  Axiomatic approach has demonstrated its power in mathematics. The main goal
of this preprint is to show that axiomatic methods are also very efficient for
computer science. It is possible to apply these methods to many problems in
computer science. Here the main modes of computer functioning and program
execution are described, formalized, and studied in an axiomatic context. The
emphasis is on three principal modes: computation, decision, and acceptation.
Now the prevalent mode for computers is computation. Problems of artificial
intelligence involve decision mode, while communication functions of computer
demand accepting mode. The main goal of this preprint is to study properties of
these modes and relations between them. These problems are closely related to
such fundamental concepts of computer science and technology as computability,
decidability, and acceptability. In other words, we are concerned with the
question what computers and software systems can do working in this or that
mode. Consequently, results of this preprint allow one to achieve higher
understanding of computations and in such a way, to find some basic properties
of computers and their applications. Classes of algorithms, which model
different kinds of computers and software, are compared with respect to their
computing, accepting or deciding power. Operations with algorithms and machines
are introduced. Examples show how to apply axiomatic results to different
classes of algorithms and machines in order to enhance their performance.
