The Upper Semi-Lattice of Degrees of Recursive Unsolvability

Abstract

The concept 'degree of recursive unsolvability' was introduced briefly in Post [16]. In his abstract [17] the concept was formulated precisely via an extension of [15], and a resulting partial scale of degrees of recursive unsolvability was applied to strengthen Theorem II of Kleene [8]. In the present paper our interest is in the abstract structure of the system of the degrees of recursive unsolvability. The concept 'degree of recursive unsolvability' is based on that of reducibility of decision problems. Three precise formulations of the latter concept have appeared, more or less completely, in the literature. In 'Turing reducibility' [19] ?4, the concept of a Turing machine [18] is generalized to that of a Turing reducibility machine. A Turing reducibility machine 9) reduces the decision problem for a set S to that for a set T, if for each positive integer n, the machine 9M applied to n terminates in the correct answer to the question whether n is in S, via ordinary Turing machine acts and the hypothetically correct answering of such questions of the form Is m in T? as may arise in the process (finite in number). In 'general recursive reducibility', the representing function of S is general recursive in that of T in the sense of Kleene [8].1 In Post's 'canonical reducibility' [17], his canonical sets [15] are generalized to T-canonical sets by hypothetically adding primitive assertions expressing the membership or non-membership of 1, 2, 3, ... in T. Then S is canonically reducible to T, if both S and its complement S with respect to the set of all positive integers are T-canonical sets. Although an infinite collection of hypothetical primitive assertions are added to an otherwise ordinary canonical system, in any derivation of an assertion in the T-canonical system but a finite number of them enter. Turing reducibility was proved equivalent to canonical reducibility by Post (unpublished), and canonical reducibility to general recursive reducibility by Martin Davis in his typed thesis [5]. A direct proof of the equivalence of Turing reducibility to general recursive reducibility was published (independently) by Kleene in his book [10] end ??68, 69. Theoretically, no increase in generality is obtained by the use of Kleene's concept of a function general recursive in other functions (as may be seen from [10] pp. 307, 291); and by talking about just sets S and T a greater simplicity of concept results. Practically, when it comes to giving rigorous demonstrations under one or another of the equivalent concepts of reducibility, there are advantages in using