Verification of liveness properties on hybrid dynamical systems

A hybrid dynamical system is a mathematical model for a part of the real world where discrete and continuous parts interact with each other. Typically such systems are complex, and it is difficult to know how they will behave for general parameters and initial conditions. However, the method of formal verification gives us the ability to prove automatically that certain behaviour does or does not happen for a range of parameters in a system. The challenge is then to define suitable methods for proving properties on hybrid systems.This thesis looks at using formal verification for proving liveness properties on hybrid systems: a liveness property says that something good eventually happens in the system. This work presents the theoretical background and practical application of various methods for proving and disproving inevitability properties (a type of liveness) in different classes of hybrid systems. The methods combine knowledge of dynamical behaviour of a system with the brute-force approach of model checking, in order to make the most of the benefits of both sides. The work on proving liveness properties is based on abstraction of dynamical systems to timed automata. This thesis explores the limits of a pre-defined abstraction method, adds some dynamical knowledge to the method, and shows that this improvement makes liveness properties provable in certain continuous dynamical systems. The limits are then pushed further to see how this method can be used for piecewise-continuous dynamical systems. The resulting algorithms are implemented for both classes of systems.In order to disprove liveness properties in hybrid systems a novel framework is proposed, using a new property called deadness. Deadness is a dynamically-aware property of the hybrid system which, if true, disproves the liveness property by means of a finite execution: we usually require an infinite execution to disprove a liveness property. An algorithm is proposed which uses dynamical properties of hybrid systems to derive deadness properties automatically, and the implementation of this algorithm is discussed and applied to a simplified model of an oilwell drillstring.

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