A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

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

Deciding Reachability for Piecewise Constant Derivative Systems on Orientable Manifolds

© 2019 Springer-Verlag. This is a post-peer-review, pre-copyedit version of a paper published in Reachability Problems: 13th International Conference, RP 2019, Brussels, Belgium, September 11–13, 2019, Proceedings. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-030-30806-3_14A hybrid automaton is a finite state machine combined with some k real-valued continuous variables, where k determines the number of the automaton dimensions. This formalism is widely used for modelling safety-critical systems, and verification tasks for such systems can often be expressed as the reachability problem for hybrid automata. Asarin, Mysore, Pnueli and Schneider defined classes of hybrid automata lying on the boundary between decidability and undecidability in their seminal paper ‘Low dimensional hybrid systems - decidable, undecidable, don’t know’ [9]. They proved that certain decidable classes become undecidable when given a little additional computational power, and showed that the reachability question remains unsolved for some 2-dimensional systems. Piecewise Constant Derivative Systems on 2-dimensional manifolds (or PCD2m) constitute a class of hybrid automata for which decidability of the reachability problem is unknown. In this paper we show that the reachability problem becomes decidable for PCD2m if we slightly limit their dynamics, and thus we partially answer the open question of Asarin, Mysore, Pnueli and Schneider posed in [9]

Model checking polygonal differential inclusions using invariance kernels

Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we identify and compute an important object of such systems’ phase portrait, namely invariance kernels. An invariant set is a set of initial points of trajectories which keep rotating in a cycle forever and the invariance kernel is the largest of such sets. We show that this kernel is a non-convex polygon and we give a non-iterative algorithm for computing the coordinates of its vertices and edges. Moreover, we present a breadth-first search algorithm for solving the reachability problem for such systems. Invariance kernels play an important role in the algorithm.peer-reviewe

IST Austria Technical Report

As hybrid systems involve continuous behaviors, they should be evaluated by quantitative methods, rather than qualitative methods. In this paper we adapt a quantitative framework, called model measuring, to the hybrid systems domain. The model-measuring problem asks, given a model M and a specification, what is the maximal distance such that all models within that distance from M satisfy (or violate) the specification. A distance function on models is given as part of the input of the problem. Distances, especially related to continuous behaviors are more natural in the hybrid case than the discrete case. We are interested in distances represented by monotonic hybrid automata, a hybrid counterpart of (discrete) weighted automata, whose recognized timed languages are monotone (w.r.t. inclusion) in the values of parameters.The contributions of this paper are twofold. First, we give sufficient conditions under which the model-measuring problem can be solved. Second, we discuss the modeling of distances and applications of the model-measuring problem

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

Hybrid Automata in Systems Biology: How far can we go?

We consider the reachability problem on semi-algebraic hybrid automata. In particular, we deal with the effective cost that has to be afforded to solve reachability through first-order satisfiability. The analysis we perform with some existing tools shows that even simple examples cannot be efficiently solved. We need approximations to reduce the number of variables in our formulae: this is the main source of time computation growth. We study standard approximation methods based on Taylor polynomials and ad-hoc strategies to solve the problem and we show their effectiveness on the repressilator case study

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

Discrete Semantics for Hybrid Automata

Many natural systems exhibit a hybrid behavior characterized by a set of continuous laws which are switched by discrete events. Such behaviors can be described in a very natural way by a class of automata called hybrid automata. Their evolution are represented by both dynamical systems on dense domains and discrete transitions. Once a real system is modeled in a such framework, one may want to analyze it by applying automatic techniques, such as Model Checking or Abstract Interpretation. Unfortunately, the discrete/continuous evolutions not only provide hybrid automata of great flexibility, but they are also at the root of many undecidability phenomena. This paper addresses issues regarding the decidability of the reachability problem for hybrid automata (i.e., "can the system reach a state a from a state b?") by proposing an "inaccurate" semantics. In particular, after observing that dense sets are often abstractions of real world domains, we suggest, especially in the context of biological simulation, to avoid the ability of distinguishing between values whose distance is less than a fixed \u3b5. On the ground of the above considerations, we propose a new semantics for first-order formul\ue6 which guarantees the decidability of reachability. We conclude providing a paradigmatic biological example showing that the new semantics mimics the real world behavior better than the precise one