Introduction to Dialectical Nets

TL;DR

This paper introduces dialectical nets, a new generalization of traditional nets based on dialectical systems and the fundamental contradiction in closed preorders, unifying various concepts like Petri nets and transition systems.

Contribution

It proposes a novel dialectical approach to net theory, defining dialectical nets as a unifying framework for logical dynamics and transition systems.

Findings

Dialectical nets generalize Petri nets, Kan quantification, and transition systems.

They are defined via the fundamental contradiction in closed preorders.

Dialectical nets serve as a unifying concept for various net theories.

Abstract

This paper initiates the dialectical approach to net theory. This approach views nets as special, but very important and natural, dialectical systems. By following this approach, a suitably generalized version of nets, called dialectical nets, can be defined in terms of the "fundamental contradiction" inherent in the structure of closed preorders. Dialectical nets are the least conceptual upper bound subsuming the notions of Petri nets, Kan quantification and transition systems. The nature of dialectical nets is that of logical dynamics, and is succinctly defined and summarized in the statement that "dialectical nets are transition systems relativized to closed preorders, and hence are general predicate transformers".