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
2027649310 1552505815 1815155750 105974170
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. The paper presents an approach for shape analysis based on predicate abstraction. Using a predicate base that involves reachability relations between program variables pointing into the heap, we are able to analyze functional properties of programs with destructive heap updates, such as list reversal and various in-place list sorts. The approach allows verification of both safety and liveness properties. The abstraction we use does not require any abstract representation of the heap nodes (e.g. abstract shapes), only reachability relations between the program variables. The computation of the abstract transition relation is precise and automatic yet does not require the use of a theorem prover. Instead, we use a small model theorem to identify a truncated (small) finite-state version of the program whose abstraction is identical to the abstraction of the unbounded-heap version of the same program. The abstraction of the finite-state version is then computed by BDD techniques. For proving liveness properties, we augment the original system by a well-founded ranking function, which is abstracted together with the system. Well-foundedness is then abstracted into strong fairness (compassion). We show that, for a restricted class of programs that still includes many interesting cases, the small model theorem can be applied to this joint abstraction. Independently of the application to shape-analysis examples, we demonstrate the utility of the ranking abstraction method and its advantages over the direct use of ranking functions in a deductive verification of the same property. We present a novel approach for generalizing the IC3 algorithm for invariant checking from finite-state to infinite-state transition systems, expressed over some background theories. The procedure is based on a tight integration of IC3 with Implicit (predicate) Abstraction, a technique that expresses abstract transitions without computing explicitly the abstract system and is incremental with respect to the addition of predicates. In this scenario, IC3 operates only at the Boolean level of the abstract state space, discovering inductive clauses over the abstraction predicates. Theory reasoning is confined within the underlying SMT solver, and applied transparently when performing satisfiability checks. When the current abstraction allows for a spurious counterexample, it is refined by discovering and adding a sufficient set of new predicates. Importantly, this can be done in a completely incremental manner, without discarding the clauses found in the previous search. 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 .