Full Axiomatisation of Timed Processes of Interval-Timed Petri Nets (original) (raw)

Timed Processes of Interval-Timed Petri Nets

HAL (Le Centre pour la Communication Scientifique Directe), 2016

In this paper we use partial order semantics to express the truly concurrent behaviour of interval-timed Petri nets (ITPNs) in their most general setting, i.e. with autoconcurrency and zero duration, as studied with its standard maximal step semantics in [8]. First we introduce the notion of timed processes for ITPNs inductively. Then we investigate if the equivalence of inductive and axiomatic process semantics-true for classical Petri nets-could hold for ITPNs too. We will see that the notions of independence and immediate firing obligation seem to be antagonistic ones, and that local axioms, adequate to define processes of classical Petri nets, are not sufficient to caracterize timed Processes of IITPNs. We propose several original "global" axioms which reveal to be an effective solution. Thus we yield finally a full axiomatic definition of timed processes for ITPNs.

Interval-Timed Petri Nets with Auto-concurrent Semantics and their State Equation

2015

In this paper we consider Interval-Timed Petri nets (ITPN), an extension of Timed Petri nets in which the discrete time delays of transitions are allowed to vary within fixed intervals including possible zero durations. These nets will be analyzed for the first time under some maximal step semantics with auto-concurrency. This matches well with the reality of time critical systems which could be modeled and analyzed with our model. We introduce in particular the notion of global firing step which regroups all what happens inbetween two time ticks. Full algebraic representations of the semantics are proposed. We introduce time-dependent state equations for a sequence of global firing steps of ITPNs which are analogous to the state equation for a firing sequence in standard Petri nets and we prove its correctness using linear algebra. Our result delivers a necessary condition for reachability which is also a sufficient condition for non-reachability of an arbitrary marking in an ITPN.

Comparison of Different Semantics for Time Petri Nets

Lecture Notes in Computer Science, 2005

In this paper we study the model of Time Petri Nets (TPNs) where a time interval is associated with the firing of a transition, but we extend it by considering general intervals rather than closed ones. A key feature of timed models is the memory policy, i.e. which timing informations are kept when a transition is fired. The original model selects an intermediate semantics where the transitions disabled after consuming the tokens, as well as the firing transition, are reinitialised. However this semantics is not appropriate for some applications. So we consider here two alternative semantics: the atomic and the persistent atomic ones. First we present relevant patterns of discrete event systems which show the interest of these semantics. Then we compare the expressiveness of the three semantics w.r.t. weak timed bisimilarity, establishing inclusion results in the general case. Furthermore we show that some inclusions are strict with unrestricted intervals even when nets are bounded. Then we focus on bounded TPNs with upper-closed intervals and we prove that the semantics are equivalent. Finally taking into account both the practical and the theoretical issues, we conclude that persistent atomic semantics should be preferred.

Time Process Equivalences for Time Petri Nets

2014

In the core of every theory of systems lies a notion of equivalence between systems: it indicates which particular aspects of systems behaviors are considered to be observable. In concurrency theory, a variety of observational equivalences has been promoted, and the relationships between them have been quite wellunderstood. In order to investigate the performance of systems (e.g. the maximal time used for the execution of certain activities and average waiting time for certain requests), many time extensions have been de ned for a non-interleaving model of Petri nets. On the other hand, there are few mentions of a fusion of timing and partial order semantics, in the Petri net literature. In [9], processes of timed Petri nets (under the asap hypothesis) have been de ned by an algebra of the so-called weighted pomsets. The paper [8] has provided and compared timed step sequence and timed process semantics for timed Petri nets. A method to compute all valid timings for a causal net pro...

“Truly concurrent” and nondeterministic semantics of discrete-time Petri nets

Programming and Computer Software, 2016

In the paper, a "truly concurrent" and nondeterministic semantics is defined in terms of branching processes of discrete-time Petri nets (DTPNs). These nets may involve infinite numbers of transitions and places, infinite number of tokens in places, and (maximal) steps of concurrent transitions, which allows us to consider this class of DTPNs to be the most powerful class of Petri nets. It is proved that the unfolding (maximal branching process) of the DTPN is the greatest element of a complete lattice constructed on branching processes of DTPNs with step semantics. Moreover, it is shown that this result is true also in the case of maximal transition steps if additional restrictions are imposed on the structure and behavior of the DTPN.

