String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.
Graphs are often used for specification and reasoning, both formally and informally. They have both an appealing visual nature as well as the ability to naturally abstract structure. In this paper, we will focus on "string diagrams", the graphical structures that arise in monoidal theories. Well known examples include proof-nets in linear logic [Girard, 1996], Penrose's tensor notation [Penrose, 1971], Feynman diagrams, diagrammatic notations for logic circuits, and high level languages for quantum information processing [Coecke and Duncan, 2008]. A common feature of these graphical languages is that they can be understood as describing a computational process, and they support reasoning by manipulating the graphical presentation. However, such manipulation is both tedious and error prone to do by hand. In this paper, we address this difficulty by providing a generic, but also concrete and computable, account of graphical reasoning in monoidal-theories. Our long-term goal is to support automation for graphical reasoning about computational structures.
The main concept we introduce is a formal theory of open-graphs. Like graph-based drawings of circuits, the visual presentation of open-graphs consists of vertices connected by edges. Crucially, edges in an opengraph need not be attached to vertices. They may be unconnected at one or both ends, or even connected to themselves to form a “circle”. In terms of a computational process, the unconnected ends of edges represent the inputs and outputs of a process. A diagram in this graphical language is interpreted as a compound computation with vertices as the atomic operations and wires defining the flow of information. For example, an electronic circuit that defines the compound logical operation of an or-gate, using not-gates around an and-gate, can be drawn as:
Open-graphs have a rich compositional structure and a convenient algebraic language. We introduce methods for plugging graphs together, merging over common subgraphs, and cutting out pieces of a graph. Using these tools, we develop rewriting for open-graphs. In this regard, our formalism functions analogously to a type-system in a programming language: we ensure that the interface of a process is maintained by rewriting. In particular, we show that rewriting also has a compositional nature: the decomposition of graphs by cutting their edges enables rewriting to be performed in parallel on the separated components, with a guarantee that the separate rewritten parts can be recomposed appropriately. Moreover, the compositional properties of open-graphs allow rewrite rules themselves to be rewritten using the same machinery.
To formalise the process of rewriting, we use a well-behaved embedding of the category of open-graphs into its ambient category of typed graphs. This embedding is an instance of a more general notion which we introduce as selective adhesive functors. In particular, these functors reflect pushouts, so many results about pushouts in an adhesive category are true of so-called adhesive pushouts, i.e. the pushouts reflected by a selective adhesive functor.
We also parameterise the category of open-graphs by a graphical signature. This defines a collection of vertex and edge types and assigns to each vertex type its input and output types. We construct a type graph from such a signature and form the category of typed open-graphs by slicing over this type graph. Combined with a collection of graphical rules, these typed open-graphs provide a formal way to reason with a graphical theory of some algebraic or dynamical system. We demonstrate the generality of our construction by showing that typed open-graphs can be used to construct free symmetric monoidal categories, PROPs, and a wide range of more general monoidal theories. Unlike many other (topological) constructions for diagrammatic accounts of monoidal categories, our construction involves finite data. Thus our construction enables the development of software tools that work with graphical theories. In particular, it provides the basis for employing techniques from automated reasoning, such as completion-based methods [Knuth and Bendix, 1970], to mechanise working with string diagrams.
The rest of the paper is structured as follows. In section 2, we introduce and motivate graphical theories with boolean circuits and tensor networks. We also note key challenges in working with these systems using traditional graph-based methods. After reviewing some of these methods in section 3, we define selective adhesive functors in section 4. These give an abstract characterisation for categories that sit inside an ambient adhesive category, and inherit enough properties to support rewriting. We define open-graphs in section 5 and show that they have a selective adhesive functor into a slice category over Graph. In section 6, we demonstrate how open-graphs can be composed and decomposed, and use these operations for rewriting open-graphs in section
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