Hierarchical models for service-oriented systems

We present our approach to the denotation and representation of hierarchical graphs: a suitable algebra of hierarchical graphs and two domains of interpretations. Each domain of interpretation focuses on a particular perspective of the graph hierarchy: the top view (nested boxes) is based on a notion of embedded graphs while the side view (tree hierarchy) is based on gs-graphs. Our algebra can be understood as a high-level language for describing such graphical models, which are well suited for defining graphical representations of service-oriented systems where nesting (e.g. sessions, transactions, locations) and linking (e.g. shared channels, resources, names) are key aspects

Bisimilarity congruences for open terms and term graphs via tile logic

The definition of sos formats ensuring that bisimilarity on closed terms is a congruence has received much attention in the last two decades. For dealing with open terms, the congruence is usually lifted from closed terms by instantiating the free variables in all possible ways; the only alternatives considered in the literature are Larsen and Xinxin’s context systems and Rensink’s conditional transition systems. We propose an approach based on tile logic, where closed and open terms are managed uniformly, and study the ‘bisimilarity as congruence’ property for several tile formats, accomplishing different concepts of open system

Dependently-Typed Formalisation of Typed Term Graphs

We employ the dependently-typed programming language Agda2 to explore formalisation of untyped and typed term graphs directly as set-based graph structures, via the gs-monoidal categories of Corradini and Gadducci, and as nested let-expressions using Pouillard and Pottier's NotSoFresh library of variable-binding abstractions.Comment: In Proceedings TERMGRAPH 2011, arXiv:1102.226

Tiles for Concurrent and Located Calculi

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Free gs-monoidal categories and free Markov categories

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams ($=$term graphs) as a combinatorial data structure. We formulate the appropriate $2$-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.Comment: 35 pages. v2: references update

Deformation theory of representations of prop(erad)s

We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.Comment: Version 4 : Statement about the properad of (non-commutative) Frobenius bialgebras fixed in Section 4. [82 pages

Rewriting modulo symmetric monoidal structure

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory. An important role in many such approaches is played by equational theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation for this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure. We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids