On the inductive inference of recursive real-valued functions

This paper presents a metod of inductive inference of realvalued functions from given pairs of observed data of (x,h(x)), where h is a target function to be inferred. Eac of suc observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers sow te approximate value and te error bound, respectively. On the other hand, a real number called a recursive real number can be represented by a pair of two of rational numbers, which converges to the real number and converges to zero, respectively. These sequeces show an approximate value of te real number and an error bound at each point. Such a real number can also be represented by a sequence of closed intervals wit rational end points converges to a singleton interval with the real number as both end points.In this paper, we propose a notion of recursive real-valued functions that can enjoy the merits of the both representations of the recursive real numbers. Then we present an algorithm which approximately infers real-valued functions from numerical data with some error bounds, and show that there exists a rich set of real-valued functions which is approximately inferable in the limit from such numerical data. We also discuss the precision of the guesses from the machine when sufficient data have not yet given.

A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

4 - Recursive Enumerability

is. The study of partially ordered sets is a kind of abstract mathematics of the most general sort, compared with the theory of algebraic integers which gave rise to ideal theory. I should like to say a few words about the distinction which is often made between abstract mathematics and the other kind, which may be called classical mathematics. Classical mathematics is based on the real number system, or equivalently on the system of positive integers or on the complex number system. Abstract mathematics is based on set theory. It studies the consequences of quite arbitrary postulate systems. From the point of view of the abstract mathematician, the real number system is a particular ordered field, or a particular locally compact topological ring; in other words, one of a class of similar abstract systems, and not necessarily more interesting than the others. From this same point of view, classical mathematics is a special case of a more general abstract theory. Actually, all mathematics is equally abstract. The real number system is no more concrete than, say, the theory of groups. Neither deals with the physical universe. To be sure, classical mathematics has seemed to be the kind most useful in applications to the sciences. However, perhaps this is an illusion, due to the time lag between the development of a theory and its application. The classical mathematics of the last century is being applied now. The abstract mathematics now being developed may turn out to be just as useful in the next century. Perhaps it will be found that you can kill just as many people just as efficiently using the theory of quasi-groups or p-adic fields as you can with the real number system. Of course many mathematicians will disagree with this view. But it is true now, just as it has been found to be true in the past, that mathematical results arrived at with no thought of their possible use are being applied to situations not imagined by their originators.

We consider a learning model in which each element of a class of recursive functions is to be identified in the limit by a computable strategy. Given gradually growing initial segments of the graph of a function, the learner is supposed to generate a sequence of hypotheses converging to a correct hypothesis. The term correct means that the hypothesis is an index of the function to be learned in a given numbering. Restriction of the basic definition of learning in the limit yields several inference criteria, which have already been compared with respect to their learning power.The scope of uniform learning is to synthesize appropriate identification strategies for infinitely many classes of recursive functions by a uniform method, i.e. a kind of meta-learning is considered. In this concept we can also compare the learning power of several inference criteria. If we fix a single numbering to be used as a hypothesis space for all classes of recursive functions, we obtain results similar to the non-uniform case. This hierarchy of inference criteria changes, if we admit different hypothesis spaces for different classes of functions. Interestingly, in uniform identification most of the inference criteria can be separated by collections of finite classes of recursive functions.

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

A point of Euclidean 1-space R1 has just one coordinate x, and thus it should be denoted by (x), in accordance with the general terminology introduced in I.2.1. However, as a matter of convenience, we shall speak of the point x instead of the point (x). If this terminology is adopted, then the points of R1 are merely the real numbers, and if x′, x″ are two real numbers then their (Euclidean) distance is equal to ∣x″ - x′∣, the absolute value of their difference (see I.2.1). While, in general, there exists a significant conceptual difference between an object x and the set consisting of the single object x, it should be noted that in the present case we could operate directly with the space whose points are the real numbers, the distance of two real numbers x′, x″ being defined as ∣x″ - x′∣. The space so obtained is the real number-line of Analysis. In other words, the notational agreement of writing x instead of (x) has the effect that R1 becomes formally identical with the real number-line. Let now U be a non-empty subset of R1, and consider a transformation T: U → R1. Then T associates with every real number u ∈ U a definite real number x = T(u). Thus T may be thought of as a real-valued function x = T(u) of the real variable u ∈ U.

