The present paper deals with the learnability of indexed families of uniformly recursive languages from positive data as well as from both, positive and negative data. We consider the influence of various monotonicity constraints to the learning process, and provide a thorough study concerning the influence of several parameters. In particular, we present examples pointing to typical problems and solutions in the field. Then we provide a unifying framework for learning. Furthermore, we survey results concerning learnability in dependence on the hypothesis space, and concerning order independence. Moreover, new results dealing with the efficiency of learning are provided. First, we investigate the power of iterative learning algorithms. The second measure of efficiency studied is the number of mind changes a learning algorithm is allowed to perform. In this setting we consider the problem whether or not the monotonicity constraints introduced do influence the efficiency of learning algo...

The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomono-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.

In this paper we survey the use of different AI methods for algorithmic composition, present their advantages and disadvantages, discuss some important general issues and propose desirable future prospects.

The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resource-oriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive P-specific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.

In designing learning algorithms it seems quite reasonable to construct them in a way such that all data the algorithm already has obtained are correctly and completely reflected in the hypothesis the algorithm outputs on these data. However, this approach may totally fail, i.e., it may lead to the unsolvability of the learning problem, or it may exclude any efficient solution of it. In particular, we present a natural learning problem and prove that it can be solved in polynomial time if and only if the algorithm is allowed to ignore data.

Kleene's Second Recursion Theorem provides a means for transforming any program p into a program e(p) which first creates a quiescent self copy and then runs p on that self copy together with any externally given input. e(p), in effect, has complete (low level) self knowledge, and p represents how e(p) uses its self knowledge (and its knowledge of the external world). Infinite regress is not required since e(p) creates its self copy outside of itself. One mechanism to achieve this creation is a self replication trick isomorphic to that employed by single-celled organisms. Another is for e(p) to look in a mirror to see which program it is. In 1974 the author published an infinitary generalization of Kleene's theorem which he called the Operator Recursion Theorem. It provides a means for obtaining an (algorithmically) growing collection of programs which, in effect, share a common (also growing) mirror from which they can obtain complete low level models of themselves and the other prog...

Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppersemilattice. The local theory concerns

We extend Angluin's (1980) theorem to characterize identifiability of indexed families of r.e. languages, as opposed to indexed families of recursive languages. We also prove some variants characterizing conservativity and two other similar restrictions, paralleling Zeugmann, Lange, and Kapur's (1992, 1995) results for indexed families of recursive languages. 1 Introduction A significant portion of the work of recent years in the field of inductive inference of formal languages, as initiated by Gold 1967, stems from Angluin's (1980b) theorem, which characterizes when an indexed family of recursive languages is identifiable in the limit from positive data in the sense of Gold. Up until around 1980, a prevalent view had been that inductive inference from positive data is too weak to be of much theoretical interest. This misconception was due to the negative result in Gold's original paper, which says that any class of languages that contains every finite language and at least one infini...

Machine learning of limit programs (i.e., programs allowed finitely many mind changes about their legitimate outputs) for computable functions is stud-ied. Learning of iterated limit programs is also studied. To partially motivate these studies, it is shown that, in some cases, interesting global properties of computable functions can be proved from suitable (n + 1)-iterated limit pro-grams for them which can not be proved from any n-iterated limit programs for them. It is shown that learning power is increased when (n + 1)-iterated limit programs rather than n-iterated limit programs are to be learned. Many trade-off results are obtained regarding learning power, number (possibly zero) of limits taken, program size constraints and information, and number of errors tolerated in final programs learned.

Induction is the process by which we reason from the particular to the general; In this paper we use ideas from the theory of abstract machines and recursion theory to study this process. We focus on pure induction in which the conclusions "go beyond the information given " in the premises from which they are derived and on simple induction, which is rather a stark kind of induction that deals with computable predicates on the integers in rather straightforward ways. Our basic question is "What are the relationships between the kinds of abstract machinery we bring to bear on the job of doing induction and our ability to do that job well? " Our conclusions are as follows: (1) If we use only the abstract machinery of the digital computer in a computing center (which we assume to be capable of only evaluating totally computable functionals or functionals in 210 of the Arithmetic Hierarchy) then a single inductive procedure can only develop finitely many sound theories. (2) If we use only the abstract machinery of the mathematician (which we assume to be the machinery required to evaluate a functional in 271 of the Arithmetic Hierarchy) then we can develop inductive