Tue. Oct 22nd, 2019

A Matrix Vector Transition Net Implementation

Aims: Classic Petri nets also known as place transition nets provide many interesting and useful features for system modeling. They are however limited by the place types that are used. A novel approach is presented in this work. A matrix vector transition net model is created and is used to model complex system behavior. This solution extends the modeling power of normal Petri nets.

Proposed Solution: A traditional Petri net is modified to create a matrix vector transition net (MVTN).  The idea is to combine the ideas from normal Petri net semantics with a matrix vector approach.

Implementing the Matrix Vector Transition Net: Ordinary  Petri  net  places  are  replaced  with  matrices  or vectors.  The  input  and  output  arcs  must  have  a  specific  function  matrix  that  determines  firing.  Firing  and behavior remain conceptually and functionally similar to that of a Petri net. It is possible to interchange row and column vectors. The behavior of matrix transition nets must elicit similar behavior to that of a place transition net.  Instead  of  normal  tokens,  matrix  elements  are  used.  The  matrix  vector  type  of  structure  increases  the modeling power, abstraction capacity and the complexity of the net.

Case Study: To illustrate this work a toy case of an abstract network structure containing processing elements is used to illustrate the use of the matrix vector transition net structure.

Results and Findings: The behavior of matrix transition nets is shown to be similar in principle to that of a place transition net. However instead of tokens, matrix elements are used. It is possible to construct a symbolic marking  graph  or  reach ability  graph  for  the  system  This  type  of  structure  definitely  increases  the  modeling power, abstraction capacity and the complexity of the net. The matrix transition net could be useful for certain types of communication system problems and complex system interfacing.  Several other uses can be found for this approach in both computing and modeling.

Keywords: Complex systems; matrices and vectors; matrix vector transition net; modeling; Petri nets; colored Petri nets; symbolic modeling.

Conclusion

The main properties of the MVT nets along with the current availability of Petri net theory, give the possibility for constructing more complex models of modern information and computer systems or structures. Because of the strong influence, matrix theory has on normal Petri net representation and analysis, similar concepts have been considered for the MVTN net. The ability of matrices for expressing system composition and behavior, should not be overlooked. The use of matrices should follow intuitively, because matrices represent complexity using compaction. It might seem that all these concepts can be contained in CPNs however intrinsically these are  not  focused  on  matrix  representation  that  would  undoubtedly  simplify  certain  expressions  of  abstract behavior. The models developed can be enhanced and formalized. Extensions of the models can be developed and  more  complex combinations are  possible. The  visual  expressivity,  decomposability, solvability  has to be properly understood when constructing representational structures.

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