The aim of this paper is the study of applications of a group in a general fuzzy automaton. For this purpose, we de ne the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative. Then we obtain the relationships between them...

In this paper, the concepts of somewhat fuzzy Automata continuous functions and somewhat fuzzy Automata open functions in fuzzy Automata topological spaces are introduced and some interesting properties of these functions are studied. In this connection, the concepts of fuzzy Automata resolvable spaces and fuzzy Automata irresolvable spaces are also introduced and their properties are studied.

In the present research, a type of permutation optimization was introduced. It is assumed that the cost function has an unknown probability distribution function. Since the solution space is inherently large, solving the problem of finding the optimal permutation is complex and this assumption increases the complexity. In the present study, an algorithm based on distributed learning Automata was presented to solve the problem by searching in the permutation answer space and sampling random values. In the present research, in addition to the mathematical analysis of the behavior of the proposed new algorithm, it was shown that by choosing the appropriate values of the parameters of the learning algorithm, this new method can find the optimal solution with a probability close to 100% and by targeting the search using the distributed learning algorithms. The result of adopting this policy is to decrease the number of samplings in the new method compared to methods based on standard sampling. In the following, the problem of finding the minimum spanning tree in the stochastic graph was evaluated as a random permutation optimization problem and the proposed solution based on learning Automata was used to solve it.

The current study aims to establish a connection between graphs and Automata theory, which apparently demonstrate di erent mathematical structures. Through searching out some properties of one of these structures, we try to  nd some new properties of the other structure as well. This will result in obtaining some unknown properties. At  rst, a novel automaton called zero-forcing (Z-F)  nite Automata is de ned according to the notion of a zero-forcing set of a graph. It is shown that for a given graph and for some zero forcing sets, various Z-F- nite Automata will be obtained. In addition, the language and the closure properties of Z-F- nite Automata, in particular; union, connection, and serial connection are studied. Moreover, considering some properties of graphs such as the closed trail, connected and complete; some new features for Z-F- nite Automata are presented. Further, it is shown that there is not any  nite graph such that f be a part of the language of its Z-F- nite Automata. Actually, it is proved that for every given graph, the Z-F- nite Automata of it does not show any closed trail containing all edges for every zero forcing set, but if the graph G has been a closed trail containing all edges, then the Z-F- nite Automata of it has a weak closed trail containing all edges. Some examples are also given to clarify these new notions.

In this paper, we define the notion of a bipolar general fuzzy automaton, then we construct some closure operators on the set of states of a bipolar general fuzzy automaton. Also, we construct some topologies on the set of states of a bipolar general fuzzy automaton. Then we obtain some relationships between them.

In this paper, the authors have represented the genetic structures in terms of Automata. With the algebraic structure defined on the genetic code authors defined an automaton on those codons as Σ,= (CG, P, AM, F, G) where P is the set of the four bases A, C, G, U as a set of alphabets or inputs, CG is the set of all 64 codons, obtained from the ordering of the elements of P, as the set of states, AM is the set of the 20 amino acids as the set of outputs that produce during the process. F and G are transition function and output function respectively. Authors observed that M(Σ, ) = ({fa: a ∈,P}, ◦, ) defined on the Automata Σ,where fa(q) = F(q, a) = qa, q ∈,CG, a ∈,P is a monoid called the syntactic monoid of Σ, , with fa ◦,fb = fba ∀, a, b ∈,P. Studying the structure defined in terms of Automata it is also observed that the algebraic structure (M(CG), +, ·, ) forms a Near-Ring with respect to the two operations ′,+ ′,and ′, ·, ′,where M(CG) = {f|f: CG →,CG}.