Coreflective Concurrent Semantics for Single-Pushout Graph Grammars

Net processes correspond to derivation processes in graph grammars

Graph Rewriting Systems are a powerful formalism for the specification of parallel and distributed systems, and the corresponding theory is rich of results concerning parallelism and concurrency. I will review the main results of the theory of concurrency for the algebraic approach to graph rewriting, emphasizing the relationship with the theory of Petri nets. In fact, graph rewriting systems can be regarded as a proper generalization of Petri nets, where the current state of a system is described by a graph instead of by a collection of tokens. Recently, this point of view allowed for the generalization to graph rewriting of some interesting results and constructions of the concurrent semantics of nets, including processes, unfoldings, and categorical semantics based on pair of adjoint functors.KeywordsDirect DerivationGraph TransformationGraph GrammarAdjoint FunctorHost GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Place/transition (PT) Petri nets are one of the most widely used models of concurrency. However, they still lack, in our view, a satisfactory semantics: on the one hand the ‘token game’ is too intensional, even in its more abstract interpretations in terms of nonsequential processes and monoidal categories; on the other hand, Winskel's basic unfolding construction, which provides a coreflection between nets and finitary prime algebraic domains, works only for safe nets. In this paper we extend Winskel's result to PT nets. We start with a rather general category PTNets of PT nets, we introduce a category DecOcc of decorated (nondeterministic) occurrence nets and we define adjunctions between PTNets and DecOcc and between DecOcc and Occ, the category of occurrence nets. The role of DecOcc is to provide natural unfoldings for PT nets, i.e., acyclic safe nets where a notion of family is used to relate multiple instances of the same place. The unfolding functor from PTNets to Occ reduces to Winskel's when restricted to safe nets. Moreover, the standard coreflection between Occ and Dom, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions between PTNets and Dom.

In the area of specification and modelling of concurrent systems, Petri nets have become a standard tool, and they still work behind the scenes in tools for graph transformation systems (cf. [1]). Moreover there is still potential for crossfertilization between the graph transformation and Petri net community. Even a better understanding of adhesive categories [2] and the related concepts of [3] seems possible in light of the proposed notion of weakly adhesive categories [4], which has emerged during work on the generalization of the co-reflective semantics of Petri nets to the realm of graph transformation and adhesive rewriting systems.KeywordsType ObjectGraph TransformationConcurrent SystemGraph GrammarGraph Transformation SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we deal with the problem of critical observability for discrete event systems modeled by labeled safe Petri nets (PNs). Critical observability is a property originated from the safety-critical applications of cyber-physical systems. For the purpose to check this property of a PN model, it is necessary to detect whether the current state of the net system is, or is not in a set of critical states representing dangerous operations. The main results of the work is to propose a necessary and sufficient condition for checking the critical observability in safe PNs when the set of critical states is modeled by an arbitrary set of reachable markings. The proposed method exploits the solution of integer linear programming problems. Finally, several examples are discussed to demonstrate the efficiency of the proposed approach.

A method to find the minimized state machine cover of live and safe Petri nets (PNs) is proposed in this paper. It is required that a cover be obtained in some implementation methods of PN-based control algorithms and other applications such as data mining. Most of the known methods of its computation have at least exponential time and space complexity; this is true not only for the methods which guarantee a minimal cover is obtained but also for some methods of finding approximate solutions. The proposed algorithm is an approximation algorithm and has polynomial computational complexity. The experimental verification shows a very high efficiency of the presented technique. The proposed method is especially valuable in the case of large systems where a solution is obtained hundreds of times quicker than by other methods.

In this paper we propose a technique for verification of mobile systems. We translate finite control processes, which are a well-known subset of π-Calculus, into Petri nets, which are subsequently used for model checking. This translation always yields bounded Petri nets with a small bound, and we develop a technique for computing a non-trivial bound by static analysis. Moreover, we introduce the notion of safe processes, which are a subset of finite control processes, for which our translation yields safe Petri nets, and show that every finite control process can be translated into a safe one of at most quadratic size. This gives a possibility to translate every finite control process into a safe Petri net, for which efficient unfolding-based verification is possible. Our experiments show that this approach has a significant advantage over other existing tools for verification of mobile systems in terms of memory consumption and runtime.Keywordsfinite control processessafe processesπ-Calculusmobile systemsmodel checkingPetri net unfoldings