Fat-Shattering and the Learnability of Real-Valued Functions

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.

On some open problems in reflective inductive inference

If / is a function on Ft1 of A-bounded variation and period 2ir, then its nth Fourier coefficient f(n) = 0(1/E 1/Xj) and its integral modulus of continuity wi(/;<5) = 0(l/s(1/i) 1/X,-). The result on /(n) is best possible in a sense. These results can be extended to certain other classes of functions of generalized variation. For various classes of functions, estimates of the magnitude of the Fourier coeffi- cients and of the integral moduli of continuity can be made. Here we shall do this for certain classes of functions of generalized bounded variation. 1. Let us suppose that / is a real-valued function on an interval 7 of R1. If In = (an, bn) C 7, set f(In) = f(bn) — f(an). We suppose the intervals 7n, n = 1,2,..., to be nonoverlapping. If A = {Xn} is a nondecreasing sequence of positive real numbers such that ^2 1/Xn = oo, we have said (5) that / is of A-bounded variation (ABV) if, for every {7},

Quantum inductive inference by finite automata

This paper examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both “generalized concave” and “generalized convex” is closely related to the class of monotone functions of a single variable. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. Several results characterizing the extreme values of generalized concave functions are given.

The mutation operator is the only source of variation in Evolutionary Programming. In the past these have been human nominated and included the Gaussian, Cauchy, and the Levy distributions. We automatically design mutation operators (probability distributions) using Genetic Programming. This is done by using a standard Gaussian random number generator as the terminal set and and basic arithmetic operators as the function set. In other words, an arbitrary random number generator is a function of a randomly (Gaussian) generated number passed through an arbitrary function generated by Genetic Programming. Rather than engaging in the futile attempt to develop mutation operators for arbitrary benchmark functions (which is a consequence of the No Free Lunch theorems), we consider tailoring mutation operators for particular function classes. We draw functions from a function class (a probability distribution over a set of functions). The mutation probability distribution is trained on a set of function instances drawn from a given function class. It is then tested on a separate independent test set of function instances to confirm that the evolved probability distribution has indeed generalized to the function class. Initial results are highly encouraging: on each of the ten function classes the probability distributions generated using Genetic Programming outperform both the Gaussian and Cauchy distributions.

The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied from two different but complementary points of view. The first approach is concerned with individuating those numeric functions that are effectively calculable. This approach reached its systematization with the theory of the recursive functions (Gödel, Church Kleene).This theory is not directly concerned with computing devices or computations. Rather, the effective calculability of a recursive function is guaranteed by the algorithmic nature of its definition. In contrast, the second approach focuses on a family of abstract mechanisms, which are then typically used to compute or recognize numeric functions, sets of numbers, or numbers. These devices can be divided into two broad categories: automata or machines (Turing and Post), and systems of rules for symbol manipulation (Post). The mechanisms that have been studied include: a. Automata or Machines 1. gate-nets and McCulloch-Pitts nets 2. finite automata (Mealy and Moore machines) 3. push-down automata 4. stack automata 5. Turing machines 6. register machines 7. wang machines 8. cellular automata b. Systems of Rules 9. monogenic production systems in general 10. monogenic Post canonical systems 11. monogenic Post normal systems 12. tag systems. I call any device studied by computation theory a computational system. Computation theory is traditionally interested in studying the relations between each type of computational system and the others, and in establishing what class of numeric functions each type can compute. Accordingly one proves two kinds of theorem: (1) that systems of a given type emulate systems of another type (examples: Turing machines emulate register machines and cellular automata; cellular automata emulate Turing machines, etc.), and (2) that a certain type of system is complete relative to the class of the (partial) recursive functions or, in other words, that this type of system can compute all and only the (partial) recursive functions (examples of complete systems: Turing machines, register machines, cellular automata, tag systems, etc.). All different types of computational systems have much in common. Nevertheless, it is not at all clear exactly which properties these mechanisms share.