Rewriting in Free Hypergraph Categories

Zanasi, Fabio

doi:10.4204/eptcs.263.2

Cited by 4 publications

(3 citation statements)

References 35 publications

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“…Hypergraphs have been used before as a graphical language for compact closed categories, with interfaces defined via a cospan structure [3,26,5]. The cospan structure fits naturally in this case as the hypergraphs come equipped with a Frobenius monoid which allows vertices to be identified arbitrarily.…”

Section: Related Workmentioning

confidence: 99%

Rewriting Graphically with Symmetric Traced Monoidal Categories

Kaye¹

2020

Preprint

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We examine a variant of hypergraphs that we call linear hypergraphs, with the aim of creating a sound and complete graphical language for symmetric traced monoidal categories (STMCs). We first define the category of linear hypergraphs as a full subcategory of conventional (simple) hypergraphs, in which each vertex is either the source or the target of exactly one edge. The morphisms of a freely generated STMC can be then interpreted as linear hypergraphs, up to isomorphism (soundness). Moreover, any linear hypergraph is the representation of a unique STMC morphism, up to the equational theory of the category (completeness). This establishes linear hypergraphs as the graphical language of STMCs. Linear hypergraphs are then shown to form a partial adhesive category which means that a broad range of equational properties of some STMC can be specified as a graph rewriting system. The graphical language of digital circuits is presented as a case study.

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Section: Related Workmentioning

confidence: 99%

Rewriting Graphically with Symmetric Traced Monoidal Categories

Kaye¹

2020

Preprint

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“…Indeed, even (one of) the earliest papers on string diagrams, namely that of Roger Penrose [Pen71], already introduces "nominal" string diagrams where the wires of his pictures are given labels. Amongst later works, a commonly seen variation to ordinary string diagrams is the notion of colored props [HR15,Zan17]. This generalisation from one-sorted to many-sorted PROPs is orthogonal to our generalisation to nominal PROPs.…”

Section: Introductionmentioning

confidence: 96%

Completeness of Nominal PROPs

Balco¹,

Kurz²

2023

Logical Methods in Computer Science

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We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.

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“…This can be made completely formal by interpreting string diagrams as (directed) hypergraphs with boundaries[2,21], where generators form hyperedges and wires connecting them are the nodes. In this context, reachability between wires (as in Definition 17 below) can be defined as the existence of a forward path between the corresponding nodes in the hypergraphs.…”

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confidence: 99%

Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits

Wilson

Zanasi

2021

Electron. Proc. Theor. Comput. Sci.

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We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.Reverse Derivative Ascent 2. We define Reverse Derivative Ascent as 'gradient'-based algorithm working for arbitrary morphisms of reverse differential categories.3. We can then apply Reverse Derivative Ascent to boolean circuits, and demonstrate its empirical value by giving experimental results for benchmark datasets.

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Rewriting in Free Hypergraph Categories

Cited by 4 publications

References 35 publications

Rewriting Graphically with Symmetric Traced Monoidal Categories

Rewriting Graphically with Symmetric Traced Monoidal Categories

Completeness of Nominal PROPs

Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits

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