Abstract—Functionalities and control behavior are both primary requirements in design of a complex system. Automata theory plays an important role in modeling behavior of a system. Z is an ideal notation which is used for describing state space of a system and then defining operations over it. Consequently, an integration of automata and Z will be an effective tool for increasing modeling power for a complex system. Further, nondeterministic finite automata (NFA) may have different implementations and therefore it is needed to verify the transformation from diagrams to a code. If we describe formal specification of an NFA before implementing it, then confidence over transformation can be increased. In this paper, we have given a procedure for integrating NFA and Z. Complement of a special type of NFA is defined. Then union of two NFAs is formalized after defining their complements. Finally, formal construction of intersection of NFAs is described. The specification of this relationship is analyzed and validated using Z/EVES tool. Keywords—Modeling, Nondeterministic finite automata, Z notation, Integration of approaches, Validation. I. INTRODUCTION N this paper, a relationship between automata and Z notation is investigated. Automata have various applications in many areas of computer science and engineering. Modeling control behavior, compiler constructions, modeling of finite state systems, defining a regular set of finite words are some of the traditional applications of automata. Automata have emerged with several modern applications, for example, optimization of logic based programs, verification of protocols [1] and human computer interaction. The Z notation [2] is a model oriented approach based on set theory and first order predicate logic. It is used for specifying the abstract data types and sequential programs. Z notation can also be used to define state of a system and then defining operations over it. The design of a complex system, not only requires the techniques for capturing functionalities but it also needs to N. A. Zafar is with the Faculty of Information Technology, University of Central Punjab, Lahore, on leave from Pakistan Institute of Engineering Applied Sciences, Islamabad, Pakistan (phone: +92-51-9290273-4; fax: +92- 51-2208070; e-mail: nazafar@pieas.edu.pk). N. Sabir is in Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan (e-mail: nabeel.bloch@ucp.edu.pk). A. Ali is a student of PhD in Faculty of Information Technology, University of Central Punjab (e-mail: amiralishahid@ucp.edu.pk). model control behavior [3]. Functions over any of the systems can be decomposed in terms of operations and the constraints, and, hence, Z notation is an ideal application for this purpose. Control over a system can be viewed in terms of visual flows in between the system’s functions. Automata theory is very powerful thereat. Consequently, it requires an integration of automata and Z to increase modeling power for a complex system, which is one of the objectives of this research. The design of a complex system, not only requires the techniques for capturing functionalities but it also needs to model control behavior [3]. Functions over any of the systems can be decomposed in terms of operations and constraints, and hence Z notation is an ideal application for this purpose. Control over a system can be viewed in terms of visual flows in between the system’s functions. Automata theory is very powerful thereat. Consequently, it requires an integration of automata and Z to increase modeling power for a complex system, which is one of the objectives of this research. As we know that nondeterministic and deterministic finite automata are equivalent in power, in a sense, that if a language is recognized by one, it is also recognized by the other. Nondeterministic finite automata (NFA) are sometimes useful because constructing an NFA is easier than constructing deterministic finite automata (DFA). This is because the complexity of mathematical work is reduced using NFA. Further, many important properties in automata can be established easily using NFA. For example, to prove that a union or concatenation of regular languages is regular using NFA is much easier than using DFA [4]. This is another reason that NFA is selected to be integrated with Z notation. Nondeterministic finite automata are abstract models of machines which can be represented using diagrams. These models can be used to perform computations on inputs by moving through a sequence of configurations. An NFA consumes the entire input of symbols and for each input symbol it transforms to a new state until all symbols have been consumed. If we are able to reach any of the accepting state after consuming whole input then the input is accepted. At this level of integration, we have defined two NFAs and their complements are described. As we know that complement of an NFA is not well defined in general therefore, in this paper, we have proposed it only for particular cases. Union of the complemented NFAs is constructed and formal specification of their relationships is given. Finally intersection of the given NFAs is constructed by taking complement of the resultant. Formal specification of the whole Construction of Intersection of Nondeterministic Finite Automata using Z Notation Nazir Ahmad Zafar, Nabeel Sabir, and Amir Ali I World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:2, No:4, 2008 1057 International Scholarly and Scientific Research & Innovation 2(4) 2008 scholar.waset.org/1307-6892/12799 International Science Index, Computer and Information Engineering Vol:2, No:4, 2008 waset.org/Publication/12799