Category Theoretic Understandings of Universal Algebra and its Dual: Monads and Lawvere Theories, Comonads and What?

Behrisch, Mike; Kerkhoff, Sebastian; Power, John

doi:10.1016/j.entcs.2012.08.002

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…The study of algebraic theories and their role in modelling computing systems [30,3] is a recurring theme of John Power's research, and the subject of some of his most influential contributions. In a series of articles [49,51,34,25], he and others developed an enriched category theoretic generalisation of Lawvere theories and explored their applications, particularly in the study of computational effects of programming languages.…”

Section: Introductionmentioning

confidence: 99%

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

Bonchi¹,

Gadducci²,

Kissinger³

et al. 2021

Preprint

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Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thoughts as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

show abstract

Section: Introductionmentioning

confidence: 99%

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

Bonchi¹,

Gadducci²,

Kissinger³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The study of algebraic theories and their role in modelling computing systems (Behrisch et al 2012;Hyland and Power 2007) is a recurring theme of John Power's research, and the subject of some of his most influential contributions. In a series of articles (Garner and Power 2018;Lack and Power 2009;Power 1999Power , 2005, he and his coauthors developed an enriched category theoretic generalisation of Lawvere theories and explored their applications, particularly in the study of computational effects of programming languages.…”

Section: Introductionmentioning

confidence: 99%

String diagram rewrite theory II: Rewriting with symmetric monoidal structure

Bonchi

Gadducci

Kissinger

et al. 2022

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

show abstract

Category Theoretic Understandings of Universal Algebra and its Dual: Monads and Lawvere Theories, Comonads and What?

Cited by 2 publications

References 13 publications

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

String diagram rewrite theory II: Rewriting with symmetric monoidal structure

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