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]

STORMED hybrid systems

Abstract. We introduce STORMED hybrid systems, a decidable class which is similar to o-minimal hybrid automata in that the continuous dynamics and constraints are described in an o-minimal theory. However, unlike o-minimal hybrid automata, the variables are not initialized in a memoryless fashion at discrete steps. STORMED hybrid systems require flows which are monotonic with respect to some vector in the continuous space and can be characterised as bounded-horizon systems in terms of their discrete transitions. We demonstrate that such systems admit a finite bisimulation, which can be effectively constructed provided the o-minimal theory used to describe the system is decidable. As a consequence, many verification problems for such systems have effective decision algorithms

Computation in Economics

This is an attempt at a succinct survey, from methodological and epistemological perspectives, of the burgeoning, apparently unstructured, field of what is often – misleadingly – referred to as computational economics. We identify and characterise four frontier research fields, encompassing both micro and macro aspects of economic theory, where machine computation play crucial roles in formal modelling exercises: algorithmic behavioural economics, computable general equilibrium theory, agent based computational economics and computable economics. In some senses these four research frontiers raise, without resolving, many interesting methodological and epistemological issues in economic theorising in (alternative) mathematical modesClassical Behavioural Economics, Computable General Equilibrium theory, Agent Based Economics, Computable Economics, Computability, Constructivity, Numerical Analysis

Hybridization based CEGAR for Hybrid Automata with Affine Dynamics

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Relaxing goodness is still good for SPDIs

Polygonal hybrid systems (SPDIs) are planar hybrid systems, whose dynamics are defined in terms of constant differential inclusions, one for each of a number of polygonal regions partitioning the plane. The reachability problem for SPDIs is known to be decidable, but depends on the goodness assumption — which states that the dynamics do not allow a trajectory to both enter and leave a region through the same edge. In this paper we extend the decidability result to generalised SPDIs (GSPDI), SPDIs not satisfying the goodness property, and give an algorithmic solution to decide reachability of such systems.peer-reviewe

Relaxing goodness is still good for SPDIs

Polygonal hybrid systems (SPDIs) are planar hybrid systems, whose dynamics are defined in terms of constant differential inclusions, one for each of a number of polygonal regions partitioning the plane. The reachability problem for SPDIs is known to be decidable, but depends on the goodness assumption — which states that the dynamics do not allow a trajectory to both enter and leave a region through the same edge. In this paper we extend the decidability result to generalised SPDIs (GSPDI), SPDIs not satisfying the goodness property, and give an algorithmic solution to decide reachability of such systems.peer-reviewe

Relaxing goodness is still good

Polygonal hybrid systems (SPDIs) are planar hybrid systems, whose dynamics are defined in terms of constant differential inclusions, one for each of a number of polygonal regions partitioning the plane. The reachability problem for SPDIs is known to be decidable, but depends on the goodnessassumption -- which states that the dynamics do not allow a trajectory to both enter and leave a region through the same edge. In this paper we extend the decidability result to generalised SPDIs(GSPDI), SPDIs not satisfying the goodness assumption, and give an algorithmic solution to decide reachability of such systems.peer-reviewe

Remedies for building reliable cyber-physical systems

Cyber-physical systems (CPS) are systems that are tight integration of computer programs as controllers or cyber parts, and physical environments. The interaction is carried out by obtaining information about the physical environment through reading sensors and responding to the current knowledge through actuators. Examples of such systems are autonomous automobile systems, avionic systems, robotic systems, and medical devices. Perhaps the most common feature of all these systems is that they are all safety critical systems and failure most likely causes catastrophic consequences. This means that while testing continues to increase confidence in cyber-physical systems, formal or mathematical proofs are needed at the very least for the safety requirements of these systems. Hybrid automata is the main modeling language for cyber-physical systems. However, verifying safety properties is undecidable for all but very restricted known classes of these automata. Our first result introduces a new subclass of hybrid automata for which bounded time safety model checking problem is decidable. We also prove that unbounded time model checking for this subclass is undecidable which suggests this is the best one can hope for the new class. Our second result in this thesis is a counter-example guided abstraction refinement algorithm for unbounded time model checking of non- linear hybrid automata. Clearly, this is an undecidable problem and that is the main reason for using abstraction refinement techniques. Our CEGAR framework for this class is sound but not complete, meaning the algorithm never incorrectly says a system is safe, but may output unsafe incorrectly. We have also implemented our algorithm and compared it with seven other tools. There are multiple inherent problems with traditional model checking approaches. First, it is well-known that most models do not depict physical environments precisely. Second, the model checking problem is undecidable for most classes of hybrid automata. And third, even when model checking is decidable, controller part in most models cannot be implemented. These problems suggest that current methods of modeling cyber-physical systems and problems might not be the right ones. Our last result focuses on robust model checking of cyber-physical systems. In this part of the thesis, we focus on the implementability issue and show how to solve four different robust model checking problem for timed automata. We also introduce an optimal algorithm for robust time bounded safety model checking of monotonic rectangular automata