Extended finite automata and decision problems for matrix semigroups

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 $2 \times 2$ integer matrices. We show that the emptiness problem for extended finite automata over $4 \times 4$ 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.Comment: NCMA2018 Short Pape

New Results on Vector and Homing Vector Automata

We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over $2 \times 2$ integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations.Comment: Accepted to IJFC

Extended Models of Finite Automata

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.Comment: Ph.D. Thesis, Bo\u{g}azi\c{c}i University Computer Engineering Department, 201