Petri Net Recommender System for Generating of Perfect Binary Tree

Introduction

Petri net is one of the mathematical language tools used especially for the modelling of concurrent distributed discrete event systems (Murata, 1989; Thong et al., 2015). It was first defined by Carl Adam Petri in his dissertation (Petri, 1966). Since then lots of theories have been recommended (Bobadilla et al., 2013; Lü et al., 2012; Ricci et al., 2015) and are used to implement Petri net in many areas such as biological networks and pathways (Chaouiya, 2007; Frainay et al., 2018; Marwan et al., 2011; Wan & Che, 2014), health study (Gaur & Singh, 2016), social networks, computer science, mathematical science, drug designing (Alaimo et al., 2016), transportation, workflow and business process model (Hornung et al., 2009), network optimizations, information forecasting, music system (Katarya & Verma, 2018), system modelling (Murata, 1989; Reddy et al., 1993). In Petri net theory, reachability graphs and tree plays an important role because of its reachability property. In earlier studies, authors have constructed a 1-safe Petri net that generates all the binary n-vectors, possibly with repetitions and exactly once (Kansal et al., 2010; 2011). Besides that, the problem of characterizing the 1-safe Petri nets that generate all the binary n-vectors as marking vectors once (Kansal et al., 2015) have also been shown. Petri nets have been demonstrated as a family of formalisms which can be used sometimes with the advantage of improving the communication between stages of the life cycle in manufacture process (Silva & Teruel, 1997). Pal and Skowron have shown its main concepts and its characterizing features of rough set theory with its integration to fuzzy set theory, to develop an efficient soft computing strategy for machine learning (Pal & Skowron, 1999). Set-indexers of a graph and set-graceful graphs have also been developed and discussed (Acharya, 2001). Salimifard and Wright proposed the application of coloured Petri nets in the modelling of workflow management systems using an integration method to show the flow (Salimifard & Wright, 2001). It has been shown that fuzzy graph theory has an increasing application in the modelling real-time systems in which the level of information is inherent and varies with different levels of precision values (Pasi et al., 2004). Various methods and applications of Petri net for the modelling and analysis of molecular networks and system biology have been explained recently (Chaouiya, 2007; Koch, 2015). Class of finite and infinite binary trees have been used to study in many hierarchical structures (Clempner, 2014). In this article, a mathematical theorem-based method for Petri net recommender system to study and generate the levels and nodes of a perfect binary tree as the reachability tree is discussed. This type of study can be recommended to study model-based approach in decision science to explain the variables involved in decision-making processes (Cho et al., 2002; Bouza et al, 2008).