**Theory**

20 Will-be-set-by-IN-TECH

[29] Marsan MA, Bobbio, A, Donatelli S (1998) Petri Nets in Performance Analysis: an

[30] Merlin, P., Faber, D. (1976) Recoverability on communication protocols - implications of a theoretical study, IEEE Trans. on Communications, vol. 4, no. 9, pp. 1036-1043. [31] Murata, T (1989) Petri Nets: Properties, Analysis and Applications, Proceedings of IEEE,

[32] Patrice B (2010) A New Control Synthesis Approach of P-Time Petri Nets, in Petri Nets

[33] Pawlewski P (2010) Using Petri Nets to Model and Simulation Production Systems in Process Reengineering, in Petri Nets Applications, Pawel Pawlewski (ed.), InTech, 752

[34] Ramchandani, C. (1973) Analysis of Asynchronous Concurrent Systems by Timed Petri

[35] Riviere, N., Valette, R., Pradin-Chezalviel, B. and Ups, I. A. . (2001). Reachability and temporal conflicts in t-time petri nets, *PNPM '01: Proceedings of the 9th international Workshop on Petri Nets and Performance Models (PNPM'01)*, IEEE Computer Society,

[36] Roux, O. H. and Déplanche, A. M. (2002) . A T-time petri net extension for real time-task

[37] Salmon, A. Z. O., Miralles, J. A. S. P., del Foyo, P. M. G., Silva, J. R. (2011) Towards a Unified View of Modeling and Design with GHENeSys, Proc. of COBEM 2011. [38] Sifakis, J. (1980). Performance evaluation of systems using nets, *Proceedings of the Advanced Course on General Net Theory of Processes and Systems*, Springer-Verlag, London,

[39] Tierry-Mieg, Y, Hilah, L-M (2008) UML Behavioral Consistency Checking Using Instantiable Petri Nets, Workshop UML in Formal Methods, 10th. Int. Conf. on Formal

[40] Vaquero, T.S., Silva, J.R., Ferreira, M., Tonidandel, F., Bech, J.C. (2009) From Requirements and Analysis to PDDL in itSIMPLE 3.0, Proc. of Int. Conf. in Artificial

[41] Vaquero, T.S., Silva, J.R., Bech, J.C. (2011) A Brief Review of Tools and Methods for Knowledge Engineering for Planning and Scheduling, Proc. of Int. Conf. in Artificial

[42] Wang, J (1998) Timed Petri Nets: Theory and Applications, Kluwer Academic Pub. 281

[43] Wang, J, Deng, Y, Xu, G (2000) Reachability Analysis of Real-time Systems Using Timed Petri Nets, IEEE Trans. on Syst. Man and Cybernetics, vol 30 no. 5, pp. 725-736. [44] Yoneda, T. and Ryuba, H. (1998). CTL model checking of time petri nets using geometric

[45] Zuberek, W. M. (1980). Timed petri nets and preliminary performance evaluation, *ISCA '80: Proceedings of the 7th annual symposium on Computer Architecture*, ACM Press, New

regions, *IEICE Trans. on Information and Systems* E81-D(3): 297–396.

scheduling modeling, *European Journal of Automation* 36: 973–987.

Introduction, LNCS 1491, pp. 211-256.

Applications, Pawel Pawlewski (ed.), InTech, 752 p.

vol. 77, pp 541-580.

Nets, PHD thesis, MIT, 220 p.

Washington, DC, USA, p. 229.

UK, pp. 307–319.

p.

Engineering Methods.

Planning and Scheduling, AAAI.

Planning and Scheduling, AAAI.

York, NY, USA, pp. 88–96.

p.

**Chapter 0**

**Chapter 17**

**Boolean Petri Nets**

http://dx.doi.org/10.5772/50354

**1. Introduction**

Sangita Kansal, Mukti Acharya and Gajendra Pratap Singh

Petri net is a graphical tool invented by Carl Adam Petri [13]. These are used for describing, designing and studying discrete event-driven dynamical systems that are characterized as being concurrent, asynchronous, distributed, parallel, random and/or nondeterministic. As a graphical tool, Petri net can be used for planning and designing a system with given objectives, more practically effective than flowcharts and block diagrams. As a mathematical tool, it enables one to set up state equations, algebraic equations and other mathematical models which govern the behavior of discrete dynamical systems. Still, there is a drawback inherent in representing discrete event-systems. They suffer from the state explosion problem as what will happen when a system is highly populated, i.e., initial state consists of a large number of places that are nonempty. This phenomenon may lead to an exponential growth of its reachability graph. This makes us to study the safe systems. The aim of this chapter is to present some basic results on 1-safe Petri nets that generate the elements of a Boolean hypercube as marking vectors. Complete Boolean hypercube is the most popular interconnection network with many attractive and well known properties such as regularity, symmetry, strong connectivity, embeddability, recursive construction, etc. For brevity, we shall call a 1-safe Petri net that generates all the binary *n*-vectors as marking vectors a *Boolean Petri net*. *Boolean Petri nets* are not only of theoretical interest but also are of practical importance, required in practice to construct control systems [1]. In this chapter, we will consider the problems of characterizing the class of Boolean Petri nets as also the class of *crisp* Boolean Petri nets, viz., the Boolean Petri nets that generate all the binary *n*-vectors exactly once. We show the existence of a disconnected Boolean Petri net whose reachability tree is homomorphic to the *n*-dimensional complete lattice *Ln*. Finally, we observe that

We begin by showing that a 1-safe *Star Petri net Sn* [5], with |*P*| = *n* and |*T*| = *n* + 1, having a central transition, is a Boolean Petri net; here, *P* is the set of its places and *T* is the set of its transitions. Often, it is desirable to have a crisp Boolean Petri net because one may possibly explore for existence of certain sequences of enabled transitions to fire toward initiating and

> ©2012 Kansal et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Kansal et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

completing a prescribed process that uses specified nodes of the Boolean lattice.

Additional information is available at the end of the chapter

characterizing a Boolean Petri net is rather intricate.

cited.
