Functional System

an important concept of mathematical cybernetics. A functional system consists of a set of functions and a set of operations that are applied to the functions. It constitutes a formalized reflection of the following principal aspects of real and abstract control systems: (1) the functioning of a system (the functions in a functional system), (2) rules for constructing more complex control systems from given systems, and (3) a description of the functioning of the complex systems in terms of the functioning of their components. (The second and third aspects are reflected in the operations of a functional system.) Examples of functional systems are many-valued logics, automata algebras, and algebras of recursive functions.

The specific character of a functional system depends on the standpoint from which the problems and approaches arising in the study of the functional system are viewed. For example, from the perspective of mathematical cybernetics, functional systems are regarded as languages describing the functioning of complex systems. From the perspective of mathematical logic, functional systems are treated as models of logics—that is, as systems of propositions on which logical operations are defined. From the perspective of algebra, functional systems are algebraic systems. An important characteristic distinguishing a functional system from the general class of algebraic systems is that a functional system has a meaningful connection with real cybernetic models of control systems. This connection, on the one hand, determines the range of essential requirements placed on a functional system and, on the other hand, gives rise to a series of important problems that have both theoretical and applied significance.

The study of functional systems began with particular logic models, one of the first of which was two-valued logic. Subsequently, a number of specific functional systems were studied; these systems also contributed to the formation of the concept of a functional system. The set of problems associated with functional systems is vast and has much in common with the problems of many-valued logic. Among the most important problems for functional systems are completeness, the complexity of expressing functions in terms of other functions, identity transformations, and synthesis and analysis. Considerable success has been achieved in solving these problems for many specific functional systems.