Interrupt Timed Automata: verification and expressiveness

We introduce the class of Interrupt Timed Automata (ITA), a subclass of hybrid automata well suited to the description of timed multi-task systems with interruptions in a single processor environment. While the reachability problem is undecidable for hybrid automata we show that it is decidable for ITA. More precisely we prove that the untimed language of an ITA is regular, by building a finite automaton as a generalized class graph. We then establish that the reachability problem for ITA is in NEXPTIME and in PTIME when the number of clocks is fixed. To prove the first result, we define a subclass ITA- of ITA, and show that (1) any ITA can be reduced to a language-equivalent automaton in ITA- and (2) the reachability problem in this subclass is in NEXPTIME (without any class graph). In the next step, we investigate the verification of real time properties over ITA. We prove that model checking SCL, a fragment of a timed linear time logic, is undecidable. On the other hand, we give model checking procedures for two fragments of timed branching time logic. We also compare the expressive power of classical timed automata and ITA and prove that the corresponding families of accepted languages are incomparable. The result also holds for languages accepted by controlled real-time automata (CRTA), that extend timed automata. We finally combine ITA with CRTA, in a model which encompasses both classes and show that the reachability problem is still decidable. Additionally we show that the languages of ITA are neither closed under complementation nor under intersection

On the decidability of linear bounded periodic cyber-physical systems

Cyber-Physical Systems (CPSs) are integrations of distributed computing systems with physical processes via a networking with actuators and sensors, where feedback loops among the components allow the physical processes to affect the computations and vice versa. Although CPSs can be found in several complex and sometimes critical real-world domains, their verification and validation often relies on simulation-test systems rather then automatic methodologies to formally verify safety requirements. In this work, we prove the decidability of the reachability problem for discrete-time linear CPSs whose physical process in isolation has a periodic behavior, up to an initial transitory phase

Hybridization based CEGAR for Hybrid Automata with Affine Dynamics

Ope

CHARDA: Causal Hybrid Automata Recovery via Dynamic Analysis

We propose and evaluate a new technique for learning hybrid automata automatically by observing the runtime behavior of a dynamical system. Working from a sequence of continuous state values and predicates about the environment, CHARDA recovers the distinct dynamic modes, learns a model for each mode from a given set of templates, and postulates causal guard conditions which trigger transitions between modes. Our main contribution is the use of information-theoretic measures (1)~as a cost function for data segmentation and model selection to penalize over-fitting and (2)~to determine the likely causes of each transition. CHARDA is easily extended with different classes of model templates, fitting methods, or predicates. In our experiments on a complex videogame character, CHARDA successfully discovers a reasonable over-approximation of the character's true behaviors. Our results also compare favorably against recent work in automatically learning probabilistic timed automata in an aircraft domain: CHARDA exactly learns the modes of these simpler automata.Comment: 7 pages, 2 figures. Accepted for IJCAI 201

SPeeDI - a verification tool for polygonal hybrid systems

Hybrid systems combining discrete and continuous dynamics arise as mathematical models of various artificial and natural systems, and as an approximation to complex continuous systems. A very important problem in the analysis of the behavior of hybrid systems is reachability. It is well-known that for most non-trivial subclasses of hybrid systems this and all interesting verification problems are undecidable. Most of the proved decidability results rely on stringent hypothesis that lead to the existence of a finite and computable partition of the state space into classes of states which are equivalent with respect to reachability. This is the case for classes of rectangular automata [1] and hybrid automata with linear vector fields [2]. Most implemented computational procedures resort to (forward or backward) propagation of constraints, typically (unions of convex) polyhedra or ellipsoids [3, 4, 5]. In general, these techniques provide semi-decision procedures, that is, if the given final set of states is reachable, they will terminate, otherwise they may fail to. Maybe the major drawback of set-propagation, reachset approximation procedures is that they pay little attention to the geometric properties of the specific (class of) systems under analysis.peer-reviewe

Realizability of embedded controllers: from hybrid models to correct implementations

