1-Safe Petri nets and special cube complexes: equivalence and applications

Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we proved that Thiagarajan's conjecture is true for regular event structures whose domains are principal filters of universal covers of (virtually) finite special cube complexes. In the current paper, we prove the converse: to any finite 1-safe Petri net $N$ one can associate a finite special cube complex ${X}_N$ such that the domain of the event structure $\mathcal{E}_N$ (obtained as the unfolding of $N$) is a principal filter of the universal cover $\widetilde{X}_N$ of $X_N$. This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that the monadic second order logic of a 1-safe Petri net is decidable if and only if its unfolding is grid-free. Our counterexample is the trace regular event structure $\mathcal{\dot E}_Z$ which arises from a virtually special square complex $\dot Z$. The domain of $\mathcal{\dot E}_Z$ is grid-free (because it is hyperbolic), but the MSO theory of the event structure $\mathcal{\dot E}_Z$ is undecidable

A Forward On-The-Fly Approach in Controller Synthesis of Time Petri Nets

P. Heidira, H. Boucheneb ?A Forward On-The-Fly Approach in Controller Synthesis of Time Petri Nets?, Petri Nets - Manufacturing and Computer Science, Pawel Pawlewski (Ed.), ISBN: 978-953-51-0700-2, InTech, August 2012

Improving the construction of the DBM over approximation of the state spce of real-time preemptive systems

We present in this paper an algorithm allowing an efficient computation of the tightest DBM over-approximation of the state space of preemptive systems modeled by using Time Petri Nets with inhibitor arcs. First of all, we propose an algorithm that reduces the effort of computing the tightest DBM over-approximated graph. For this effect, each class of this graph is expressed as a pair (M, D), where M is a marking and D is the system of all DBM inequalities even the redundant ones. We thereby make it possible to compute the system D straightforwardly in its normal form, without requiring to compute the intermediary polyhedra. Hence, we succeed to remove the errors reported in the implementation of other DBM approximations. Then we show that by relaxing a bit in the precision of the DBM approximation, we can achieve to construct more compact graphs while reducing still more the cost of their computation. We provide for this abstraction a suitable equivalence relation that contract yet more the graphs. The experimental results comparing the defined constructions with other approaches are reported

Compositional approach to design of digital circuits

PhD ThesisIn this work we explore compositional methods for design of digital circuits with the aim of improving existing methodoligies for desigh reuse. We address compositionality techniques looking from both structural and behavioural perspectives. First we consider the existing method of handshake circuit optimisation via control path resynthesis using Petri nets, an approach using structural composition. In that approach labelled Petri net parallel composition plays an important role and we introduce an improvement to the parallel composition algorithm, reducing the number of redundant places in the resulting Petri net representations. The proposed algorithm applies to labelled Petri nets in general and can be applied outside of the handshake circuit optimisation use case. Next we look at the conditional partial order graph (CPOG) formalism, an approach that allows for a convenient representation of systems consisting of multiple alternative system behaviours, a phenomenon we call behavioural composition. We generalise the notion of CPOG and identify an algebraic structure on a more general notion of parameterised graph. This allows us to do equivalence-preserving manipulation of graphs in symbolic form, which simplifies specification and reasoning about systems defined in this way, as displayed by two case studies. As a third contribution we build upon the previous work of CPOG synthesis used to generate binary encoding of microcontroller instruction sets and design the corresponding instruction decoder logic. The proposed CPOG synthesis technique solves the optimisation problem for the general case, reducing it to Boolean satisfiability problem and uses existing SAT solving tools to obtain the result.This work was supported by a studentship from Newcastle University EECE school, EPSRC grant EP/G037809/1 (VERDAD) and EPSRC grant EP/K001698/1 (UNCOVER). i

On the Semantics of Petri Nets: Processes, Unfoldings and Infinite Computations.

