Exact symbolic models for verification

The evolution of physical quantities such as position, temperature, humidity, etc, is usually described by differential equations with solutions evolving on $${\mathbb R}^n$$ or appropriate subsets. The infinite cardinality of $${\mathbb R}^n$$ prevents a direct application of the verification methods described in Chapter 5. However, verification algorithms are still applicable whenever suitable finite-state abstractions of these infinite-state systems can be constructed. In recent years, several methods have been proposed for the construction of these abstractions based on a very interesting blend of different mathematical techniques. We present several of these methods starting with timed automata to illustrate the general principles of the abstraction process. Most of the abstraction techniques described in this chapter require linear differential equations. For this reason, we discuss as a special topic how to transform a class of nonlinear differential equations into linear di_erential equations in larger state spaces.