Fundamental Properties of Real Computers

Abstract

This paper treats fundamental properties of real computers. Unsolvable problems (the halting problem is a well-known example) are known to exist on theoretical computers such as Turing machines, random access machines, etc. This leads to a corresponding question regarding real computers: are there problems concerning computations performed on a real computer and important from viewpoint of its user which are at the same time unsolvable by the computer? We give affirmative answer here showing e.g., that the halting problem concerning a real computer and a collection of real programs satisfying a weak closure property is unsolvable by the computer. Unsolvability of another problem dealing with approximation of results of computations performed by programs from given collection is proved as well.

Any unsolvable problem is relativized with respect to a concrete. It is shown moreover that for every computer some more powerful computer can solve the problems which are unsolvable on the first computer. On the other hand there are problems concerning the capabilities of this more powerful computer that are unsolvable by it.

Thus, a hierarchy of computers is shown to exist. Each computer is more powerful then computers preceding it in the hierarchy. At the same time there are problems unsolvable by a computer from the hierarchy but solvable by computers lying in the hierarchy above it.

Consequences to computer construction as well as possible applications to computer science, analog computers, heuristic reasoning and intelligent systems are mentioned.