Rule Algebras for Adhesive Categories

We demonstrate that the most well-known approach to rewriting graphical structures, the Double-Pushout (DPO) approach, possesses a notion of sequential compositions of rules along an overlap that is associative in a natural sense. Notably, our results hold in the general setting of $\mathcal{M}$-adhesive categories. This observation complements the classical Concurrency Theorem of DPO rewriting. We then proceed to define rule algebras in both settings, where the most general categories permissible are the finitary (or finitary restrictions of) $\mathcal{M}$-adhesive categories with $\mathcal{M}$-effective unions. If in addition a given such category possess an $\mathcal{M}$-initial object, the resulting rule algebra is unital (in addition to being associative). We demonstrate that in this setting a canonical representation of the rule algebras is obtainable, which opens the possibility of applying the concept to define and compute the evolution of statistical moments of observables in stochastic DPO rewriting systems

Interacting Hopf Algebras

We introduce the theory IH of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IH are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid-comonoid pairs. This construction is instrumental in showing that IH is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R

The Axiom of Choice in Cartesian Bicategories

We argue that cartesian bicategories, often used as a general categorical algebra of relations, are also a natural setting for the study of the axiom of choice (AC). In this setting, AC manifests itself as an inequation asserting that every total relation contains a map. The generality of cartesian bicategories allows us to separate this formulation from other set-theoretically equivalent properties, for instance that epimorphisms split. Moreover, via a classification result, we show that cartesian bicategories satisfying choice tend to be those that arise from bicategories of spans

Graphical Conjunctive Queries

The Calculus of Conjunctive Queries (CCQ) has foundational status in database theory. A celebrated theorem of Chandra and Merlin states that CCQ query inclusion is decidable. Its proof transforms logical formulas to graphs: each query has a natural model - a kind of graph - and query inclusion reduces to the existence of a graph homomorphism between natural models. We introduce the diagrammatic language Graphical Conjunctive Queries (GCQ) and show that it has the same expressivity as CCQ. GCQ terms are string diagrams, and their algebraic structure allows us to derive a sound and complete axiomatisation of query inclusion, which turns out to be exactly Carboni and Walters\u27 notion of cartesian bicategory of relations. Our completeness proof exploits the combinatorial nature of string diagrams as (certain cospans of) hypergraphs: Chandra and Merlin\u27s insights inspire a theorem that relates such cospans with spans. Completeness and decidability of the (in)equational theory of GCQ follow as a corollary. Categorically speaking, our contribution is a model-theoretic completeness theorem of free cartesian bicategories (on a relational signature) for the category of sets and relations

Connector algebras for C/E and P/T nets interactions

A quite fourishing research thread in the recent literature on component based system is concerned with the algebraic properties of different classes of connectors. In a recent paper, an algebra of stateless connectors was presented that consists of five kinds of basic connectors, namely symmetry, synchronization, mutual exclusion, hiding and inaction, plus their duals and it was shown how they can be freely composed in series and in parallel to model sophisticated "glues". In this paper we explore the expressiveness of stateful connectors obtained by adding one-place buffers or unbounded buffers to the stateless connectors. The main results are: i) we show how different classes of connectors exactly correspond to suitable classes of Petri nets equipped with compositional interfaces, called nets with boundaries; ii) we show that the difference between strong and weak semantics in stateful connectors is reflected in the semantics of nets with boundaries by moving from the classic step semantics (strong case) to a novel banking semantics (weak case), where a step can be executed by taking some "debit" tokens to be given back during the same step; iii) we show that the corresponding bisimilarities are congruences (w.r.t. composition of connectors in series and in parallel); iv) we show that suitable monoidality laws, like those arising when representing stateful connectors in the tile model, can nicely capture concurrency aspects; and v) as a side result, we provide a basic algebra, with a finite set of symbols, out of which we can compose all P/T nets, fulfilling a long standing quest

A categorical approach to open and interconnected dynamical systems

In his 1986 Automatica paper Willems introduced the influential behavioural approach to control theory with an investigation of linear time-invariant (LTI) discrete dynamical systems. The behavioural approach places open systems at its centre, modelling by tearing, zooming, and linking. We show that these ideas are naturally expressed in the language of symmetric monoidal categories.Our main result gives an intuitive sound and fully complete string diagram algebra for reasoning about LTI systems. These string diagrams are closely related to the classical notion of signal flow graphs, endowed with semantics as multi-input multi-output transducers that process discrete streams with an infinite past as well as an infinite future. At the categorical level, the algebraic characterisation is that of the prop of corelations.Using this framework, we derive a novel structural characterisation of controllability, and consequently provide a methodology for analysing controllability of networked and interconnected systems. We argue that this has the potential of providing elegant, simple, and efficient solutions to problems arising in the analysis of systems over networks, a vibrant research area at the crossing of control theory and computer science.<br/

Rule Algebras for Adhesive Categories

We demonstrate that the most well-known approach to rewriting graphical structures, the Double-Pushout (DPO) approach, possesses a notion of sequential compositions of rules along an overlap that is associative in a natural sense. Notably, our results hold in the general setting of $\mathcal{M}$-adhesive categories. This observation complements the classical Concurrency Theorem of DPO rewriting. We then proceed to define rule algebras in both settings, where the most general categories permissible are the finitary (or finitary restrictions of) $\mathcal{M}$-adhesive categories with $\mathcal{M}$-effective unions. If in addition a given such category possess an $\mathcal{M}$-initial object, the resulting rule algebra is unital (in addition to being associative). We demonstrate that in this setting a canonical representation of the rule algebras is obtainable, which opens the possibility of applying the concept to define and compute the evolution of statistical moments of observables in stochastic DPO rewriting systems

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

Refinement for Signal Flow Graphs

The symmetric monoidal theory of Interacting Hopf Algebras provides a sound and complete axiomatisation for linear relations over a given field. As is the case for ordinary relations, linear relations have a natural order that coincides with inclusion. In this paper, we give a presentation for this ordering by extending the theory of Interacting Hopf Algebras with a single additional inequation. We show that the extended theory gives rise to an abelian bicategory - a concept due to Carboni and Walters - and highlight similarities with the algebra of relations. Most importantly, the ordering leads to a well-behaved notion of refinement for signal flow graphs

Diagrammatic Algebra of First Order Logic

We introduce the calculus of neo-Peircean relations, a string diagrammatic extension of the calculus of binary relations that has the same expressivity as first order logic and comes with a complete axiomatisation. The axioms are obtained by combining two well known categorical structures: cartesian and linear bicategories