Linear algebra in net theory

Memmi, Gérard; Roucairol, Gérard

doi:10.1007/3-540-10001-6_24

Cited by 119 publications

(54 citation statements)

References 4 publications

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“…It is known that the reproducibility of a firing sequence in a Petri net indicates the existence of a T-invariant (Memmi & Roucairol, 1980 (Alaiwan, 1985;Krukeberg & Jaxy, 1987;Silva & Colom, 1991;Takano et al, 2001). Algorithms for the calculation of T-invariants are implemented in many Petri net software tools such as INA (Roch & Starke, 2001); GreatSPN (Chiola et al, 1995), TimeNET (German et al, 1995), and QPN (Bause & Kemper, 1994), to mention only a few.…”

Section: Petri Net: Theory and Applications 438mentioning

confidence: 99%

“…The method is applicable for reachability analysis of a particular class of place/transition Petri nets having no transition invariants, or T-invariants. Algebraically, T-invariants of a Petri net with incidence matrix D are nonnegative integer (1 × m) vectors F such that FD = 0 (Memmi & Roucairol, 1980). According to the scheme proposed in (Kostin, 2003), given a Petri net with an initial and a target markings, a so called complemented Petri net is created that consists of the given Petri net and an additional, complementary transition with some input and output places of the original Petri net, which are uniquely determined by the initial and target markings.…”

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confidence: 99%

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Using Transition Invariants for Reachability Analysis of Petri Nets

Kostin¹

2008

Petri Net, Theory and Applications

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Section: Petri Net: Theory and Applications 438mentioning

confidence: 99%

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confidence: 99%

Using Transition Invariants for Reachability Analysis of Petri Nets

Kostin¹

2008

Petri Net, Theory and Applications

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“…', also referred to as the structural boundedness problem. It has been shown in [10] that this structural boundedness problem for parallel-composition-VASS is co-$ & % -complete, unlike for standard Petri nets where it is polynomial [12,6].…”

Section: Level 2: Parallel-composition-vassmentioning

confidence: 99%

A Scalable Incomplete Test for Message Buffer Overflow in Promela Models

Leue

Mayr

Wei

2004

Model Checking Software

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Abstract. In Promela, communication buffers are defined with a fixed length, and buffer overflows can be handled in two different ways: block the send statement or lose the message. Both solutions change the semantics of the system, compared to one with unbounded channels. The question arises, if such buffer overflows can ever occur in a given system and what buffer lengths are sufficient to avoid them. We describe a scalable incomplete boundedness test for the communication buffers in Promela models, which is based on overapproximation and static analysis. We first reduce Promela models to systems of communicating finite state machines (CFSMs) and then apply further abstractions that leave us with a system of linear inequalities. Those represent the message sending and receiving effect that the control flow cycles of every process have on any message buffer. The test tries to establish the existence of a linear combination of the effect vectors so that at least one message can occur an unbounded number of times. If no such linear combination exists then the system is bounded. We discuss the complexity of this test and present experimental results using our implementation in the IBOC system. Scalability of the test is in part due to the fact that it is polynomial for the type of sparse control flow graphs derived from Promela models. Also, the analysis is local, i.e., it avoids the combinatorial state space explosion due to concurrency of the models. We also present a method to derive upper bound estimates for the maximal occupancy of each individual message buffer. Previously, we have applied this approach to UML RT models, while in this paper we focus on the additional problems specific to Promela code: determining the potential message types of any channel, tracking potential contents of variables, channels passed as arguments to processes, channel assignments, channel arrays and parallel process creation.

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“…Obviously, every unbounded system is also not structurally bounded. It will be shown in Section 7 that this structural boundedness problem for parallel-composition-VASS is co-N P-complete, unlike for standard Petri nets where it is polynomial [23,12]. (The reason for this difference is that an encoding of control-states by Petri net places does not preserve structural boundedness, because it is not assured that only one of these places is marked at any time.)…”

Section: Level 2: Parallel-composition-vassmentioning

confidence: 99%

A Scalable Incomplete Test for the Boundedness of UML RT Models

Leue

Mayr

Wei

2004

Tools and Algorithms for the Construction and Analysis of Systems

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Abstract. We describe a scalable incomplete boundedness test for the communication buffers in UML RT models. UML RT is a variant of the UML modeling language, tailored to describing asynchronous concurrent embedded systems. We reduce UML RT models to systems of communicating finite state machines (CFSMs). We propose a series of further abstractions that leaves us with a system of linear inequalities. Those represent the message sending and receiving effect that the control flow cycles of every process have on the overall message buffer. The test tries to establish the existence of a linear combination of the effect vectors so that at least one message can occur an unbounded number of times. We discuss the complexity of this test and present experimental results using the IBOC system that we are implementing. Scalability of the test is in part due to the fact that it is polynomial for the type of sparse control flow graphs that are derived from UML RT models. Also, the analysis is local, i.e., it avoids the combinatorial state space explosion due to concurrency of the models. We also present a method to derive upper bound estimates for the maximal occupancy of each individual message buffer. While we focus on the analysis of UML RT models, the analysis can directly be applied to any type of CFSM models.

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Linear algebra in net theory

Cited by 119 publications

References 4 publications

Using Transition Invariants for Reachability Analysis of Petri Nets

Using Transition Invariants for Reachability Analysis of Petri Nets

A Scalable Incomplete Test for Message Buffer Overflow in Promela Models

A Scalable Incomplete Test for the Boundedness of UML RT Models

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