Harro Wimmel

A uniform approach to true-concurrency and interleaving semantics for Petri nets

This paper continues a research on universal contexts and semantics for Petri nets started by Nielsen, Priese and Sassone (1995). We consider generalized, labelled Petri nets N where some transitions and places are distinguished as public. They form the interface of N that may communicate with a Petri net context. An elementary calculus, E, is introduced in which one can construct any Petri net with an interface from trivial constants (single places, single transitions) by drawing arcs, adding tokens, and hiding public places and transitions. We prove the existence of a universal context U such that two Petri nets behave the same in any context if their behaviour is equal in the universal context. Let B(U[N]) be the behaviour of N embedded in its universal context, where B may be the interleaving language, step-language, or true-concurrent pomset-language. In any of these cases, B(U[N]) (in contrast to B(N)) turns out to be a compositional semantics of N with respect to the algebra E. In addition, the interleavingand step-semantics are just special cases of the true-concurrency pomset-semantics.

Report on the Model Checking Contest at Petri Nets 2011

This article presents the results of the Model Checking Contest held within the SUMo 2011 workshop, a satellite event of Petri Nets 2011. This contest aimed at a fair and experimental evaluation of the performances of model checking techniques applied to Petri nets.

Applying CEGAR to the Petri Net State Equation

We propose a reachability verification technique that combines the Petri net state equation (a linear algebraic overapproximation of the set of reachable states) with the concept of counterexample guided abstraction refinement. In essence, we replace the search through the set of reachable states by a search through the space of solutions of the state equation. We demonstrate the excellent performance of the technique on several real-world examples. The technique is particularly useful in those cases where the reachability query yields a negative result: While state space based techniques need to fully expand the state space in this case, our technique often terminates promptly. In addition, we can derive some diagnostic information in case of unreachability while state space methods can only provide witness paths in the case of reachability.

Reducing k-safe Petri nets to pomset-equivalent 1-safe Petri nets

It is a well-known fact that for every k-safe Petri net, i.e. a Petri net in which no place contains more than k ∈ N tokens under any reachable marking, there is a 1-safe Petri net with the same interleaving behaviour. Indeed these types of Petri nets generate regular languages. In this paper, we show that this equivalence of k-safe and 1-safe Petri nets holds also for their pomset languages, a true-concurrency semantics.

New algorithms for deciding the siphon-trap property

The siphon-trap property, also known as Commoner-Hack property, establishes a relation between structural entities within a Petri net – the eponymous siphons and traps. The property is linked to the behavior of a Petri net, for instance to deadlock freedom and liveness of the net. It is nevertheless nontrivial to decide the property as a net can have exponentially many siphons and traps even if only minimal siphons are considered. Consequently, the value of the property depends on the availability of powerful decision procedures. We contribute to this issue by proposing two new methods for deciding the siphon-trap property. One is a plain translation of the property into a Boolean satisfiability (SAT) problem, which exploits the fact that incredibly powerful SAT solvers are available. The second procedure has a divide-and-conquer nature which builds upon a decomposition of a Petri net into open nets and projects information about siphons and traps onto the interfaces of the components.

Separability in conflict-free petri nets

We study whether transition sequences that transform markings with multiples of a number k on each place can be separated into k sequences, each transforming one k-th of the original marking. We prove that such a separation is possible for marked graph Petri nets, and present an inseparable sequence for a free-choice net.

Applying CEGAR to the petri net state equation

We propose a reachability verification technique that combines the Petri net state equation (a linear algebraic overapproximation of the set of reachable states) with the concept of counterexample guided abstraction refinement. In essence, we replace the search through the set of reachable states by a search through the space of solutions of the state equation. We demonstrate the excellent performance of the technique on several real-world examples. The technique is particularly useful in those cases where the reachability query yields a negative result: While state space based techniques need to fully expand the state space in this case, our technique often terminates promptly. In addition, we can derive some diagnostic information in case of unreachability while state space methods can only provide witness paths in the case of reachability.

Algebraic Characterization of Petri Net Pomset Semantics

Sets of pomsets are frequently used as a true-concurrency, linear-time semantics of Petri nets. For a Petri net N, let P(N), a set of pomsets, denote the pomset behaviour of N, and let P:=P(N)|N is a Petri net} denote the class of pomset behaviours of Petri nets, a generalization of L, the class of all Petri net languages. We present here an algebraic characterization for P, similar to the known algebraic characterizations for L.

Characterising Petri Net Solvable Binary Words

A word is called Petri net solvable if it is isomorphic to the reachability graph of an unlabelled Petri net. In this paper, the class of finite, two-letter, Petri net solvable words is studied. A linear time, necessary condition allows for an educated guess at which words are solvable and which are not. A full decision procedure with a time complexity of \(O(n^2)\) can be built based on letter counting. The procedure is fully constructive and can either yield a Petri net solving a given word or determine why this fails. Algorithms solving the same problem based on systems of integer inequalities reflecting the potential Petri net structure are only known to be in \(O(n^3)\). Finally, the decision procedure can be adapted from finite to cyclic words.

Structure Theory of Petri Nets

The aim of this tutorial is to give a concise, but nonetheless not too narrow, overview of definitions and results pertaining centrally to Petri net structure theory. The Petri net model considered in these notes are the classical place/transition nets, as they have been defined in the First Advanced Course held in 1979 and originate back to the late Sixties. Structure theory asks what behavioural properties of a Petri net can be derived from its structural properties. Other aspects of Petri nets are neglected to a large extent in the present notes, such as various extensions and generalisations of central notions and results, as well as almost all algorithmic and complexity-theoretic consequences that accompany the structure-theoretic results. Because full proofs can easily be retrieved from the literature, they are not given, unless they are small and perhaps somewhat characteristic for Petri net oriented reasoning. Proof ideas are often sketched, however, and the sharpness of various results is accentuated by means of examples and counterexamples. A list for further reading is also provided.