Towards a Notion of Distributed Time for Petri Nets

Lecture Notes in Computer Science, 2001

We set the ground for research on a timed extension of Petri nets where time parameters are associated with tokens and arcs carry constraints that qualify the age of tokens required for enabling. The novelty is that, rather than a single global clock, we use a set of unrelated clocks-possibly one per place-allowing a local timing as well as distributed time synchronisation. We give a formal definition of the model and investigate properties of local versus global timing, including decidability issues and notions of processes of the respective models.

Formalization of petri nets with clocks

Journal of Computational Methods in Sciences and Engineering, 2005

PN (PN) are tools for the analysis and design of concurrent systems. There is a formal theory, which supports PN. An extension of PN is Petri Nets with Clocks (PNwC). PNwC are useful to model systems with temporal requirements via specification of clocks, using temporal invariants for the places and temporal conditions in the transitions. Using invariants in places allows the specifications of hard deadlines constrains (upper bound constrains): when a deadline is reached the progress of time is blocked by the invariant and the action becomes urgent. An algorithm for the analysis of a PNwC has been proposed in [1]. The algorithm is oriented to the verification and correction of errors in the modelling of the time variable. The algorithm generates information about temporal unreachable states and process deadlocks with temporal blocks. Also, it corrects places invariants and transitions conditions. We present here a formalism for PNwC based on Timed Graphs. The analysis algorithm is presented here using the formalism. We show here how Petri Net theory can be joined with Timed Graph theory to construct a formalism, which supports a tool for the analysis of models of concurrent process with real time specification.

True Concurrent Equivalences in Time Petri Nets*

Fundamenta Informaticae, 2016

The intention of the paper is towards a framework for developing, studying and comparing observational equivalences in the setting of a real-time true concurrent model. In particular, we introduce a family of trace and bisimulation equivalences in interleaving, step, partial order and causal net semantics in the setting of time Petri nets (elementary net systems whose transitions are labeled with time firing intervals, can fire only if their lower time bounds are attained, and are forced to fire when their upper time bounds are reached) [3]. We deal with the relationships between the equivalences showing the discriminating power of the approaches of the linear-time-branching-time and interleaving-partial order spectra and construct a hierarchy of equivalent classes of time Petri nets. This allows studying in complete detail the timing behaviour in addition to the degrees of relative concurrency and nondeterminism of processes.

A theory of implementation and refinement in timed Petri nets

Theoretical Computer Science, 1998

We define formally the notion of implementation for time critical systems in terms of provability of properties described abstractly at the specification level. We characterize this notion in terms of formulas of the temporal logic TRIO and operational models of timed Petri nets, and provide a method to prove that two given nets are in the implementation relation. Refinement steps are often used as a means to derive in a systematic way the system design starting from its abstract specification. We present a method to formally prove the correctness of refinement rules for timed Petri nets and apply it to a few simple cases. We show how the possibility to retain properties of the specification in its implementation can simplify the verification of the designed systems by performing incremental analysis at various levels of the specification/implementation hierarchy.

Some results on timed petri-nets

Optimization Techniques

Petri-nets have been found an adequate tool to describe the state transitions of rather complicated systems (as asynchronous systems). Many cocrdinatio~ problems have been modeled successfully with them. However, these models need more information in order to study some quantitative aspects as utilisation rates , delays ..... which are of main interest for a practical point of view. 8o, we are interested in more sophisticated models called Timed Petri-Nets (TPN) in which the time dimension is introduced. In this paper, we first give a formal and rigorous definition of the execution of a TPN ; then, we give some general results on what we call "a program" ; finally, we extend Ramachandani previous results on strongly periodic event graphs to general Petri-nets.