An Algebraic Approach to Timed Petri Nets with Applications to Communication Networks -- Extended Version (original) (raw)

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.

Timed-Arc Petri Nets vs. Networks of Timed Automata

Lecture Notes in Computer Science, 2005

We establish mutual translations between the classes of 1safe timed-arc Petri nets (and its extension with testing arcs) and networks of timed automata (and its subclass where every clock used in the guard has to be reset). The presented translations are very tight (up to isomorphism of labelled transition systems with time). This provides a convenient characterization from the theoretical point of view but is not always satisfactory from the practical point of view because of the possible non-polynomial blow up in the size (in the direction from automata to nets). Hence we relax the isomorphism requirement and provide efficient (polynomial time) reductions between networks of timed automata and 1-safe timed-arc Petri nets preserving the answer to the reachability question. This makes our techniques suitable for automatic translation into a format required by tools like UPPAAL and KRONOS. A direct corollary of the presented reductions is a new PSPACE-completeness result for reachability in 1-safe timed-arc Petri nets, reusing the region/zone techniques already developed for timed automata.

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.

Full Axiomatisation of Timed Processes of Interval-Timed Petri Nets

Fundamenta Informaticae, 2018

Timed Mobility in process algebra and Petri nets

The Journal of Logic and Algebraic Programming, 2011

We present a process algebra called TiMo in which timeouts of interactions and adaptable migrations in a distributed environment with explicit locations can be specified. Timing constraints allow to control the communication between co-located mobile processes, and a migration action with variable destination supports flexible movement from one location to another. The model of time is based on local clocks rather than a global clock. We provide a structural translation of TiMo into behaviourally equivalent high level timed Petri nets. As a result, we obtain a formal net semantics for timed interaction and migration which is both structural and allows one to deal directly with concurrency and causality.

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...

Comparing the Expressiveness of Timed Automata and Timed Extensions of Petri Nets

Lecture Notes in Computer Science

Time dependant models have been intensively studied for many reasons, among others because of their applications in software verification and due to the development of embedded platforms where reliability and safety depend to a large extent on the time features. Many of the time dependant models were suggested as real-time extensions of several well-known untimed models. The most studied formalisms include Networks of Timed Automata which extend the model of communicating finite-state machines with a finite number of real-valued clocks, and timed extensions of Petri nets where the added time constructs include e.g. time intervals that are assigned to the transitions (Time Petri Nets) or to the arcs (Timed-Arc Petri Nets). In this paper, we shall semiformally introduce these models, discuss their strengths and weaknesses, and provide an overview of the known results about the relationships among the models.

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.

Comparison of the Expressiveness of Timed Automata and Time Petri Nets

Lecture Notes in Computer Science, 2005

In this paper we consider the model of Time Petri Nets (TPN) where time is associated with transitions. We also consider Timed Automata (TA) as defined by Alur & Dill, and compare the expressiveness of the two models w.r.t. timed language acceptance and (weak) timed bisimilarity. We first prove that there exists a TA A s.t. there is no TPN (even unbounded) that is (weakly) timed bisimilar to A. We then propose a structural translation from TA to (1-safe) TPNs preserving timed language acceptance. Further on, we prove that the previous (slightly extended) translation also preserves weak timed bisimilarity for a syntactical subclass T Asyn(≤, ≥) of TA. For the theory of TPNs, the consequences are: 1) TA, bounded TPNs and 1-safe TPNs are equally expressive w.r.t. timed language acceptance; 2) TA are strictly more expressive than bounded TPNs w.r.t. timed bisimilarity; 3) The subclass T Asyn(≤, ≥), bounded and 1-safe TPNs "à la Merlin" are equally expressive w.r.t. timed bisimilarity.

Time-based expressivity of time Petri nets for system specification

Theoretical Computer Science, 1999

Various models of time Petri Nets have been successfully used to adequately specify timecritical systems. For such systems correctness depends not only on the actions that are performed, but also on the times when they are performed. Therefore, the semantics must take explicitly into account the timings of actions, and also concepts of time-bused expressivity are needed to compare the expressive power of the various models. In the paper we introduce a general framework that includes many Petri net models, present in the literature, which differ from one another with respect to timing location, timing strength and time domain. Then we introduce an operational semantics that takes into account both sequentialization and timing of actions. When abstracting time away we reobtain previously reported results, which are useful when one is interested in modelling systems that are not time-critical. On the other hand, when abstracting the sequentialization of actions away we define a new kind of expressivity in terms of which we compare the various models we have considered.

