1,004 research outputs found
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
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
Petri nets for systems and synthetic biology
We give a description of a Petri net-based framework for
modelling and analysing biochemical pathways, which uni¯es the qualita-
tive, stochastic and continuous paradigms. Each perspective adds its con-
tribution to the understanding of the system, thus the three approaches
do not compete, but complement each other. We illustrate our approach
by applying it to an extended model of the three stage cascade, which
forms the core of the ERK signal transduction pathway. Consequently
our focus is on transient behaviour analysis. We demonstrate how quali-
tative descriptions are abstractions over stochastic or continuous descrip-
tions, and show that the stochastic and continuous models approximate
each other. Although our framework is based on Petri nets, it can be
applied more widely to other formalisms which are used to model and
analyse biochemical networks
Forward Analysis and Model Checking for Trace Bounded WSTS
We investigate a subclass of well-structured transition systems (WSTS), the
bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete
deterministic ones, which we claim provide an adequate basis for the study of
forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth.
Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered
previously for the termination of forward analysis, boundedness is decidable.
Boundedness turns out to be a valuable restriction for WSTS verification, as we
show that it further allows to decide all -regular properties on the
set of infinite traces of the system
Vector Addition System Reversible Reachability Problem
The reachability problem for vector addition systems is a central problem of
net theory. This problem is known to be decidable but the complexity is still
unknown. Whereas the problem is EXPSPACE-hard, no elementary upper bounds
complexity are known. In this paper we consider the reversible reachability
problem. This problem consists to decide if two configurations are reachable
one from each other, or equivalently if they are in the same strongly connected
component of the reachability graph. We show that this problem is
EXPSPACE-complete. As an application of the introduced materials we
characterize the reversibility domains of a vector addition system
On functional module detection in metabolic networks
Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models
Formal and efficient verification techniques for Real-Time UML models
The real-time UML profile TURTLE has a formal semantics expressed by translation into a timed process algebra: RT-LOTOS. RTL, the formal verification tool developed for RT-LOTOS, was first used to check TURTLE models against design errors. This paper opens new avenues for TURTLE
model verification. It shows how recent work on translating RT-LOTOS specifications into Time Petri net model may be applied to TURTLE. RT-LOTOS to TPN translation patterns are presented. Their formal proof is the subject of another paper. These patterns have been implemented in a RT-LOTOS to TPN translator which has been interfaced with TINA, a Time Petri Net Analyzer which implements several reachability analysis procedures depending on the class of property to be verified. The paper illustrates the benefits of the TURTLE->RT-LOTOS->TPN transformation chain on an avionic case study
A new approach for diagnosability analysis of Petri nets using Verifier Nets
In this paper, we analyze the diagnosability properties of labeled Petri nets. We consider the standard notion of diagnosability of languages, requiring that every occurrence of an unobservable fault event be eventually detected, as well as the stronger notion of diagnosability in K steps, where the detection must occur within a fixed bound of K event occurrences after the fault. We give necessary and sufficient conditions for these two notions of diagnosability for both bounded and unbounded Petri nets and then present an algorithmic technique for testing the conditions based on linear programming. Our approach is novel and based on the analysis of the reachability/coverability graph of a special Petri net, called Verifier Net, that is built from the Petri net model of the given system. In the case of systems that are diagnosable in K steps, we give a procedure to compute the bound K. To the best of our knowledge, this is the first time that necessary and sufficient conditions for diagnosability and diagnosability in K steps of labeled unbounded Petri nets are presented
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