Dimiter Skordev

Computability of Real Numbers by Using a Given Class of Functions in the Set of the Natural Numbers

Given a class ℱ oft otal functions in the set oft he natural numbers, one could study the real numbers that have arbitrarily close rational approximations explicitly expressible by means of functions from ℱ. We do this for classes ℱsatisfying certain closedness conditions. The conditions in question are satisfied for example by the class of all recursive functions, by the class of the primitive recursive ones, by any of the Grzegorczyk classes ℰnwith n ≥ 2, by the class of all functions recursive in a given function and by the class of the functions primitive recursive in it, as well as by the class of all total functions in the set of the natural numbers.

An abstract approach to some loop detection problems

The abstract approach proposed here encompasses both the detection of some periodic loops during the execution of Prolog programs and the detection of some periodic loops during recursive computations (an attempt to look at the loop detection problem for Prolog from an abstract point of view has been done previously in a paper by the same author published in 1993).

M2-computable real numbers

The article concerns subrecursive computability of real numbers. Certain significant real numbers are shown to be M2-computable, and the set of the M2-computable real numbers is shown to be closed under the elementary functions of calculus.

M 2 -Computable Real Numbers

The article concerns subrecursive computability of real numbers. Certain significant real numbers are shown to be M2-computable, and the set of the M2-computable real numbers is shown to be closed under the elementary functions of calculus.

E2 -computability of e, π and Other Famous Constants

We show that e, @p and other remarkable real numbers are limits of E^2-computable sequences of rational numbers having a polynomial rate of convergence (as usual, E^2 denotes the second Grzegorczyk class). However, only the rational numbers are limits of E^2-computable sequences of rational numbers with an exponential rate of convergence.

Characterization of the Computable Real Numbers by Means of Primitive Recursive Functions

One usually defines the notion of a computable real number by using recursive functions. However, there is a simple way due to A. Mostowski to characterize the computable real numbers by using only primitive recursive functions.We prove Mostowskis result differently and apply it to get other simple characterizations of this kind. For instance, a real number is shown to be computable if and only if it belongs to all members of some primitive recursive sequence of nested intervals with rational end points and with lengths arbitrarily closely approaching 0.

A subrecursive refinement of the fundamental theorem of algebra

Let us call an approximator of a complex number α any sequence γ0,γ1,γ2,... of rational complex numbers such that

On Some Computability Notions for Real Functions

|\gamma_t-\alpha|\le \frac{1}{t+1},\ \ t=0,1,2,\ldots