Some Contribution To Finite State Machine on The Basis of Soft Set

Abstract

The present work in this thesis is written in the theoretical background of finite state machines and soft sets. It contains the necessary part of automata theory and shows to formulate in an elegant way variousaconcepts and facts about the soft finite stateamachine. Prerequisites are minimal and the work is self-contained. In this thesis, we have six chapters. Chapter 1 provides some basic material needed for an understanding of finite state machine and algebra. In second chapter, using the notion of soft sets, we introduced the concept of soft finite state machines (SFSM) as a generalization of fuzzy finite state machines. Study of soft finite state machines is interesting and worthwhile because there are results which hold for fuzzy finite state machine but do not hold for soft finite machine. For example, [20], in fuzzy finite state machines if p is successor of q and r is a successor of p, then r is a successor of q. In general this result holds no longer in soft finite machines. In this chapter, we also introduce a congruence relation which can be naturally established in such a way that each associates a semigroup with a soft finite state machine (SFSM). We introduce a (strong) homomorphism of SFSM and then we investigate the related properties. Using a SFSM we make three finite semigroups with identity and show that they are isomorphic. We defined soft admissible relation and establish a relation between SFSM and the quotient structure of another soft finite state machine. Finally soft transformation semigroups are defined and related properties investigated. Concept of covering, cascade product, and wreath product play an important role in the study of automata and their associated semigroups. In 3rd chapter, we examine these concepts for soft finite state machines. In chapter 4, we continue our study of a soft finite state machine utilizing algebraic techniques. We defined the concept of soft submachine, separability, connectivity and decomposition of soft finite state machine. With the help of these concepts, we will prove Decomposition Theorem for soft finite state machine. In 5th chapter concepts of soft subsystem, strong soft subsystem, switching, commutative and soft finite switchboard state machine are introduced and some of its properties are discussed. In last chapter we study a new product of two soft finite machines M₁ and M₂, written M₁⋅M₂ and called the Cartesian composition of M₁ and M₂. We also define the concept of soft admissible partitions and construct the quotient structure of SFSM with the help of soft admissible partitions. Finally we discuss the associativity of wreath product, sum and cascade products of soft finite state machines.