An Introduction to String Diagrams for Computer Scientists

This document is an elementary introduction to string diagrams. It takes a computer science perspective: rather than using category theory as a starting point, we build on intuitions from formal language theory, treating string diagrams as a syntax with its semantics. After the basic theory, pointers are provided to contemporary applications of string diagrams in various fields of science

A Finite Axiomatisation of Finite-State Automata Using String Diagrams

We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks.Comment: arXiv admin note: text overlap with arXiv:2009.1457

A String Diagrammatic Axiomatisation of Finite-State Automata

We develop a fully diagrammatic approach to the theory of finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. Moreover, we provide an equational theory that completely axiomatises language equivalence in this new setting. This theory has two notable features. First, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks. Second, the proposed axiomatisation is finitary -- a result which is provably impossible to obtain for the one-dimensional syntax of regular expressions.Comment: Minor corrections, in particular in the proof of completeness (including the ordering of the steps of Brzozowski's algorithm

String Diagram Rewriting Modulo Commutative Monoid Structure

We characterise freely generated props with a chosen commutative monoid structure as certain categories of hypergraphs with interfaces. We use this result to give a sound and complete interpretation of rewriting modulo commutative monoid equations in a prop in terms of double-pushout rewriting of hypergraphs

A Finite Axiomatisation of Finite-State Automata Using String Diagrams

We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks

Bialgebraic Semantics for String Diagrams

Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their bialgebraic semantics in terms of a distributive law over that monad. As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, \`a la Hoare and \`a la Milner.Comment: Accepted for publications in the proceedings of the 30th International Conference on Concurrency Theory (CONCUR 2019

Picturing resources in concurrency

Inspired by the pioneering work of Petri and the rise of diagrammatic formalisms to reason about networks of open systems, we introduce the resource calculus---a graphical language for distributed systems. Like process algebras, the resource calculus is modular, with primitive connectors from which all diagrams can be built. We characterise its equational theory by proving a full completeness result for an interpretation in the symmetric monoidal category of additive relations---a result that constitutes the main contribution of this thesis. Additive relations are frequently exploited by model-checking algorithms for Petri nets. In this thesis, we recognise them as a fundamental algebraic structure of concurrency and use them as an axiomatic framework. Surprisingly, the resource calculus has the same syntax as that of interacting Hopf algebras, a diagrammatic formalism for linear (time-invariant dynamical) systems. Indeed, the approach stems from the simple but fruitful realisation that, by replacing values in a field with values in the semiring of non-negative integers, concurrent behaviour patterns emerge. This change of model reflects the interpretation of diagrams as systems manipulating limited and discrete resources instead of continuous signals. We also extend the resource calculus in two orthogonal directions. First, by adding an affine primitive to express access to a constant quantity of resources. The extended calculus is remarkably expressive and allows the formulation of non-additive patterns of behaviour, like mutual exclusion. Once more, we characterise it---this time as the equational theory of the symmetric monoidal category of polyhedral relations, discrete analogues of polyhedra in convex geometry. Secondly, we add a synchronous register to model stateful systems. The stateful resource calculus is expressive enough to faithfully capture the behaviour of Petri nets while being strictly more expressive. It is also shown to axiomatise a category of open Petri nets, in the style of the connector algebras of nets with boundaries first studied by Bruni, Melgratti, Montanari and SobociÅski.</p

Picturing resources in concurrency

Inspired by the pioneering work of Petri and the rise of diagrammatic formalisms to reason about networks of open systems, we introduce the resource calculus---a graphical language for distributed systems. Like process algebras, the resource calculus is modular, with primitive connectors from which all diagrams can be built. We characterise its equational theory by proving a full completeness result for an interpretation in the symmetric monoidal category of additive relations---a result that constitutes the main contribution of this thesis. Additive relations are frequently exploited by model-checking algorithms for Petri nets. In this thesis, we recognise them as a fundamental algebraic structure of concurrency and use them as an axiomatic framework. Surprisingly, the resource calculus has the same syntax as that of interacting Hopf algebras, a diagrammatic formalism for linear (time-invariant dynamical) systems. Indeed, the approach stems from the simple but fruitful realisation that, by replacing values in a field with values in the semiring of non-negative integers, concurrent behaviour patterns emerge. This change of model reflects the interpretation of diagrams as systems manipulating limited and discrete resources instead of continuous signals. We also extend the resource calculus in two orthogonal directions. First, by adding an affine primitive to express access to a constant quantity of resources. The extended calculus is remarkably expressive and allows the formulation of non-additive patterns of behaviour, like mutual exclusion. Once more, we characterise it---this time as the equational theory of the symmetric monoidal category of polyhedral relations, discrete analogues of polyhedra in convex geometry. Secondly, we add a synchronous register to model stateful systems. The stateful resource calculus is expressive enough to faithfully capture the behaviour of Petri nets while being strictly more expressive. It is also shown to axiomatise a category of open Petri nets, in the style of the connector algebras of nets with boundaries first studied by Bruni, Melgratti, Montanari and Sobociński.</p

A Complete Diagrammatic Calculus for Boolean Satisfiability

We propose a calculus of string diagrams to reason about satisfiability ofBoolean formulas, and prove it to be sound and complete. We then showcase ourcalculus in a few case studies. First, we consider SAT-solving. Second, weconsider Horn clauses, which leads us to a new decision method forpropositional logic programs equivalence under Herbrand model semantics