Non-interference characterises the absence of undesired information flows in a computing system, by asking that actions with higher level of confidentiality do not cause any observable effect at the lower levels. In many concrete applications, this requirement is too strict and the abstract model is enriched with some form of downgrading, namely with the possibility of declassifying information, thus allowing for a controlled form of leakage. This paper focuses on BINI (Bisimilarity-based Intransitive non-interference), a formalisation of non-interference with downgrading in the setting of Petri nets. Generalising some previous works, we provide a causal characterisation of BINI in terms of the unfolding semantics, a true concurrent semantics of Petri nets. Building on this, we design an algorithm for checking BINI on safe Petri nets which relies on the construction of suitable complete prefixes of the unfolding. The algorithm is implemented in a prototype tool and some preliminary tests are quite encouraging as they suggest that the management of downgrading does not cause any significant performance decay.

Firstly, net theory, which is a well established theory of distributed systems and processes [I] relies heavily on a graphical notation. One of the strengths of this theory is that many of the fundamental notions concerning concurrent systems and in particular, the dynamics of a concurrent systemcan be made transparent with the help of simple net diagrams and transformations of such diagrams. It is important to provide a solid formal foundation to this aspect of net theory and graph grammars appear to be the obvious tool for doing so. The second motivation is that in the study of concurrency problems that arise within the area of graph grammars [2] we think it is possible and fruitful to apply the concepts and techniques of net theory.

Petri nets have been proposed initially to model concurrency and distributed systems. The ordinary formalism is so simple with a smart formal background, but it fails in modeling mobility aspects. In spite of this, searchers in mobility modeling domain are attracted by Petri nets, and many extensions have been proposed to treat mobility aspects. Among these, we find Labeled Reconfigurable Nets (LRN), This new formalism allows explicit modeling of computational environments and processes mobility between them. It allows, in a simple and an intuitive approach, modeling mobile code paradigms (mobile agent, code on demand, remote evaluation). In this paper, we propose an approach based on the combined use of Meta-modeling and Graph Grammars to automatically generate a visual modeling tool for LRN, for analysis and simulation purposes. This tool produces graphical models, highly-structured to reach this goal. In our approach, the UML Class diagram formalism is used to define a meta-model of LRN. The meta-modeling tool ATOM3 is used to generate a visual modeling tool according to the proposed LRN meta-model. To generate R-Maude specification of the graphically specified LRN models, we have proposed a graph grammar. Then the reconfigurable rewriting logic language R-Maude can be used to perform the simulation of the resulted R-Maude specification. Our approach is illustrated through examples.

Relating SPO and DPO graph rewriting with Petri nets having read, inhibitor and reset arcs

In the framework of graph transformation systems with Negative Application Conditions (NACs) the classical notion of switch equivalence of derivations is extended to permutation equivalence, because there are intuitively equivalent derivations which are not switch-equivalent if NACs are considered. By definition, two derivations are permutation-equivalent, if they respect the NACs and disregarding the NACs they are switch equivalent. A direct analysis of permutation equivalence is very complex in general, thus we propose a much more efficient analysis technique. For this purpose, we construct a Place/Transition Petri net, called dependency net, which encodes the dependencies among rule applications of the derivation, including the inhibiting effects of the NACs. The analysis of permutation equivalence is important for analysing simulation runs within development environments for systems modelled by graph transformation. The application of the technique is demonstrated by a graph transformation system within the context of workflow modelling. We show the effectiveness of the approach by comparing the minimal costs of a direct analysis with the costs of the efficient analysis applied to a derivation of our example system.

Petri nets generating all the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> binary n-vectors as their marking vectors are not only of theoretical interest but also are of practical importance. In this note, we demonstrate the existence of a disconnected 1-safe Petri net whose reachability tree is homomorphic to the n-dimensional complete lattice L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . This makes the problem of characterizing the 1-safe Petri nets that generate all the binary n-vectors as marking vectors exactly once appear more intricate.

A definition of “action refinement” is given for an operational model of concurrent processes based on safe Petri Nets, generalizing previous work of Vogler and van Glabbeek/Goltz. A failure-style denotational semantics is described for process nets. The semantics is fully abstract for Hennessy Testing-equivalence on nets acting as refinement operators as well as operands. The semantics embodies the notions of deadlock, failures and divergences found in the Hoare/CSP and Hennessy Testing-equivalence theories, as well as some of the basic ideas of “pomset runs” and “causal” partial orders of Net theory.

A set of necessary conditions for a Petri net to be plain, pure and safe is given. Some applications of these conditions both in practice (for Petri net synthesis), and in theory (e.g., as part of a characterisation of the reachability graphs of live and safe marked graphs) are described.