119 research outputs found
Decision Problems for Petri Nets with Names
We prove several decidability and undecidability results for nu-PN, an
extension of P/T nets with pure name creation and name management. We give a
simple proof of undecidability of reachability, by reducing reachability in
nets with inhibitor arcs to it. Thus, the expressive power of nu-PN strictly
surpasses that of P/T nets. We prove that nu-PN are Well Structured Transition
Systems. In particular, we obtain decidability of coverability and termination,
so that the expressive power of Turing machines is not reached. Moreover, they
are strictly Well Structured, so that the boundedness problem is also
decidable. We consider two properties, width-boundedness and depth-boundedness,
that factorize boundedness. Width-boundedness has already been proven to be
decidable. We prove here undecidability of depth-boundedness. Finally, we
obtain Ackermann-hardness results for all our decidable decision problems.Comment: 20 pages, 7 figure
The Semilinear Home-Space Problem Is Ackermann-Complete for Petri Nets
A set of configurations H is a home-space for a set of configurations X of a Petri net if every configuration reachable from (any configuration in) X can reach (some configuration in) H. The semilinear home-space problem for Petri nets asks, given a Petri net and semilinear sets of configurations X, H, if H is a home-space for X. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when X is a singleton and H is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. In particular, we prove that for any Petri net and any linear set of configurations L we can effectively compute a semilinear set C of configurations, called a non-reachability core for L, such that for every set X the set L is not a home-space for X if, and only if, C is reachable from X. We show that the established relation to the reachability problem yields the Ackermann-completeness of the (semilinear) home-space problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time
Structural liveness of petri nets is ExpSpace-hard and decidable
Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is verification of safety and liveness properties in this model; despite an extensive research effort, some basic problems remain open, which is exemplified by the complexity status of the reachability problem that is still not fully clarified. The liveness problems are known to be closely related to the reachability problem, and various structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem of determining whether a net is structurally live, i.e. whether there is an initial marking for which it is live, remained open for some time; e.g. Best and Esparza (Inf Process Lett 116(6):423–427, 2016. https://doi.org/10.1016/j.ipl.2016.01.011) emphasize this open question. Here we show that the structural liveness problem for Petri nets is ExpSpace-hard and decidable. In particular, given a net N and a semilinear set S, it is decidable whether there is an initial marking of N for which the reachability set is included in S; this is based on results by Leroux (28th annual ACM/IEEE symposium on logic in computer science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, IEEE Computer Society, pp 23–32, 2013. https://doi.org/10.1109/LICS.2013.7)
Forward Analysis for WSTS, Part III: Karp-Miller Trees
This paper is a sequel of "Forward Analysis for WSTS, Part I: Completions"
[STACS 2009, LZI Intl. Proc. in Informatics 3, 433-444] and "Forward Analysis
for WSTS, Part II: Complete WSTS" [Logical Methods in Computer Science 8(3),
2012]. In these two papers, we provided a framework to conduct forward
reachability analyses of WSTS, using finite representations of downward-closed
sets. We further develop this framework to obtain a generic Karp-Miller
algorithm for the new class of very-WSTS. This allows us to show that
coverability sets of very-WSTS can be computed as their finite ideal
decompositions. Under natural effectiveness assumptions, we also show that LTL
model checking for very-WSTS is decidable. The termination of our procedure
rests on a new notion of acceleration levels, which we study. We characterize
those domains that allow for only finitely many accelerations, based on ordinal
ranks
Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma
Dickson's Lemma is a simple yet powerful tool widely used in termination
proofs, especially when dealing with counters or related data structures.
However, most computer scientists do not know how to derive complexity upper
bounds from such termination proofs, and the existing literature is not very
helpful in these matters.
We propose a new analysis of the length of bad sequences over (N^k,\leq) and
explain how one may derive complexity upper bounds from termination proofs. Our
upper bounds improve earlier results and are essentially tight
On the Home-Space Problem for Petri Nets and its Ackermannian Complexity
A set of configurations H is a home-space for a set of configurations X of a
Petri net if every configuration reachable from (any configuration in) X can
reach (some configuration in) H. The semilinear home-space problem for Petri
nets asks, given a Petri net and semilinear sets of configurations X, H, if H
is a home-space for X. In 1989, David de Frutos Escrig and Colette Johnen
proved that the problem is decidable when X is a singleton and H is a finite
union of linear sets with the same periods. In this paper, we show that the
general (semilinear) problem is decidable. This result is obtained by proving a
duality between the reachability problem and the non-home-space problem. In
particular, we prove that for any Petri net and any semilinear set of
configurations H we can effectively compute a semilinear set C of
configurations, called a non-reachability core for H, such that for every set X
the set H is not a home-space for X if, and only if, C is reachable from X. We
show that the established relation to the reachability problem yields the
Ackermann-completeness of the (semilinear) home-space problem. For this we also
show that, given a Petri net with an initial marking, the set of minimal
reachable markings can be constructed in Ackermannian time
Structural computation of alignments of business processes over partial orders
Relating event data and process models is becoming an important element for organizations. This paper presents a novel approach for aligning traces and process models. The approach is based on the structural theory of Petri nets (the marking equation), applied over an unfolding of the initial process model. Given an observed trace, the approach adopts an iterative optimization mechanism on top of the unfolding, computing at each iteration part of the resulting alignment. In contrast to the previous work that is primarily grounded in the marking equation, this approach is guaranteed to provide real solutions, and tries to mimic as much as possible the events observed in the trace. Experiments witness the significance of this approach both in quality and execution time perspectives.Peer ReviewedPostprint (author's final draft
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