Extended Models of Finite Automata

TL;DR

This thesis explores extended finite automaton models with additional storage, specifically automata over groups and homing vector automata, revealing their language recognition capabilities and relationships to decision problems.

Contribution

It introduces and analyzes the homing vector automaton model and investigates the language recognition power of automata over various matrix groups, establishing new theoretical links.

Findings

Finite automata over matrix groups have diverse language recognition powers.

Homing vector automata can simulate classical automata under certain restrictions.

New relationships between matrix semigroup decision problems and automata are established.

Abstract

Many of the numerous automaton models proposed in the literature can be regarded as a finite automaton equipped with an additional storage mechanism. In this thesis, we focus on two such models, namely the finite automata over groups and the homing vector automata. A finite automaton over a group $ G $ is a nondeterministic finite automaton equipped with a register that holds an element of the group $ G $. The register is initialized to the identity element of the group and a computation is successful if the register is equal to the identity element at the end of the computation after being multiplied with a group element at every step. We investigate the language recognition power of finite automata over integer and rational matrix groups and reveal new relationships between the language classes corresponding to these models. We examine the effect of various parameters on the language recognition power. We establish a link between the decision problems of matrix semigroups and the corresponding automata. We present some new results about valence pushdown automata and context-free valence grammars. We also propose the new homing vector automaton model, which is a finite automaton equipped with a vector that can be multiplied with a matrix at each step. The vector can be checked for equivalence to the initial vector and the acceptance criterion is ending up in an accept state with the value of the vector being equal to the initial vector. We examine the effect of various restrictions on the model by confining the matrices to a particular set and allowing the equivalence test only at the end of the computation. We define the different variants of the model and compare their language recognition power with that of the classical models.