A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine - a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'Church-Turing thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by (...) Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of _nonclassical_ computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a number of foundational arguments that are commonly rehearsed in cognitive science, and gesture towards a new class of cognitive models. (shrink)

We investigate computability of a self-similar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a self-similar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the self-similar set is a recursive compact set. A further result on the case that the self-similar set forms a curve is also discussed.

We study some representations of real numbers. We compare these representations, on the one hand from the viewpoint of recursive functionals, and of complexity on the other hand.The impossibility of obtaining some functions as recursive functionals is, in general, easy. This impossibility may often be explicited in terms of complexity: - existence of a sequence of low complexity whose image is not a recursive sequence, - existence of objects of low complexity but whose images have arbitrarily high time- complexity .Moreover, (...) some representations of real numbers that are equivalent from the viewpoint of recursive functionals, are very distinct from the viewpoint of complexity.We make a particular study of representations via continued fractions . We precise exactly what part of information available in the x's dfc is equivalent to the information available in its Dedekinds cut. We show that the sum of two reals whose dfcs are polynomial-time computable may be a real whose dfc has time complexity arbitrarily high.This work confirms that the unique representation of real numbers suitable for the ordinary calculus is via explicit Cauchy sequences of rationals.RésuméNous étudions différentes manières de présenter les nombres réels. Nous comparons ces présentations du point de vue des fonctionnelles récursives d'une part, et de celui des classes de complexité d'autre part.L'impossibilité d'obtenir certaines fonctions sous forme de fonctionnelles récursives est en général facile à établir. Cette impossiblité peut souvent être explicitée en termes de complexité: - il existe une suite de faible complexité dont l'image est une suite non récursive, - il existe des objets de faible complexité mais dont les images sont des objets de complexité arbitrairement grande .En outre, certaines présentations des réels équivalentes du point de vue des fonctionnelles récursives se distinguent nettement du point de vue de la complexité.Nous faisons une étude particulière concernant les développements en fraction continue . Nous précisions exactement quelle est la partie de l'information disponible dans le dfc d'un réel x qui équivaut à l'information disponible dans sa coupure de Dedekind. Nous montrons également que la somme de deux réels dont le dfc est calculable en temps polynomial peut être un réel dont le dfc est de complexité arbitrairement grande.Ce travail confirme que seule une présentation des réels via des suites de rationnels explicitement de Cauchy est adaptée aux calculs avec les réels. (shrink)

Let h : ℕ → ℚ be a computable function. A real number x is called h-monotonically computable if there is a computable sequence of rational numbers which converges to x h-monotonically in the sense that h|x – xn| ≥ |x – xm| for all n andm > n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h : ℕ → ℚ, we show that the class h-MC coincides (...) with the classes of computable and semi-computable real numbers if and only if Σi∈ℕ) = ∞and the sum Σi∈ℕ) is a computable real number, respectively. On the other hand, if h ≥ 1 and h converges to 1, then h-MC = SC no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h-MC = c-MC. Finally, if h is monotone and unbounded, then h-MC contains all ω-mc real numbers which are g-mc for some computable function g. (shrink)

We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.

In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding (...) sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel. (shrink)