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Open Graphs and Computational Reasoning
"... We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of halfedges (edges which are drawn with an unconnected end) and enjo ..."
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We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of halfedges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these halfedges. In particular, this allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information. 1
Diagram rewriting and operads
, 2009
"... We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for prac ..."
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We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for practical computations, but also for theoretical results. Moreover, rewriting is strongly related to homotopy theory. For instance, it can be used to compute homological invariants of algebraic structures, or to prove coherence results.
Open Graphs and Monoidal Theories
, 2010
"... 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 interpret ..."
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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 opengraph. While the category of opengraphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor
PROs and diagram rewriting ∗
, 2010
"... We give a survey of a diagrammatic syntax for PROs and PROPs, which are related to the theory of bialgebras. Using diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. Except for Sections 4 and 7, most of the mater ..."
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We give a survey of a diagrammatic syntax for PROs and PROPs, which are related to the theory of bialgebras. Using diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. Except for Sections 4 and 7, most of the material presented here comes from [La03], which was inspired by [Bu93].