A generalization of the notions of a machine and computability

In this paper we define the notion of an abstract (Z, Q)-machine, which is a mathematical model for a device uniquely extending the functions of a real variable on the set Z to the set Q, where Z, Q are some subsets of the set of all nonnegative real numbers and Z ⊊ Q. Every such extension is called a computation of the machine. Any function which is a computation of some (Z, Q)machine is called (Z, Q)-computable. Similarly, a set of functions is called (Z, Q)-computable if it is the set of all computations of some (Z, Q)-machine. We examine the basic properties of these notions. It is proved that the theory of (Z, Q)-machines contains as special cases the theory of some discrete, continuous, as well as hybrid computers. Consequently, the paper contains one general mathematical model for a uniform theory of computability, closely related to existing computer practice