamics on discrete finitely generated infinite groups, we define a model of multi-headed finite automata that walk on Cayley graphs, and use it to define subshifts. We characterize the torsion groups (also known as periodic groups) as those on which the group-walking automata are strictly weaker than Turing machines.

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.

IOSR Journals , 2019

Let (Q, *, Ʃ, δ, q0, F) be a Finite Group Automaton. Let (S, *, E, γ, q0, T) be a Finite Sub-group Automaton of Q. Then (Q/S, o , Ʃ, ˄ , q0*S, F’) is a finite group automaton. This finite group automaton is known as the Quotient Finite Group Automaton (Quotient FGA) corresponding to the Finite Subgroup Automaton (S, *, E, γ, qs , T). If a string w is accepted by (Q, *, Ʃ, δ, q0, F), then w is accepted by (Q/S, o , Ʃ, ˄ , q0*S, F’). If L is a language accepted by a finite group automaton (Q, *, Ʃ, δ, q0, F) , then L is accepted by (Q/S, o , Ʃ, ˄ , q0*S, F’).

arXiv (Cornell University), 2018

We make a connection between the subgroup membership and identity problems for matrix groups and extended finite automata. We provide an alternative proof for the decidability of the subgroup membership problem for 2 × 2 integer matrices. We show that the emptiness problem for extended finite automata over 4 ×4 integer matrix semigroups is undecidable. We prove that the decidability of the universe problem for extended finite automata is a sufficient condition for the decidability of the subgroup membership and identity problems.

2013

A finite automaton is one of the classic models of recognition devices, which is used to determine the type of language a string belongs to. A string is said to be recognized by a finite automaton if the automaton "reads" the string from the left to the right starting from the initial state and finishing at a final state. Another type of automata which is a counterpart of sticker systems, namely Watson-Crick automata, is finite automata which can scan the double-stranded tapes of DNA strings using the complimentary relation. The properties of groups have been extended for the recognition of finite automata over groups. In this paper, two variants of automata, modified deterministic finite automata and modified deterministic Watson-Crick automata are used in the study of Abelian groups. Moreover, the relation between finite automata diagram over Abelian groups and the Cayley table is introduced. In addition, some properties of Abelian groups are presented in terms of automata.