DS-1998-01: Computability and Measure

DS-1998-01: Terwijn, Sebastiaan A. (1998) Computability and Measure. Doctoral thesis, University of Amsterdam.

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Abstract

n this dissertation, we discuss several issues in recursion theory that relate to measure and randomness. The central concept in recursion theory is the concept of a "recursive set." A set is recursive if an algorithm exists to determine whether something is an element of this set. When studying subclasses of the class of recursive sets, one generally speaks of complexity theory.
In Chapter 1, we introduce and discuss the central concepts of this dissertation. In particular, we discuss some elementary measure theory and its presentation using so-called martingales. These functions, which can be viewed as betting strategies, are used throughout a large part of this dissertation to describe constructive measure theory, following the work of Schnorr and Lutz. In this theory, various concepts from classical measure theory are made constructive by imposing the requirement that they be computable. The degree of computability acts here as a parameter \Delta, which can be thought of as a limit on the permitted methods. For this reason, this form of constructive measure theory is also called "resource-bounded measure theory."
The goal of resource-bounded measure theory is twofold. First, by choosing \Delta sufficiently strictly, it becomes possible to apply ideas from classical measure theory to the study of various complexity classes. This provides information about how "most" elements of a complexity class behave. Second, studying resource-bounded measure provides insight into the behavior of random or stochastic sets. We can think of these as sets generated by a random process, such as flipping a coin. A set A is \Delta-random if {A} does not have measure zero in the resource-bounded measure theory with parameter \Delta. Intuitively, this means that an algorithm from class \Delta cannot discover any regularity in the set A.
The ideas above are applied in Chapter 2 to the complexity class E of sets that are computable in exponential time. In the first part of this chapter, so-called generic sets are studied and used to prove a generalization of a theorem by Juedes and Lutz. In section 2.5, randomness and genericity are compared and it becomes clear that in this context generic sets can be understood as a weak variant of random sets. In the second part of Chapter 2, random sets are used to answer a question from Lutz regarding the existence of weakly complete sets that are not complete.
In Chapter 3, Lutz’s design for resource-bounded measure theory is slightly adapted to define a measure suitable for the study of recursively enumerable (r.e.) sets. This is a class of sets that plays a prominent role in recursion theory. We study weak notions of completeness and obtain a full picture of the relations between the different notions of weak and "ordinary" completeness. The study of these notions also has consequences for a question that does not mention the concept of measure, namely the question of to what extent an incomplete set can resemble a complete set.
In Chapter 4, we study classes of martingales corresponding to the classes from the arithmetical hierarchy. In particular, we study the resource-bounded measure defined by taking the set of recursively enumerable functions for the class \Delta. The associated random sets are exactly the sets originally introduced by Martin-Löf as a proposal for a general definition of randomness. The existence of universal recursively enumerable sets results in the r.e.-random sets having elegant properties. We describe the distribution of these sets in terms of the well-known reducibility relations from recursion theory. We locate the class R(r.e.) of sets constructed by so-called r.e.-constructors. Unlike the classes R(\Delta) from Lutz’s resource-bounded measure theory, the class R(r.e.) does not correspond to a known class from recursion theory. Finally, we address analogous questions for the measures belonging to the levels \Delta_n of the arithmetical hierarchy, and we prove that these measures coincide with the measures belonging to the levels \Pi_n.
Chapter 5 deals with sets that are "low" for two classes of random sets: the class R of Martin-Löf from Chapter 4 and the class S, originally introduced by Schnorr as a more constructive version of R. A set A is low for a class C if for the relativized version C^A of C it holds that C = C^A. Intuitively: if A relative to C does not contribute additional computing power. Recursive sets are trivially low for both R and S. We prove that in both cases, non-recursive sets also exist that are low. This shows that substantial amounts of information can be encoded in a set in such a way that they are not accessible to elements of R and S respectively. The cases R and S differ considerably. In the case of R, we construct a non-recursive, recursively enumerable low set, and we do not know if such sets exist outside of \Delta_2. In the case of S, we construct 2^{\aleph_0} non-recursive low sets and show that these must necessarily lie outside of \Delta_2. The results for the sets that are low for S are obtained via a characterization of these sets in purely recursion-theoretic terms. According to this characterization, the functions that are recursive in a low set are recursively traceable. We further show that sets that are low for S are hyper-immune, and that the converse of this statement does not hold in general.
In Chapter 6, finally, we briefly discuss a number of themes relating to recursive martingales. Topics include reducibility to random sets, relations between recursive randomness and Kolmogorov complexity, the relation between the measure defined by recursive martingales and the Schnorr measure, partial recursive martingales, and measure in \Delta_2.

Item Type:	Thesis (Doctoral)
Report Nr:	DS-1998-01
Series Name:	ILLC Dissertation (DS) Series
Year:	1998
Depositing User:	Dr Marco Vervoort
Date Deposited:	14 Jun 2022 15:16
Last Modified:	09 Apr 2026 22:20
URI:	https://eprints.illc.uva.nl/id/eprint/2004

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