Petri nets for systems and synthetic biology

We give a description of a Petri net-based framework for modelling and analysing biochemical pathways, which uni¯es the qualita- tive, stochastic and continuous paradigms. Each perspective adds its con- tribution to the understanding of the system, thus the three approaches do not compete, but complement each other. We illustrate our approach by applying it to an extended model of the three stage cascade, which forms the core of the ERK signal transduction pathway. Consequently our focus is on transient behaviour analysis. We demonstrate how quali- tative descriptions are abstractions over stochastic or continuous descrip- tions, and show that the stochastic and continuous models approximate each other. Although our framework is based on Petri nets, it can be applied more widely to other formalisms which are used to model and analyse biochemical networks

A <i>P</i>- and <i>T</i>-invariant characterization of product form and decomposition in stochastic Petri nets

Structural product form and decomposition results for stochastic Petri nets are surveyed,unifed and extended. The contribution is threefold. First, the literature on structural results for product form over the number of tokens at the places is surveyed and rephrased completely in terms of T-invariants. Second, based on the underlying concept of group-local-balance, the product form results for stochastic Petri nets are demarcated and an intuitive explanation is provided of these results based on T-invariants, only. Third, a decomposition result is provided that is completely formulated in terms of both T-invariants and P-invariants

Traps characterize home states in free choice systems

AbstractFree choice nets are a subclass of Petri nets allowing to model concurrency and nondeterministic choice, but with the restriction that choices cannot be influenced externally. Home states are ground markings which can be reached from any other reachable marking of a system. A trap is a structurally defined part of a net with the property that once it is marked (that is, carries at least one token), it will remain remarked in any successor marking.The main result of this paper characterizes the home states of a live and bounded free choice system by the property that all traps are marked. This characterization leads to a polynomial-time algorithm for deciding the home state property. Other consequences include the proof that executing all parts of a net at least once necessarily leads to a home state; this has been a long standing conjecture

The complexity of Petri net transformations

Bibliography: pages 124-127.This study investigates the complexity of various reduction and synthesis Petri net transformations. Transformations that preserve liveness and boundedness are considered. Liveness and boundedness are possibly the two most important properties in the analysis of Petri nets. Unfortunately, although decidable, determining such properties is intractable in the general Petri net. The thesis shows that the complexity of these properties imposes limitations on the power of any reduction transformations to solve the problems of liveness and boundedness. Reduction transformations and synthesis transformations from the literature are analysed from an algorithmic point of view and their complexity established. Many problems regarding the applicability of the transformations are shown to be intractable. For reduction transformations this confirms the limitations of such transformations on the general Petri net. The thesis suggests that synthesis transformations may enjoy better success than reduction transformations, and because of problems establishing suitable goals, synthesis transformations are best suited to interactive environments. The complexity of complete reducibility, by reduction transformation, of certain classes of Petri nets, as proposed in the literature, is also investigated in this thesis. It is concluded that these transformations are tractable and that reduction transformation theory can provide insight into the analysis of liveness and boundedness problems, particularly in subclasses of Petri nets

Lifted structural invariant analysis of Petri net product lines

Petri nets are commonly used to represent concurrent systems. However, they lack support for modelling and analysing system families, like variants of controllers, different variations of a process model, or the possible configurations of a flexible assembly line. To facilitate modelling potentially large collections of similar systems, in this paper, we enrich Petri nets with variability mechanisms based on product line engineering. Moreover, we present methods for the efficient analysis of the place and transition invariants in all defined versions of a Petri net. Efficiency is achieved by analysing the system family as a whole, instead of analysing each possible net variant separately. For this purpose, we lift the notion of incidence matrix to the product line level, and rely on constraint solving techniques. We present tool support and evaluate the benefits of our techniques on synthetic and realistic examples, achieving in some cases speed-ups of two orders of magnitude with respect to analysing each net variant separatelyThis work has been funded by the Spanish Ministry of Science (PID2021-122270OB-I00) and the R&D programme of Madrid (P2018/TCS-4314

Synthesis and Analysis of Product-form Petri Nets

For a large Markovian model, a "product form" is an explicit description of the steady-state behaviour which is otherwise generally untractable. Being first introduced in queueing networks, it has been adapted to Markovian Petri nets. Here we address three relevant issues for product-form Petri nets which were left fully or partially open: (1) we provide a sound and complete set of rules for the synthesis; (2) we characterise the exact complexity of classical problems like reachability; (3) we introduce a new subclass for which the normalising constant (a crucial value for product-form expression) can be efficiently computed.Comment: This is a version including proofs of the conference paper: Haddad, Mairesse and Nguyen. Synthesis and Analysis of Product-form Petri Nets. Accepted at the conference Petri Nets 201