PhD Thesi

A Forward On-The-Fly Approach for Safety and Reachability Controller Synthesis of Timed Systems

RÉSUMÉ Cette thèse s’intéresse à la synthèse de contrôleurs pour des systèmes temps réel (systèmes temporisés). Partant d’un système temps réel modélisé par un réseau de Petri temporel composé de transitions contrôlables et non contrôlables (TPN), le contrôle vise à forcer, en restreignant les intervalles de franchissement des transitions contrôlables, le système à satisfaire les propriétés souhaitées. Nous proposons, dans cette thèse, un algorithme pour synthétiser de tels contrôleurs pour des propriétés de sûreté et d’accessibilité. Cet algorithme, basé sur la méthode de graphe de classes d’états, calcule à la volée les classes d’états atteignables du TPN tout en collectant progressivement les sous-intervalles de tir à éviter, afin de satisfaire les propriétés souhaitées. Avec cet algorithme, il n’est plus nécessaire de calculer les prédécesseurs contrôlables et de partitionner récursivement les classes d’états jusqu’à atteindre un point fixe, comme c’est le cas dans les autres approches basées sur l’exploration, en avant et en arrière, de l’espace des états du système. Nous prouvons formellement la correction de l’algorithme, puis nous montrons que dans la catégorie des contrôleurs basés sur la restriction des intervalles de tir, l’algorithme, proposé dans cette thèse, synthétise un contrôleur optimal (le plus permissif possible). Afin d’atténuer davantage le problème d’explosion combinatoire, nous montrons comment combiner cette approche avec une abstraction par l’inclusion, par union-convexe ou par enveloppe-convexe. Nous montrons également comment exploiter cet algorithme pour générer des contrôleurs décentralisés. Enfin, nous proposons d’appliquer cet algorithme pour contrôler des TPN par des chronomètres. Notre algorithme permet de partitionner les intervalles des transitions en “bons” et “mauvais” sous-intervalles (à éviter). L’idée est d’utiliser des chronomètres pour suspendre les tâches (transitions) durant leurs mauvais sous-intervalles et les activer dans leurs “bons sous-intervalles”. Il s’agit donc de contrôler les réseaux de Petri temporels en associant des chronomètres aux transitions contrôlables, pour obtenir ainsi des réseaux de Petri temporels contrôlés.----------ABSTRACT This thesis deals with controller synthesis for real time systems (timed systems). Given a real time system modeled as a Time Petri Net (TPN) with controllable and uncontrollable transitions, the control aims at forcing the system to satisfy properties of interest, by limiting the firing intervals of controllable transitions. We propose, in this thesis, an algorithm to synthesize such controllers for safety / reachability properties. This algorithm, based on the state class graph method, computes on-the-fly the reachable state classes of the TPN while collecting progressively firing subintervals to be avoided so that the property is satisfied. It does not need to compute controllable predecessors and then split state classes until reaching a fixpoint, as it is the case for other approaches based on backward and forward exploration of state space of the system. We prove formally the correctness of the algorithm and show that, in the category of state dependent controllers based on the restriction of firing intervals, the algorithm proposed in this thesis, synthesizes maximally permissive controllers. In order to attenuate the state explosion problem, we show how to combine efficiently this approach with an abstraction by inclusion, convex union or convex hull. Afterwards, we discuss the compatibility of this method with distributed systems and decentralized controllers. Finally, we apply this algorithm to control TPN with controllable and uncontrollable transitions by stopwatch. In this approach, we find the subintervals violating the given properties and our objective is to suspend the tasks (transitions) during their bad subintervals and to resume them later. The controller is synthesized through the same algorithm already introduced. In this approach, we suggest to control time Petri nets by associating stopwatches to controllable transitions and to achieve a controlled time Petri nets

Proof Theoretic Concepts for the Semantics of Types and Concurrency

We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. The proof of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions)

Third Workshop on Modelling of Objects, Components, and Agents

This booklet contains the proceedings of the Third International Workshop on Modelling of Objects, Components, and Agents (MOCA'04), October 11-13, 2004. The workshop is organised by the CPN group at the Department of Computer Science, University of Aarhus, Denmark and the "Theoretical Foundations of Computer Science" group at the University of Hamburg. The home page of the workshop is: http://www.daimi.au.dk/CPnets/workshop0