Un controller embedded \ue8 un dispositivo (ovvero, un'opportuna combinazione di componenti hardware e software) che, immerso in un ambiente dinamico, deve reagire alle variazioni ambientali in tempo reale. I controller embedded sono largamente adottati in molti contesti della vita moderna, dall'automotive all'avionica, dall'elettronica di consumo alle attrezzature mediche. La correttezza di tali controller \ue8 indubbiamente cruciale. Per la progettazione e per la verifica di un controller embedded, spesso sorge la necessit\ue0 di modellare un intero sistema che includa sia il controller, sia il suo ambiente circostante. La natura di tale sistema \ue8 ibrido. Esso, infatti, \ue8 ottenuto integrando processi ad eventi discreti (i.e., il controller) e processi a tempo continuo (i.e., l'ambiente). Sistemi di questo tipo sono chiamati cyber-physical (CPS) o sistemi ibridi. Le dinamiche di tali sistemi non possono essere rappresentati efficacemente utilizzando o solo un modello (i.e., rappresentazione) discreto o solo un modello continuo. Diversi tipi di modelli possono sono stati proposti per descrivere i sistemi ibridi. Questi si concentrano su obiettivi diversi: modelli dettagliati sono eccellenti per la simulazione del sistema, ma non sono adatti per la sua verifica; modelli meno dettagliati sono eccellenti per la verifica, ma non sono convenienti per i successivi passi di raffinamento richiesti per la progettazione del sistema, e cos\uec via. Tra tutti questi modelli, gli Automi Ibridi (HA) [8, 77] rappresentano il formalismo pi\uf9 efficace per la simulazione e la verifica di sistemi ibridi. In particolare, un automa ibrido rappresenta i processi ad eventi discreti per mezzo di macchine a stati finiti (FSM), mentre i processi a tempo continuo sono rappresentati mediante variabili "continue" la cui dinamica \ue8 specificata da equazioni differenziali ordinarie (ODE) o loro generalizzazioni (e.g., inclusioni differenziali). Sfortunatamente, a causa della loro particolare semantica, esistono diverse difficolt\ue0 nel raffinare un modello basato su automi ibridi in un modello realizzabile e, di conseguenza, esistono difficolt\ue0 nell'automatizzare il flusso di progettazione di sistemi ibridi a partire da automi ibridi. Gli automi ibridi, infatti, sono considerati dispositivi "perfetti e istantanei". Essi adottano una nozione di tempo e di variabili basata su insiemi "densi" (i.e., l'insieme dei numeri reali). Pertanto, gli automi ibridi possono valutare lo stato (i.e., i valori delle variabili) del sistema in ogni istante, ovvero in ogni infinitesimo di tempo, e con la massima precisione. Inoltre, sono in grado di eseguire computazioni o reagire ad eventi di sincronizzazione in modo istantaneo, andando a cambiare la modalit\ue0 di funzionamento del sistema senza alcun ritardo. Questi aspetti sono convenienti a livello di modellazione, ma nessun dispositivo hardware/software potrebbe implementare correttamente tali comportamenti, indipendentemente dalle sue prestazioni. In altre parole, il controller modellato potrebbe non essere implementabile, ovvero, esso potrebbe non essere realizzabile affatto. Questa tesi affronta questo problema proponendo una metodologia completa e gli strumenti necessari per derivare da modelli basati su automi ibridi, modelli realizzabili e le corrispondenti implementazioni corrette. In un modello realizzabile, il controller analizza lo stato del sistema ad istanti temporali discreti, tipicamente fissati dalla frequenza di clock del processore installato sul dispositivo che implementa il controller. Lo stato del sistema \ue8 dato dai valori delle variabili rilevati dai sensori. Questi valori vengono digitalizzati con precisione finita e propagati al controller che li elabora per decidere se cambiare la modalit\ue0 di funzionamento del sistema. In tal caso, il controller genera segnali che, una volta trasmessi agli attuatori, determineranno il cambiamento della modalit\ue0 di funzionamento del sistema. \uc8 necessario tener presente che i sensori e gli attuatori introducono ritardi che seppur limitati, non possono essere trascurati.An embedded controller is a reactive device (e.g., a suitable combination of hardware and software components) that is embedded in a dynamical environment and has to react to environment changes in real time. Embedded controllers are widely adopted in many contexts of modern life, from automotive to avionics, from consumer electronics to medical equipment. Noticeably, the correctness of such controllers is crucial. When designing and verifying an embedded controller, often the need arises to model the controller and also its surrounding environment. The nature of the obtained system is hybrid because of the inclusion of both discrete-event (i.e., controller) and continuous-time (i.e., environment) processes whose dynamics cannot be characterized faithfully using either a discrete or continuous model only. Systems of this kind are named cyber-physical (CPS) or hybrid systems. Different types of models may be used to describe hybrid systems and they focus on different objectives: detailed models are excellent for simulation but not suitable for verification, high-level models are excellent for verification but not convenient for refinement, and so forth. Among all these models, hybrid automata (HA) [8, 77] have been proposed as a powerful formalism for the design, simulation and verification of hybrid systems. In particular, a hybrid automaton represents discrete-event processes by means of finite state machines (FSM), whereas continuous-time processes are represented by using real-numbered variables whose dynamics is specified by (ordinary) differential equation (ODE) or their generalizations (e.g., differential inclusions). Unfortunately, when the high-level model of the hybrid system is a hybrid automaton, several difficulties should be solved in order to automate the refinement phase in the design flow, because of the classical semantics of hybrid automata. In fact, hybrid automata can be considered perfect and instantaneous devices. They adopt a notion of time and evaluation of continuous variables based on dense sets of values (usually R, i.e., Reals). Thus, they can sample the state (i.e., value assignments on variables) of the hybrid system at any instant in such a dense set R 650. Further, they are capable of instantaneously evaluating guard constraints or reacting to incoming events by performing changes in the operating mode of the hybrid system without any delay. While these aspects are convenient at the modeling level, any model of an embedded controller that relies for its correctness on such precision and instantaneity cannot be implemented by any hardware/software device, no matter how fast it is. In other words, the controller is un-realizable, i.e., un-implementable. This thesis proposes a complete methodology and a framework that allows to derive from hybrid automata proved correct in the hybrid domain, correct realizable models of embedded controllers and the related discrete implementations. In a realizable model, the controller samples the state of the environment at periodic discrete time instants which, typically, are fixed by the clock frequency of the processor implementing the controller. The state of the environment consists of the current values of the relevant variables as observed by the sensors. These values are digitized with finite precision and reported to the controller that may decide to switch the operating mode of the environment. In such a case, the controller generates suitable output signals that, once transmitted to the actuators, will effect the desired change in the operating mode. It is worth noting that the sensors will report the current values of the variables and the actuators will effect changes in the rates of evolution of the variables with bounded delays

