ON SIGNED PETRI NETS

Abstract

In today’s complex scenario of technological advancement, the role of event-driven discrete dynamical systems have an impact on man’s ability to fast-forward the futuristic technologies that are likely to bring unimaginable progress in our time, and near, far futures. One such great innovation that inspired this thesis is the "Theory of Petri nets", a modeling tool for event-driven discrete dynamical systems. The study of Petri nets and its various extensions that have developed over time is one of the most active and vibrant areas of research in current time, owing to its applications in the fields of engineering and sciences. The structure of Petri nets is a directed bipartite graph. They can be used as a graphical as well as a mathematical tool. As a graphical tool, they are easier to understand and interactive in nature while as a mathematical tool, they can be used to formulate state and algebraic equations for easier calculation and analysis. The notion of Petri net was discovered by Carl Adam Petri at a mere age of 13 to describe chemical processes. More formally, it was described in his thesis "Communication with Automata" in 1962, submitted to the Science Faculty of Darmstadt Technical University. Due to their dynamic nature, Petri nets soon became useful in xi xii CONTENTS modeling of asynchronous, distributed, concurrent, parallel, nondeterministic, and/or stochastic systems. Thus, various extensions to Petri nets have been introduced viz. Continuous Petri nets, Stochastic Petri nets, Timed Petri nets, Object Petri nets, Hybrid Petri nets, Workflow nets, Fuzzy Petri nets, Lending Petri nets, Multidimensional Petri nets etc., in order to better incorporate the characteristics of the system to be modeled. The Petri nets and their extensions have been widely used in various fields. Logic Petri nets (LPNs) have been defined as high level Petri nets, to describe batch processing functions and passing value indeterminacy in cooperative systems. The thesis entitled “On Signed Petri Nets” contains seven chapters. Chapter 1 titled "General Introduction" provides a brief review of Petri net theory. It provides the contributions of various researchers who have extended the theory of Petri nets after its introduction by Carl Adam Petri. A brief survey of the Petri nets research is given. The various extensions and applications of Petri nets in some of the fields have been discussed in brief. Thus, this chapter builds up a background and motivation behind the thesis work along with the tools required to achieve the goals. Chapter 2 titled "Signed Petri net" describes the extension of Petri nets called Signed Petri net and the related terminology. The proposed concept is inspired from signed graphs and Petri nets and can be considered as an amalgamation of the two, utilizing the properties of both. The basic properties and terms associated with signed Petri nets are defined. The applications of signed Petri nets in message transmission system and production unit are discussed. Lastly, it is CONTENTS xiii shown how a Logic Petri nets can be simulated using a signed Petri nets by merely changing the execution rules of a signed Petri nets. These modified signed Petri nets are called a Logic signed Petri nets. These concepts clearly demonstrate the advantages of the proposed approach of signed Petri nets. This chapter is published as a research paper "An introduction to Signed Petri net, Journal of Mathematics, (2021)". In Chapter 3 titled "Analysis of Signed Petri nets", the work of previous chapter has been extended. Mere modeling of system is of no use unless the modeled system is interpreted, which led to the introduction of analysis techniques for analyzing signed Petri nets. Two techniques for analysis are provided: Reachability Tree and Matrix equations with main focus on matrix equations. An actual case scenario of a restaurant model is given and analyzed using the techniques given in the chapter. The benefits of using a signed Petri nets to model the restaurant system rather than using traditional Petri nets are also given. This chapter has been accepted with title "Analysis of Signed Petri net, International Journal of Computing Science and Mathematics, (2020)". In Chapter 4 titled "Structural Matrices for Signed Petri nets", several matrices which show the relationship between transitions and places have been introduced. Three different matrices are defined by different products of the adjacency matrix of signed Petri nets with its transpose and with itself. In fact, if all these matrices are given, the signed Petri net structure can be obtained after analyzing them. Such matrices are useful because while creating algorithms for various procedures and results, it is not possible to extract data from a graph xiv CONTENTS rather all the information can be provided in the form of matrices. The matrices are further utilized to find a directed cycle in signed Petri nets. Various subclasses of signed Petri nets along with their characterizations using the structural matrices have been introduced. This chapter is published with title "Structural Matrices for Signed Petri net, AKCE International Journal of Graphs and Combinatorics, (2022)". In Chapter 5 titled "Structural and Dynamical Balanceness in Signed Petri net", the concept of structural and dynamical balanceness have been introduced. A structurally balanced signed Petri net has been defined and its characterization is given. It is shown how the dynamical balanceness approach is advantageous in analyzing social interactions, since all the signed graphs (directed or undirected) can be simulated by firing of different sequence of transitions in a single signed Petri net. Also, dynamics associated with a system can be easily represented using signed Petri nets rather than a signed graph. The equivalence between balanced signed graphs and dynamically balanced signed Petri nets is established. This chapter comprises the result from the research papers "An introduction to Signed Petri net, Journal of Mathematics, (2021) and Social Interactions through Signed Petri net, Communicated". In Chapter 6 titled "Domination in Signed Petri nets", the concept of Domination has been introduced as such a concept doesn’t exist for the dynamic systems. It can be seen with the help of the applications of finding the highest and lowest ranking officials in an institute based on a certain activity, producer- consumer problem, searching of food by bees and finding similarity in research papers, how the proposed concept is CONTENTS xv beneficial. The work in this chapter is in the research paper "Domination in Signed Petri net, https://arxiv.org/abs/2001.04374, (2020)". Chapter 7 titled "Conclusion and Future Scope" concludes the thesis work and gives a future research plan. In the research work, an extension of Petri nets called as "Signed Petri nets" has been proposed. Various results have been obtained in our present work along with some real-life applications where the proposed research can be used. The authors are of the view that this extension has a great scope in the study of various dynamic systems as well as modeling real-life applications. In future, the thesis work will be extended by utilizing the signed Petri nets in modeling various scenarios and analyzing the modeled system. Lastly, the bibliography is given to appreciate those who have made it possible to understand and use the vast theory of Petri nets. A list of author’s publications is also given at the end of the thesis.