Kolmogorov Complexity and the Recursion Theorem

Abstract

Abstract. Several classes of diagonally non-recursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PA-complete, that is, A can compute a {0, 1}-valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal C-complexity among the strings of length n. A ≥ T K iff A can compute a function F such that F (n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem.

Evolving Computability

“…On the one hand, these remarks lead to a theory of intransitive indifference relations, of which concurrency is an example, axiomatized by Petri [29,32] in relation to a finitistic view of continuity as a foundation for measurement. On the other hand, they set the stage for Holt's investigation of motion complying with the needs of communication mechanics, culminating in his unpublished contribution to the May 1981 MIT-IBM conference on Physics of Computation [14].…”

“…Synthesis of reactive systems, that is of systems that interact with an environment, started as a problem in logics. Rabin's result about the decidability of monadic second-order logic over infinite trees solved Church's question for MSO specifications [32]. Church asked for an algorithm to construct devices that transform sequences of input bits into sequences of output bits in a way required by a logical formula [9].…”

“…By [32] we have k REC = d REC , and by [16, Theorem 6.2.3] we have k 1 = r 1 where 1 : N → N is the identity function. Building on the methods of Miller [40], Hudelson [24] has shown that d REC < k f < k g ≤ r 1 holds for many pairs of unbounded recursive functions f, g. In particular, this holds whenever ∀n…”

“…6. A function Z ∈ N N is said to be diagonally nonrecursive [3,23,26,32,65] if Z ∩ ψ = ∅ where ψ is the well known diagonal function, defined by ψ(n) ϕn (n). 14 Let ϕn, n ∈ N be a fixed, standard, partial recursive enumeration of the partial recursive functions.…”

“…Within this model there are obtained several universality results; we cite only [9,31] as well as [29] for the variant of the model using splicing operation. We refer to [32] for a more comprehensive survey on this topic. We refer to [32] for a more comprehensive survey on this topic.…”