We speculate on the Gau ian function [x] as an example of a noncontinuous function which is nevertheless possessed of some properties of computability. An algorithm how to compute [x] for a single computable real number is rst described, followed by a remark that [x] does not necessarily preserve sequential computability. Second, [x] is studied in the light of the notion of upper semi-computabi...

We consider the real sequences in I = [0, 1) and real functions on I. A computability notion with respect to the uniformity {Un}, where Un(x) = [ k 2n , k+1 2n ) if x ∈ [ k 2n , k+1 2n ), will be called D-computability. An R-computable sequence from I will be shown to be approximated by a recursive sequence of rational numbers with a limiting recursive modulus of convergence with respect to {Un...

This paper introduces a new communication abstraction, called Set-Constrained Delivery Broadcast (SCD-broadcast), whose aim is to provide its users with an appropriate abstraction level when they have implement objects or distributed tasks in asynchronous message-passing system prone process crash failures. allows each broadcast messages and deliver sequence of sets such way that, if delivers s...

This paper presents the first step of an attempt in designing intuitive and interesting materials for a theory course. The materials developed cover the AL5 Basic Computability unit of the ACM/IEEE Computing Curricula 2001, and can be used in a stand-alone theory course. This paper describes a “programming approach” to basic computability. Topics include a proof of the Halting Problem and the u...

The ability to simulate a biological organism by employing a computer is related to the ability of the computer to calculate the behavior of such a dynamical system, or the "computability" of the system.* However, the two questions of computability and simulation are not equivalent. Since the question of computability can be given a precise answer in terms of recursive functions, automata theor...

The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-...

Ordinal computability uses ordinals instead of natural numbers in abstract machines like register or Turing machines. We give an overview of the computational strengths of α-β-machines, where α and β bound the time axis and the space axis of some machine model. The spectrum ranges from classical Turing computability to ∞-∞-computability which corresponds to Gödel’s model of constructible sets. ...

Computability theory applies a rigorous definition of the notion of an algorithm to determine which mathematical functions can be computed and which cannot. The main concepts in this area date back to Alan Turing, who during the 1930’s gave the definition of what is now called a Turing machine, along with its generalization, the oracle Turing machine. In the ensuing seventy years, mathematician...