Open Petri Nets

The reachability semantics for Petri nets can be studied using open Petri nets. For us an "open" Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri})$, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.Comment: 30 pages, TikZ figure

Nets, relations and linking diagrams

In recent work, the author and others have studied compositional algebras of Petri nets. Here we consider mathematical aspects of the pure linking algebras that underly them. We characterise composition of nets without places as the composition of spans over appropriate categories of relations, and study the underlying algebraic structures.Comment: 15 pages, Proceedings of 5th Conference on Algebra and Coalgebra in Computer Science (CALCO), Warsaw, Poland, 3-6 September 201

Interaction Systems and 1-safe Petri Nets

Interaction systems are a formal model for component-based systems, where components are combined via connectors to form more complex systems. We compare interaction systems (IS) to the wellstudied model of 1-safe Petri nets (1SN) by giving a translation map1: 1SN → IS and a translation map2: IS → 1SN, so that a 1-safe Petri net (an interaction system) and its according interaction system (1-safe Petri net) defined by the respective mapping are isomorphic up to some label relation R. So in some sense both models share the same expressiveness. Also, the encoding map1 is polynomial and can be used to reduce the problems of reachability, deadlock and liveness in 1SN to the problems of reachability, deadlock and liveness in IS, yielding PSPACE-hardness for these questions

A New Algorithm for Partitioned Symbolic Reachability Analysis

AbstractBinary Decision Diagrams (BDDs) and their multi-terminal extensions have shown to be very helpful for the quantitative verification of systems. Many different approaches have been proposed for deriving symbolic state graph (SG) representations from high-level model descriptions, where compositionality has shown to be crucial for the efficiency of the schemes. Since the symbolic composition schemes deliver the potential SG of a high-level model, one must execute a reachability analysis on the level of the symbolic structures. This step is the main resource of CPU-time and peak memory consumption when it comes to symbolic SG generation. In this work a new operator for zero-suppressed BDDs and their multi-terminal extensions for carrying out (partitioned) symbolic reachability analysis is presented. This algorithm not only replaces standard BDD-based schemes, it even makes symbolic composition as found in contemporary symbolic model checkers such as Prism and Caspa obsolete

A Compositional Approach for Schedulability Analysis of Distributed Avionics Systems

This work presents a compositional approach for schedulability analysis of Distributed Integrated Modular Avionics (DIMA) systems that consist of spatially distributed ARINC-653 modules connected by a unified AFDX network. We model a DIMA system as a set of stopwatch automata in UPPAAL to verify its schedulability by model checking. However, direct model checking is infeasible due to the large state space. Therefore, we introduce the compositional analysis that checks each partition including its communication environment individually. Based on a notion of message interfaces, a number of message sender automata are built to model the environment for a partition. We define a timed selection simulation relation, which supports the construction of composite message interfaces. By using assume-guarantee reasoning, we ensure that each task meets the deadline and that communication constraints are also fulfilled globally. The approach is applied to the analysis of a concrete DIMA system.Comment: In Proceedings MeTRiD 2018, arXiv:1806.09330. arXiv admin note: text overlap with arXiv:1803.1105

On Compositionality of Dinatural Transformations

Natural transformations are ubiquitous in mathematics, logic and computer science. For operations of mixed variance, such as currying and evaluation in the lambda-calculus, Eilenberg and Kelly\u27s notion of extranatural transformation, and often the even more general dinatural transformation, is required. Unfortunately dinaturals are not closed under composition except in special circumstances. This paper presents a new sufficient condition for composability. We propose a generalised notion of dinatural transformation in many variables, and extend the Eilenberg-Kelly account of composition for extranaturals to these transformations. Our main result is that a composition of dinatural transformations which creates no cyclic connections between arguments yields a dinatural transformation. We also extend the classical notion of horizontal composition to our generalized dinaturals and demonstrate that it is associative and has identities

Analysis of Mobile Agents using Invariants of Object Nets

Mobility induces new challenges for dynamic systems, which need a new conceptional treatment: systems, that deal for example with mobile agents, need extended security concepts to handle the risks, induced by foreign, untrusted agents. In this contribution we use object nets to model mobile systems. Object nets are Petri nets which have Petri nets as tokens – an approach known as the nets-withinnets paradigm. Object nets are called elementary if the net system has a two levelled structure. In this work we apply structural analysis methods for object nets – namely place invariants – to a simple case study modelling mobile agents