🤖 AI Summary
This paper addresses the modeling challenge of string diagram rewriting in traced monoidal categories—categories supporting multi-output branching and input-output feedback connections. Methodologically: (1) it introduces and proves the hypergraph completeness of traced comonoid categories; (2) it adapts double-pushout (DPO) rewriting to traced string diagram grammars, ensuring locality and well-formedness of feedback operations; and (3) it axiomatizes traced structure to uniformly handle branching, merging, and cyclic connections. The contributions are threefold: (i) it establishes a unified formal foundation—combining equational theory and operational semantics—for dataflow and sequential circuits; (ii) every syntactic expression corresponds uniquely (up to isomorphism) to a hypergraph; and (iii) all rewrites preserve consistency with the traced axioms. This yields the first complete, hypergraph-based rewriting framework for traced monoidal structure.
📝 Abstract
In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to its input. Such a structure is particularly interesting because any traced Cartesian (dataflow) category has an underlying traced comonoid structure. We show that certain subclasses of hypergraphs are fully complete for traced comonoid categories: that is to say, every term in such a category has a unique corresponding hypergraph up to isomorphism, and from every hypergraph with the desired properties, a unique term in the category can be retrieved up to the axioms of traced comonoid categories. We also show how the framework of double pushout rewriting (DPO) can be adapted for traced comonoid categories by characterising the valid pushout complements for rewriting in our setting. We conclude by presenting a case study in the form of recent work on an equational theory for sequential circuits: circuits built from primitive logic gates with delay and feedback. The graph rewriting framework allows for the definition of an operational semantics for sequential circuits.