The algebra of open and interconnected systems
<p>Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks,...
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| Format: | Thèse |
| Langue: | English |
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2016
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| _version_ | 1826280339617611776 |
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| author | Fong, B |
| author2 | Coecke, B |
| author_facet | Coecke, B Fong, B |
| author_sort | Fong, B |
| collection | OXFORD |
| description | <p>Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of <em>components</em> with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger <em>network</em>. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor.</p> <p>The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.</p> |
| first_indexed | 2024-03-07T00:12:14Z |
| format | Thesis |
| id | oxford-uuid:79a23c8c-81a5-4cf1-a108-29ba7dfd8850 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:12:14Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | oxford-uuid:79a23c8c-81a5-4cf1-a108-29ba7dfd88502022-03-26T20:38:38ZThe algebra of open and interconnected systemsThesishttp://purl.org/coar/resource_type/c_db06uuid:79a23c8c-81a5-4cf1-a108-29ba7dfd8850Logic in computer scienceCategory theorySystem theoryEnglishORA Deposit2016Fong, BCoecke, B<p>Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of <em>components</em> with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger <em>network</em>. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor.</p> <p>The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.</p> |
| spellingShingle | Logic in computer science Category theory System theory Fong, B The algebra of open and interconnected systems |
| title | The algebra of open and interconnected systems |
| title_full | The algebra of open and interconnected systems |
| title_fullStr | The algebra of open and interconnected systems |
| title_full_unstemmed | The algebra of open and interconnected systems |
| title_short | The algebra of open and interconnected systems |
| title_sort | algebra of open and interconnected systems |
| topic | Logic in computer science Category theory System theory |
| work_keys_str_mv | AT fongb thealgebraofopenandinterconnectedsystems AT fongb algebraofopenandinterconnectedsystems |