Papers

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.

This thesis details the culmination of a project to define a fully compositional theory of synchronous sequential circuits built from primitive components, motivated by applying techniques successfully used in programming languages to hardware. The first part of the thesis defines the syntactic foundations required to create sequential circuit morphisms, and then builds three different semantic theories on top of this: denotational, operational and algebraic. We characterise the denotational semantics of sequential circuits as certain causal stream functions, as well as providing a link to existing circuit methodologies by mapping between circuit morphisms, stream functions and Mealy machines. The operational semantics is defined as a strategy for applying some global transformations followed by local reductions in order to demonstrate how a circuit processes a value, leading to a notion of observational equivalence. The algebraic semantics consists of equations for bringing circuits into a pseudo-normal form, and then encoding between different state sets. This part of the thesis concludes with a discussion of some novel applications, such as those for using partial evaluation for digital circuits. While mathematically rigorous, the categorical string diagram formalism is not suited for reasoning computationally. The second part of this thesis details an extension of existing work on string diagram rewriting with hypergraphs so that it is compatible with the traced comonoid structure present in the category of digital circuits. We identify the properties that characterise cospans of hypergraphs corresponding to traced comonoid terms, and demonstrate how to identify rewriting contexts valid for rewriting modulo traced comonoid structure. We apply the graph rewriting framework to fixed point operators as well as the operational semantics from the first part, and present a new hardware description language based on these theoretical developments.

We adapt the existing work on rewriting string diagrams using hypergraphs in order to apply it to settings with a traced comonoid structure, which happens to be where we model digital circuits.

We model digital circuits with delay and feedback as morphisms in a symmetric traced monoidal category, formalise their semantics as stream functions with certain properties, and present equational theory for reasoning with them.

We examine a variant of hypergraphs with the aim of creating a sound and complete graphical language for symmetric traced monoidal categories.

We detail the development of a set of tools to aid in the research of the topological properties of linear λ-terms when they are represented as 3-valent rooted maps.