🤖 AI Summary
Compositional systems lack a structured, rewriteable mathematical foundation. Method: This work proposes a category-theoretic rewriting framework wherein structured cospans serve as the fundamental syntactic units; it introduces, for the first time, the coupling of structured cospans with double-pushout (DPO) rewriting, yielding a unified theory supporting both traced and trace-free semantics. The framework enables inductive, structure-preserving decomposition of closed systems and establishes a sound correspondence between syntax (structured cospans) and semantics (DPO rewriting). Contribution/Results: It provides the first categorical integration of structured cospans with DPO rewriting; defines two distinct rewriting paradigms—traceable and trace-free; and delivers the first mathematically rigorous, compositional, and rewriteable foundation for systems science, enabling cross-disciplinary modeling and formal analysis of complex systems.
📝 Abstract
Author(s): Cicala, Daniel Michael | Advisor(s): Baez, John | Abstract: The concept of a system has proliferated through natural and social sciences. While myriad theories of systems exist, there is no mathematical general theory of systems. In this thesis, we take a first step towards formulating such a theory. Our focus is on developing a syntax for compositional systems equipped with a rewriting theory. We pull from category theory and linguistics to accomplish this. The basic syntactical unit is a structured cospan and rewriting is introduced via the double pushout method. Two versions of rewriting are proposed: one that tracks intermediate steps and another disregards them. Benefits and drawbacks of both versions are discussed. We apply our results to the decomposition of closed systems, obtaining a structurally inductive viewpoint of rewriting such systems.