An algebraic structure of petri nets

Lecture Notes in Computer Science, 1980

The paper concerns algebraic properties of Petr± nets. A wide class of ne~s, called simple nets, is inwroduced and a lattice of these nets is defined. IV turns out tha~ nets representing sequential systems and processes are a~oms of this lat~ice~ and this fact provides the natural way of building nets representing° concurren~ systems as the superposit~on of nets representlng sequen-Zial system components. TNe notion of concurrency relatlon for large class of nets including cyclic nets is precisely defined. An influence of statmo, i.e. unmarked, structure of neZs on the class of "proper" markings is discussed. The notion of naZural markings, i.e. markings defined by the static (unmar~ed) s~ruc~ure of ne~s is introduced. Properties of safeness, compaoZness, fireability and K-densiSy of marked nets are discussed. A classification of nets is proposed and an attempZ of wne algebraic definition of net with properties requlred from "well defined" dynamlc concurren~ system is given.

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.

Comparison of Expressiveness for Timed Automata and Time Petri Nets

Combinatorial Optimization and Theoretical Computer Science, 2008

In this paper we consider the model of Time Petri Nets (TPN) "à la Merlin" where a time interval is associated with the firing of a transition, but we extend it with open intervals. We also consider Timed Automata (TA) as defined by Alur & Dill. We investigate some questions related to expressiveness for these models : we study the impact of slight variations of semantics for TPN and we compare the expressive power of TA and TPN, with respect to both time language acceptance and weak time bisimilarity. We prove that TA and bounded TPNs (enlarged with strict constraints) are equivalent w.r.t. timed language equivalence, providing an efficient construction of a TPN equivalent to a TA. We then exhibit a TA A such that no TPN (even unbounded) is weakly bisimilar to A. Because of this last result, it is natural to try and identify the (strict) subclass of TA that is equivalent to TPN w.r.t. weak timed bisimilarity. Thus we give some further results: 1) we characterize the subclass TA − of TA that is equivalent to the original model of TPN as defined by Merlin, i.e. restricted to closed intervals, 2) we show that the associated membership problem for TA − is P SP ACE-complete and 3) we prove that the reachability problem for TA − is also P SP ACE-complete.

Timed Petri Nets

2012

In the early 60'sa young researcher in Darmstadt looked for a good representation for communicating systems processes that were mathematically sound and had, at the same time, a visual intuitive flavor. This event marked the beginning of a schematic approach that become very important to the modeling of distributed systems in several and distinct areas of knowledge, from Engineering to biologic systems. Carl Adam Petri presented in 1962 his PHD which included the first definition of what is called today a Petri Net.

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.

Structural Translation from Time Petri Nets to Timed Automata

Electronic Notes in Theoretical Computer Science, 2005

In this paper, we consider Time Petri Nets (TPN) where time is associated with transitions. We give a formal semantics for TPNs in terms of Timed Transition Systems. Then, we propose a translation from TPNs to Timed Automata (TA) that preserves the behavioral semantics (timed bisimilarity) of the TPNs. For the theory of TPNs this result is two-fold: i) reachability problems and more generally TCTL model-checking are decidable for bounded TPNs; ii) allowing strict time constraints on transitions for TPNs preserves the results described in i). The practical applications of the translation are: i) one can specify a system using both TPNs and Timed Automata and a precise semantics is given to the composition; ii) one can use existing tools for analyzing timed automata (like Kronos, Uppaal or Cmc) to analyze TPNs. In this paper we describe the new feature of the tool Romeo that implements our translation of TPNs in the Uppaal input format. We also report on experiments carried out on various examples and compare the result of our method to state-of-the-art tool for analyzing TPNs.

A proposal for relative time Petri nets

Third IEEE International Conference on Software Engineering and Formal Methods (SEFM'05), 2005

Petri nets are a graph-based modelling formalism which has been widely used for the formal specification and analysis of concurrent systems. A common analysis technique is that of state space exploration (or reachability analysis). Here, every possible reachable state of the system is generated and desirable properties are evaluated for each state. This approach has the great advantage of conceptual simplicity, but the great disadvantage of being susceptible to state space explosion, where the number of states is simply too large for exhaustive exploration. Many reduction techniques have been suggested to ameliorate the problem of state space explosion. In the case of timed systems, the state space is infinite, unless analysis is restricted to a bounded time period. In this paper, we present a Petri net formalism based on the notion of relative time (as opposed to the traditional approach of dealing with absolute time). The goal is to derive a finite state space for timed systems which have repeating patterns of behaviour, even though time continues to advance indefinitely.