Discreteness, hybrid automata, and biology

Most of the observable natural phenomena exhibit a mixed discrete-continuous behavior characterized by laws changing according to a phase cycle. Such behaviors can be modeled in a very natural way by a class of automata called hybrid automata. In this class the evolution of measurable quantities, such as concentrations, is represented according to both dynamical system evolutions - on dense domains - and rules phases through a discrete transition structure. Once the real systems are modeled in such a framework, one may want to analyze them by applying automatic techniques, such as model checking or abstract interpretation. Unfortunately, the interleaving of dense and discrete evolutions soon leads to undecidability results on hybrid automata. This paper addresses questions regarding the decidability of reachability problem for hybrid automata (i.e., "can the systems reach a state a from a state b?") by proposing a more "Nature"-oriented semantics. In particular, after observing that dense domains are abstractions of real world, we suggest that, for any biological system, there should be a value ε such that if the distance of two objects are less than ε, we cannot distinguish them. Using the above considerations, we propose a new semantics for hybrid automata which guarantees the decidability of reachability. Moreover, we provide a biological example showing that the new semantics mimics the real world behaviors better than the "classical" one.