Genericity of Weakly Computable Objects

Abstract

In computability theory many results state the existence of objects that in many respects lack algorithmic structure but at the same time are effective in some sense. Friedberg and Muchnik's answer to Post's problem is one of the most celebrated results in this form. The main goal of the paper is to develop a general result that embodies a large number of these particular constructions, capturing the essential idea that is common to all of them, and expressing it in topological terms. To do so, we introduce the effective topological notions of irreversible function and directional genericity and provide two main results that identify situations when such constructions are possible, clarifying the role of topology in many arguments from computability theory. We apply these abstract results to particular situations, illustrating their strength and deriving new results. This paper is an extended version of the conference paper [Hoy14] with detailed proofs and new results.

Nearly Computable Real Numbers

“…The third observation of the kind described in the previous section (that asking a real number to be left-computable and nearly computable and to satisfy some other effectivity condition forces the number to be computable) that we wish to mention is the following result by Hoyrup [10].…”

“…For example, an argument by Downey, Hirschfeldt, and LaForte [5,Pages 105,106] shows that any strongly left-computable and nearly computable real number is even computable. In Section 8 we discuss three further results of the same type, one by Hoyrup [10] and two by Stephan and Wu [22]. Hoyrup [10] has shown that any nearly computable and left-computable number that is not computable is weakly 1-generic.…”

“…In Section 8 we discuss three further results of the same type, one by Hoyrup [10] and two by Stephan and Wu [22]. Hoyrup [10] has shown that any nearly computable and left-computable number that is not computable is weakly 1-generic. We show that in this result the assumption that the real number should be left-computable can be omitted.…”

The Typical Constructible Object

“…We apply this method to obtain a result in computable analysis related to the non-computability of the ergodic decomposition theorem. In [6] we give other applications of this method showing that many complicated constructions in recursion theory can be more easily obtained by choosing the suitable topology on the space of objects, and using the corresponding Baire Category theorem on that space.…”

Computability on Quasi-Polish Spaces

“…[10]). In a computable Polish space, a Π 0 2 -set is computably overt if and only if it contains a dense computable sequence.…”