DECIDABILITY AND UNIVERSALITY IN THE AXIOMATIC THEORY OF COMPUTABILITY AND ALGORITHMS

In this paper, we study to what extent decidability is connected to universality. A natural context for such a study is provided by the axiomatic theory of computability, automata and algorithms. In Section 2, we introduce necessary concepts, constructions, axioms, postulates, conditions and problems. Section 3 contains the main results of the paper. In particular, it is demonstrated (Theorems 1 and 2) that undecidability of algorithmic problems does not depend on existence of universal algorithms and may be caused by weaker conditions. At the same time, results of Theorems 2 and 3 demonstrate that decidability is incompatible with universality. It is also proved (Theorem 5) that sufficiently big classes of total algorithms/automata, such as the class of all primitive recursive functions, cannot have universal algorithms/automata.