## Petri-Nets: A primer for Programmers

Petri-nets are a graphical modeling tool that have been around for over 60 years, but despite their long history, they continue to be a relevant and useful tool for computer programming. Whether you're a seasoned developer or just starting out, understanding the basics of Petri-nets can greatly enhance your skills and capabilities.

In this blog post, we'll dive into what Petri nets are, how they work, and most importantly, why they're such a valuable tool for computer programming. From modeling complex systems to visualizing algorithms, Petri nets offer a unique perspective on the design and implementation of software. By the end of this post, you'll have a solid understanding of the usefulness of Petri nets in computer programming and how they can help you think about model-driven code in a new way.

## Elementary Petri Nets

Petri nets are graphical models that consist of several basic elements, including places, transitions, and arcs. These elements work together to represent and model a system, making Petri nets a useful tool for visualizing and analyzing the behavior of complex systems.

**Places**: A place in a Petri net is a container that holds tokens. Tokens are the basic units of information that flow through the system and are used to represent entities or objects in the modeled system. A place can be thought of as a buffer that holds tokens until they're needed.**Transitions**: A transition in a Petri net represents an action or event that takes place in the modeled system. When a transition is fired, it moves tokens from one or more places to one or more other places. Transitions represent the flow of control in the modeled system and can be thought of as gatekeepers that control the flow of tokens from one place to another.**Arcs**: Arcs connect places and transitions and represent the flow of tokens from places to transitions and from transitions to places. There are two types of arcs in Petri nets: input arcs and output arcs. Input arcs connect places to transitions, while output arcs connect transitions to places.

These are the three basic elements of an elementary Petri net. Understanding these elements and how they work together is the key to unlocking the power of Petri nets as a modeling tool. Petri nets are simple but powerful, and they provide a valuable tool for visualizing and analyzing complex systems.

## Petri-Nets Vs DFAs

Petri nets are a powerful tool for modeling workflows and they can model workflows that are more complex than deterministic finite automata (DFA) state machines. The key difference between Petri nets and DFA state machines is the level of expressiveness. While DFA state machines are limited to modeling simple, deterministic systems, Petri nets can model much more complex systems that involve concurrency, nondeterminism, and dynamic behavior.

**Concurrency**: Petri nets can model concurrent systems, where multiple activities occur simultaneously. This is a key advantage over DFA state machines, which can only model sequential systems.**Nondeterminism**: Petri nets can model nondeterministic systems, where the outcome of an action is uncertain. This is another advantage over DFA state machines, which can only model deterministic systems.**Dynamic behavior**: Petri nets can model dynamic behavior, where the structure of the system changes over time. This is another advantage over DFA state machines, which have a static structure.**Hierarchical modeling**: Petri nets can model hierarchical systems, where a system is composed of multiple sub-systems. This allows for a more modular approach to modeling complex systems.**Tokens:**Petri nets use tokens to represent entities or objects in the modeled system, which provides a way to model real-world systems with more accuracy and detail.

These features of Petri nets make them a more powerful tool for modeling complex workflows than DFAs. Petri nets provide a flexible and expressive way to model real-world systems and can be used to analyze and understand the behavior of these systems in a visual and intuitive way.

## Petri-Nets as Tensors

Petri nets can be represented as tensors, which is a mathematical representation that is commonly used in deep learning and artificial intelligence. Tensor representations are particularly useful for Petri nets because they allow for the use of deep learning techniques to analyze and model Petri nets.

A Petri net can be represented as a tensor by mapping the places, transitions, and arcs of the Petri net to tensors. The places are represented as matrices, the transitions are represented as vectors, and the arcs are represented as tensor operations.

In this representation, the places are used to store the state of the Petri net, the transitions represent the actions that change the state of the Petri net, and the arcs represent the flow of information between the places and transitions. The tensor operations can be used to simulate the behavior of the Petri net and analyze the system in real-time.

Using Petri nets represented as tensors has several advantages. It provides a way to use deep learning techniques to analyze and model Petri nets, making it possible to analyze complex systems and workflows in a way that was not previously possible. Additionally, tensor representations of Petri nets can be used to optimize and improve the performance of the system, making it possible to optimize the behavior of the Petri net in real-time.

## Program With Petri-Nets

As we have demonstrated, Petri nets are a powerful tool for modeling and analyzing complex systems, and their expressibility as tensors opens up new avenues for analysis and optimization of these systems. The ability to represent Petri nets as tensors makes it possible to use deep learning techniques to analyze and model these systems, making it possible to unlock new insights and improve the performance of complex workflows.

Visit github/pflow-dev to find the tools we have developed for Go, Lua, Js and Python.

Whether you're a researcher, developer, or simply interested in Petri nets, the pFlow-dev repository is a great place to start exploring the power and potential of this valuable modeling tool in several languages.