Handling Exceptions in Petri Net-Based Digital Architecture: From Formalism to Implementation on FPGAs

Leroux, Hélène; Andreu, David; Godary-Dejean, Karen

doi:10.1109/tii.2015.2435696

Cited by 10 publications

(10 citation statements)

References 27 publications

(32 reference statements)

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“…Petri nets [45], also known as vector addition systems [23], [18, cf. Section 5.1], [20], are a long established model of concurrency with extensive applications in modelling and analysis of hardware [7,28], software [6,17,21] and database [4,5] systems, as well as chemical [1], biological [2,44] and business [36,50] processes (the references on applications are illustrative). The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration.…”

Section: Introductionmentioning

confidence: 99%

The Reachability Problem for Petri Nets Is Not Elementary

Czerwiński

Lasota

Lazić

et al. 2020

J. ACM

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Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modeling and analysis of hardware, software, and database systems, as well as chemical, biological, and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and, currently, the best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from Symposium on Logic in Computer Science 2019. We establish a non-elementary lower bound, i.e., that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi, and other areas, which are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the current best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. We develop a construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R . At the heart of our proof is then a novel gadget, the so-called factorial amplifier that, assuming availability of counters that are zero testable and bounded by k , guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k . Repeatedly composing the factorial amplifier with itself by means of the former construction enables us to compute, in linear time, Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we, in fact, already establish hardness for h -exponential space for Petri nets with h + 13 counters.

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Section: Introductionmentioning

confidence: 99%

The Reachability Problem for Petri Nets Is Not Elementary

Czerwiński

Lasota

Lazić

et al. 2020

J. ACM

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show abstract

“…In other words, we are interested in building a tool-based methodology that helps to prove that the behavior of the implemented system respects a set of formally validated properties. This work therefore falls within the scope of the HILECOP (High-Level Hardware Component Programming) methodology, which was initially designed to assist in the development of safe active implantable medical devices (Andreu et al (2009); Leroux et al (2015)). These devices are implanted into human body to perform FES (Functional Electrical Stimulation) solutions which are being successfully used in an increasing number of applications, including pacemakers, deep brain stimulation, pain control and hearing restoration.…”

Section: Contextementioning

confidence: 99%

“…The programming step can introduce errors. This problem is managed in the HILECOP methodology, as the programming step is done with an automatic generation of the VHDL code corresponding to the model (Andreu et al (2008); Leroux et al (2015)). Another problem resides in the hardware target, which can has an influence on the execution of the system as it is usually not considered at the modeling level.…”

Section: Contextementioning

confidence: 99%

Interpreted synchronous extension of time Petri nets

Godary-Dejean

Leroux²,

Andreu

2021

Discrete Event Dyn Syst

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Our work is integrated into a global methodology to design synchronously executed embedded critical systems. It is used for the development of medical devices implanted into human body to perform functional electrical stimulation solutions (used in pacemakers, deep brain stimulation...). These systems are of course critical and real time, and the reliability of their behaviors must be guaranteed. These medical devices are implemented into a programmable logic circuit in a synchronous way, which allows efficient implementation (space, consumption and actual parallelism of tasks execution). This paper presents a solution that helps to prove that the behavior of the implemented system respects a set of properties, using Petri nets for modeling and analysis purposes. But one problem in formal methods is that the hardware target and the implementation strategy can have an influence on the execution of the system, but is usually not considered in the modeling and verification processes. Resolving this issue is the goal of this article. Our work has two main results: an operational one, and a theoretical one. First, we can now design critical controllers with hard safety or real time constraints, being sure the behavior is still guaranteed during the execution. Second, this work broadens the scope of expressivity and analyzability of Petri nets extensions. Until then, none managed in the same formalism, both for modeling and analysis, all the characteristics we have considered (weights on arcs, specific test and inhibitor arcs, interpretation, and time intervals, including the management of effective conflicts and the blocking of transitions).

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“…Due to their generic nature, Petri nets have found a variety of applications, ranging, for instance, from modeling of biological, chemical and business processes to the formal verification of concurrent programs, see e.g. [1,2,21,24,32]. For the analysis of algorithmic properties of Petri nets, in the contemporary literature they are often equivalently viewed as vector addition systems with states (VASS), and we will adopt this view in this article.…”

Section: Introductionmentioning

confidence: 99%

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

Blondin

EnglertMatthias

FinkelAlain

et al. 2021

J. ACM

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We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.

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Handling Exceptions in Petri Net-Based Digital Architecture: From Formalism to Implementation on FPGAs

Cited by 10 publications

References 27 publications

The Reachability Problem for Petri Nets Is Not Elementary

The Reachability Problem for Petri Nets Is Not Elementary

Interpreted synchronous extension of time Petri nets

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

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