Hybrid systems and quantum automata: Preliminary announcement

Let H denote an algebra of input symbols or events. If X is the state space for a system, then one can form the space R of observations of X. Under suitable conditions, both X and R are H-modules. Loosely speaking, the formal systems studied in this paper consists of a bialgebra H describing the input symbols and two H-modules describing the states and observations of the system. Finite automata and input-output systems are concerned with commutative R, while quantum systems, such as quantum automata, are concerned with non-commutative R arising from Hermitian operators on the state space. Of special interest are those systems consisting of interacting networks of classical systems and automata. These types of systems have become known as hybrid systems and are examples of formal systems with commutative R. In this paper, we present a number of examples of hybrid systems and quantum automata and point out some relationships between them. This is a preliminary announcement: a detailed exposition, including proofs, will appear elsewhere.