LNCS

Reachability analysis is difficult for hybrid automata with affine differential equations, because the reach set needs to be approximated. Promising abstraction techniques usually employ interval methods or template polyhedra. Interval methods account for dense time and guarantee soundness, and there are interval-based tools that overapproximate affine flowpipes. But interval methods impose bounded and rigid shapes, which make refinement expensive and fixpoint detection difficult. Template polyhedra, on the other hand, can be adapted flexibly and can be unbounded, but sound template refinement for unbounded reachability analysis has been implemented only for systems with piecewise constant dynamics. We capitalize on the advantages of both techniques, combining interval arithmetic and template polyhedra, using the former to abstract time and the latter to abstract space. During a CEGAR loop, whenever a spurious error trajectory is found, we compute additional space constraints and split time intervals, and use these space-time interpolants to eliminate the counterexample. Space-time interpolation offers a lazy, flexible framework for increasing precision while guaranteeing soundness, both for error avoidance and fixpoint detection. To the best of out knowledge, this is the first abstraction refinement scheme for the reachability analysis over unbounded and dense time of affine hybrid systems, which is both sound and automatic. We demonstrate the effectiveness of our algorithm with several benchmark examples, which cannot be handled by other tools

IST Austria Thesis

Hybrid automata combine finite automata and dynamical systems, and model the interaction of digital with physical systems. Formal analysis that can guarantee the safety of all behaviors or rigorously witness failures, while unsolvable in general, has been tackled algorithmically using, e.g., abstraction, bounded model-checking, assisted theorem proving. Nevertheless, very few methods have addressed the time-unbounded reachability analysis of hybrid automata and, for current sound and automatic tools, scalability remains critical. We develop methods for the polyhedral abstraction of hybrid automata, which construct coarse overapproximations and tightens them incrementally, in a CEGAR fashion. We use template polyhedra, i.e., polyhedra whose facets are normal to a given set of directions. While, previously, directions were given by the user, we introduce (1) the first method for computing template directions from spurious counterexamples, so as to generalize and eliminate them. The method applies naturally to convex hybrid automata, i.e., hybrid automata with (possibly non-linear) convex constraints on derivatives only, while for linear ODE requires further abstraction. Specifically, we introduce (2) the conic abstractions, which, partitioning the state space into appropriate (possibly non-uniform) cones, divide curvy trajectories into relatively straight sections, suitable for polyhedral abstractions. Finally, we introduce (3) space-time interpolation, which, combining interval arithmetic and template refinement, computes appropriate (possibly non-uniform) time partitioning and template directions along spurious trajectories, so as to eliminate them. We obtain sound and automatic methods for the reachability analysis over dense and unbounded time of convex hybrid automata and hybrid automata with linear ODE. We build prototype tools and compare—favorably—our methods against the respective state-of-the-art tools, on several benchmarks

LNCS

Template polyhedra generalize intervals and octagons to polyhedra whose facets are orthogonal to a given set of arbitrary directions. They have been employed in the abstract interpretation of programs and, with particular success, in the reachability analysis of hybrid automata. While previously, the choice of directions has been left to the user or a heuristic, we present a method for the automatic discovery of directions that generalize and eliminate spurious counterexamples. We show that for the class of convex hybrid automata, i.e., hybrid automata with (possibly nonlinear) convex constraints on derivatives, such directions always exist and can be found using convex optimization. We embed our method inside a CEGAR loop, thus enabling the time-unbounded reachability analysis of an important and richer class of hybrid automata than was previously possible. We evaluate our method on several benchmarks, demonstrating also its superior efficiency for the special case of linear hybrid automata