Dynamic properties of computably enumerable sets

A set A ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post 1944] there has been much interest in relating the deenable (especially E-deenable) properties of a c.e. set A to its information contentt, namely its Turing degree, deg(A), under T , the usual Turing reducibility. Turing 1939]. Recently, Har-rington and Soare answered a question arising from Post's program by constructing a nonemptly E-deenable property Q(A) which guarantees that A is incomplete (A < T K). The property Q(A) is of the form (9C))A m C & Q ? (A; C)], where A m C abbreviates that A is a major subset of C, and Q ? (A; C) contains the main ingredient for incompleteness. A dynamic property P(A), such as prompt simplicity, is one which is deened by considering how fast elements elements enter A relative to some simultaneous enumeration of all c.e. sets. If some set in deg(A) is promptly simple then A is prompt and otherwise tardy. We introduce here two new tardiness notions, small-tardy(A; C) and Q-tardy(A; C). We begin by proving that small-tardy(A; C) holds ii A is small in C (A s C) as deened by Lachlan 1968]. Our main result is that Q-tardy(A; C) holds ii Q ? (A; C). Therefore, the dynamic property, Q-tardy(A; C), which is more intuitive and easier to work with than the E-deenable counterpart, Q ? (A; C), is exactly equivalent and captures the same incompleteness phenomenon.

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