Reactive Systems over Cospans

The theory of reactive systems, introduced by Leifer and Milner and previously extended by the authors, allows the derivation of well-behaved labelled transition systems (LTS) for semantic models with an underlying reduction semantics. The derivation procedure requires the presence of certain colimits (or, more usually and generally, bicolimits) which need to be constructed separately within each model. In this paper, we offer a general construction of such bicolimits in a class of bicategories of cospans. The construction sheds light on as well as extends Ehrig and Konig’s rewriting via borrowed contexts and opens the way to a unified treatment of several applications

Variable binding, symmetric monoidal closed theories, and bigraphs

This paper investigates the use of symmetric monoidal closed (SMC) structure for representing syntax with variable binding, in particular for languages with linear aspects. In our setting, one first specifies an SMC theory T, which may express binding operations, in a way reminiscent from higher-order abstract syntax. This theory generates an SMC category S(T) whose morphisms are, in a sense, terms in the desired syntax. We apply our approach to Jensen and Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at Concur '0

Decidability and Expressiveness of Finitely Representable Recognizable Graph Languages

Recognizable graph languages are a generalization of regular (word) languages to graphs (as well as arbitrary categories). Recently automaton functors were proposed as acceptors of recognizable graph languages. They promise to be a useful tool for the verification of dynamic systems, for example for invariant checking. Since automaton functors may contain an infinite number of finite state sets, one must restrict to finitely representable ones for implementation reasons. In this paper we take into account two such finite representations: primitive recursive automaton functors - in which the automaton functor can be constructed on-the-fly by a primitive recursive function -, and bounded automaton functors - in which the interface size of the graphs (cf. path width) is bounded, so that the automaton functor can be explicitly represented. We show that the language classes of both kinds of automaton functor are closed under boolean operations, and compare the expressiveness of the two paradigms with hyperedge replacement grammars. In addition we show that the emptiness and equivalence problem are decidable for bounded automaton functors, but undecidable for primitive recursive automaton functors

Application Conditions for Reactive Systems with Applications to Bisimulation Theory

This paper presents generalized application conditions (GACs), a new formalism for nested application conditions. GACs are not only suitable for DPO rewriting, but for rewriting in reactive systems as well. The main theorem states that it is possible to construct an equivalent reactive system rule with a GAC for a DPO rule with application conditions under very mild conditions. The resulting reactive system rules live in the cospan category of the category C, in which the DPO rules live. It turns out that these GACs for reactive systems provide a slightly more powerful way to control the application of a rewriting rule, than it is possible in the original DPO setting. At the end, we give a short outlook on the applications of this formalism to the field of bisimulation theory, sketch our latest results and discuss future work

Full Semantics Preservation in Model Transformation – A Comparison of Proof Techniques

Model transformation is a prime technique in modern, model-driven software design. One of the most challenging issues is to show that the semantics of the models is not affected by the transformation. So far, there is hardly any research into this issue, in particular in those cases where the source and target languages are different.\ud \ud In this paper, we are using two different state-of-the-art proof techniques (explicit bisimulation construction versus borrowed contexts) to show bisimilarity preservation of a given model transformation between two simple (self-defined) languages, both of which are equipped with a graph transformation-based operational semantics. The contrast between these proof techniques is interesting because they are based on different model transformation strategies: triple graph grammars versus in situ transformation. We proceed to compare the proofs and discuss scalability to a more realistic setting.\u

Treewidth, Pathwidth and Cospan Decompositions

We will revisit the categorical notion of cospan decompositions of graphs and compare it to the well-known notions of path decomposition and tree decomposition from graph theory. More specifically, we will define several types of cospan decompositions with appropriate width measures and show that these width measures coincide with pathwidth and treewidth. Such graph decompositions of small width are used to efficiently decide graph properties, for instance via graph automata

Conditional Reactive Systems

We lift the notion of nested application conditions from graph transformation systems to the general categorical setting of reactive systems as defined by Leifer and Milner. This serves two purposes: first, we enrich the formalism of reactive systems by adding application conditions for rules; second, it turns out that some constructions for graph transformation systems (such as computing weakest preconditions and strongest postconditions and showing local confluence by means of critical pair analysis) can be done very elegantly in the more general setting

On the Construction of Sorted Reactive Systems

We develop a theory of sorted bigraphical reactive systems. Every application of bigraphs in the literature has required an extension, a sorting, of pure bigraphs. In turn, every such application has required a redevelopment of the theory of pure bigraphical reactive systems for the sorting at hand. Here we present a general construction of sortings. The constructed sortings always sustain the behavioural theory of pure bigraphs (in a precise sense), thus obviating the need to redevelop that theory for each new application. As an example, we recover Milner’s local bigraphs as a sorting on pure bigraphs. Technically, we give our construction for ordinary reactive systems, then lift it to bigraphical reactive systems. As such, we give also a construction of sortings for ordinary reactive systems. This construction is an improvement over previous attempts in that it produces smaller and much more natural sortings, as witnessed by our recovery of local bigraphs as a sorting

An Algebra for Directed Bigraphs

We study the algebraic structure of directed bigraphs, a bigraphical model of computations with locations, connections and resources previously introduced as a unifying generalization of other variants of bigraphs. We give a sound and complete axiomatization of the (pre)category of directed bigraphs. Using this axiomatization, we give an adequate encoding of the Fusion calculus, showing the utility of the added directnes

G-Reactive Systems as Coalgebras

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