This paper describes the development of finite abstractions of Max-Plus-Linear (MPL) systems using tropical operations. The idea of tropical abstraction is inspired by the fact that an MPL system is a discrete-event model updating its state with operations in the tropical algebra. The abstract model is a finite-state transition system: we show that the abstract states can be generated by operations on the tropical algebra, and that the generation of transitions can be established by tropical multiplications of matrices. The complexity of the algorithms based on tropical algebra is discussed and their performance is tested on a numerical benchmark against an existing alternative abstraction approach.
The notion of abstraction of an MPL system has been first discussed in @cite_4 . The procedure starts by transforming the MPL system characterised by @math into a PWA (piece-wise affine) model [Algorithm 2] Dieky1 , and then considering the partitions associated to the obtained PWA [Algorithm 6] Dieky1 . The abstract states associated to the partitions are represented by DBMs. The transitions are then generated using one-step forward-reachability analysis @cite_4 : first, the image of each abstract state w.r.t. the MPL system is computed; then, each image is intersected with partitions associated to other abstract states; finally, transition relations are defined for each non-empty intersection. This procedure is summarised in [Algorithm 7] Dieky1 .
@cite_4
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This work puts forward a novel technique to generate finite abstractions of autonomous and nonautonomous Max-Plus-Linear (MPL) models, a class of discrete-event systems used to characterize the dynamics of the timing related to successive events that synchronize autonomously. Nonautonomous versions of MPL models embed within their dynamics nondeterminism, namely a signal choice that is usually regarded as an exogenous control or schedule. In this paper, abstractions of MPL models are characterized as finite-state Labeled Transition Systems (LTS). LTS are obtained first by partitioning the state space (and, for the nonautonomous model, by covering the input space) of the MPL model and by associating states of the LTS to the introduced partitions, then by defining relations among the states of the LTS based on dynamical transitions between the corresponding partitions of the MPL state space, and finally by labeling the LTS edges according to the one-step timing properties of the events of the original MPL model. In order to establish formal equivalences, the finite abstractions are proven to either simulate or to bisimulate the original MPL model. This approach enables the study of general properties of the original MPL model by verifying (via model checking) equivalent logical specifications over the finite LTS abstraction. The computational aspects related to the abstraction procedure are thoroughly discussed and its performance is tested on a numerical benchmark. Max-Plus-Linear (MPL) systems are a class of discrete-event systems with a continuous state space characterizing the timing of the underlying sequential discrete events. These systems are predisposed to describe the timing synchronization between interleaved processes. MPL systems are employed in the analysis and scheduling of infrastructure networks, such as communication and railway systems, production and manufacturing lines, or biological systems. As a natural extension, Stochastic Max-Plus-Linear (SMPL) systems are MPL systems where the delays between successive events are characterized by random quantities. @PARASPLIT In practical applications SMPL systems are more realistic than simple MPL ones: for instance in a model for a railway network, train running times depend on driver behavior, on weather conditions, and on passenger numbers at stations. @PARASPLIT Verification is used to establish whether the system under consideration possesses certain properties expressed as formulae. As an example, reachability analysis is a fundamental problem in the area of formal methods, systems theory, and performance and dependability analysis. It is concerned with assessing whether a certain state of a system is attainable from given initial states of the system. @PARASPLIT Verification techniques and tools for finite-state systems have been widely investigated and developed in the past decades. However, if the system has a large number of states or even infinitely many states, in general we cannot apply such techniques directly. In this case we need to employ abstraction techniques to formally relate a concrete model to a finite abstraction of it, which is then amenable to be automatically verified by the relevant results in the literature. @PARASPLIT In this PhD thesis we develop novel abstraction techniques for MPL systems, and use them in an application to communication networks. Additionally we discuss reachability of MPL systems and abstraction techniques for SMPL systems. @PARASPLIT The abstraction and reachability algorithms for MPL systems developed in this thesis have been implemented as a MATLAB software tool, "Verification via biSimulations of MPL models'' (VeriSiMPL, as in "very simple''), which is freely available for download at http: www.sourceforge.net projects verisimpl . We use hybrid-systems techniques for the analysis of reachability properties of a class of piecewise-affine (PA) differential equations that are particularly suitable for the modeling of genetic regulatory networks. More specifically, we introduce a hyperrectangular partition of the state space that forms the basis for a discrete abstraction preserving the sign of the derivatives of the state variables. The resulting discrete transition system provides a qualitative description of the network dynamics that is well-adapted to available experimental data and that can be efficiently computed in a symbolic manner from inequality constraints on the parameters. Predicate abstraction has emerged to be a powerful technique for extracting finite-state models from infinite-state systems, and has been recently shown to enhance the effectiveness of the reachability computation techniques for hybrid systems. Given a hybrid system with linear dynamics and a set of linear predicates, the verifier performs an on-the-fly search of the finite discrete quotient whose states correspond to the truth assignments to the input predicates. To compute the transitions out of an abstract state, the tool needs to compute the set of discrete and continuous successors, and find out all the abstract states that this set intersects with. The complexity of this computation grows exponentially with the number of abstraction predicates. In this paper we present various optimizations that are aimed at speeding up the search in the abstract state-space, and demonstrate their benefits via case studies. We also discuss the completeness of the predicate abstraction technique for proving safety of hybrid systems.