Abstract
We study a countable class of real-valued functions inductively defined from a basic set of trivial functions by composition, solving first-order differential equations and the taking of infinite limits. This class is the analytical counterpart of Kleene's partial recursive functions. By counting the number of nested limits required to de. ne a function, this class can be stratified by a potentially infinite hierarchy-a hierarchy of infinite limits. In the first meaningful level of the hierarchy, we have the extensions of classical primitive recursive functions. In the next level, we find partial recursive functions, and in the following level we find the solution to the halting problem. We use methods from numerical analysis to show that the hierarchy does not collapse, concluding that the taking of infinite limits can always produce new functions from functions in the previous levels of the hierarchy.

Abstract
If we measure the position of a point particle, then we will come about with an interval [a(n), b(n)] into which the point falls. We make use of a Gedankenexperiment to find better and better values of a. and b(n), by reducing their relative distance, in a succession of intervals [a(1), b(1)] superset of [a(2),b(2)] superset of ... superset of [a(n), b(n)] that contain the point. We then use such a point as an oracle to perform relative computation in polynomial time, by considering the succession of approximations to the point as suitable answers to the queries in an oracle Turing machine. We prove that, no matter the precision achieved in such a Gedankenexperiment, within the limits studied, the Turing Machine, equipped with such an oracle, will be able to compute above the classical Turing limit for the polynomial time resource, either generating the class P/poly either generating the class BPP//log*, if we allow for an arbitrary precision in measurement or just a limited precision, respectively. We think that this result is astonishingly interesting for Classical Physics and its connection to the Theory of Computation, namely for the implications on the nature of space and the perception of space in Classical Physics. (Some proofs are provided, to give the flavor of the subject. Missing proofs can be found in a detailed long report at the address http://fgc.math.ist.utl.pt/papers/sm.pdf).

Abstract
In this article we present a strong support to real recursive function theory as a branch of computability theory rooted in mathematical analysis. This new paradiam. connects computation on reals with differential equations and infinite limits in a robust and smooth way. The results presented here are taken mainly from the article (4) of the same authors.

Abstract
We discuss combining physical experiments with machine computations and introduce a form of analogue digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

Abstract
We survey different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church-Turing thesis. We propose five theses to classify approaches to this area of investigation.

Abstract
Earlier, to explore the idea of combining physical experiments with algorithms, we introduced a new form of analogue digital (AD) Turing machine. We examined in detail a case study where an experimental procedure, based on Newtonian kinematics, is used as an oracle with classes of Turing machines. The physical cost of oracle calls was counted and three forms of AD queries were studied, in which physical parameters can be set exactly and approximately. Here, in this sequel, we complete the classi. cation of the computational power of these AD Turing machines and determine precisely what they can compute, using non- uniform complexity classes and probabilities.