There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch's Type Two Theory of Effectivity or on domain-theoretic approaches, in which case the partial objects appearing during computations are made explicit. A further, more analysis-oriented line of research is based on Grzegorczyk's work. All these approaches are claimed to be equivalent, but not in all cases proofs have been given. In this paper it is shown that a real number as well as a real-valued function are computable in Weihrauch's sense if and only if they are definable in Escardo's functional language Real PCF, an extension of the language PCF by a new ground type for (total and partial) real numbers. This is exactly the case if the number is a computable element in the continuous domain of all compact real intervals and/or the function has a computable extension to this doma...

Based on an eective theory of continuous domains, notions of computability for operators and real-valued functionals dened on the class of continuous functions are introduced. Denability and semantic characterisation of computable functionals are given. Also we propose a recursion scheme which is a suitable tool for formalisation of complex systems, such as hybrid systems. In this framework the trajectories of continuous parts of hybrid systems can be represented by computable functionals. 1

The domain-theoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is computable if it is the effective limit of a sequence of finitary partial objects of the same type as the original object. We are also provided with two different quantitative measures for approximation using the Hausdorff metric and the Lebesgue measure. In this thesis, we introduce a new data type to capture imprecise data or approximate points on the plane, given in the shape of compact convex polygons. This data type in particular includes rectangular approximation and is invariant under linear transformations of coordinate system. Based on the new data type, we define the notion of a number of partial geometric operations, including partial perpendicular bisector and partial disc and we show that these operations and the convex hull, Delaunay triangulation and Voronoi diagram are Hausdorff and Scott continuous and nestedly Hausdorff and Lebesgue computable. We develop algorithms to obtain the partial convex hull, partial Delaunay triangulation and partial Voronoi diagram. We prove that the complexity of the partial convex hull is N log N in 2D and 3D, whereas the partial Delaunay triangulation and partial Voronoi diagram algorithms for non-degenerate data have the same complexity as their classical counterparts. 2

We propose semantic characterisations of second-order computability over the reals based on -denability theory. Notions of computability for operators and real-valued functionals dened on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and real-valued functionals with validity of nite -formulas. 1

on topological spaces via domain representations

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We propose an embedding G of the unit open interval to the set f0; 1g ! ?;1 of infinite sequences of f0; 1g with at most one undefined element. This embedding is based on Gray code and it is a topological embedding with a natural topology on f0; 1g ! ?;1 . We also define a machine called an IM2 machine (indeterministic multihead type 2 machine) which input/output sequences in f0; 1g ! ?;1 , and show that the computability notion induced on real functions through the embedding G is equivalent to the one induced by the signed digit representation and Type-2 machines. We also show that basic algorithms can be expressed naturally with respect to this embedding.

We investigate the concept of generalised computability of operators and functionals dened on the set of continuous functions, rstly introduced in [9]. By working in the reals, with equality and without equality, we study properties of generalised computable operators and functionals. Also we propose an interesting application to formalisation of hybrid systems. We obtain some class of hybrid systems, which trajectories are computable in the sense of computable analysis.

Abstract. Implementations of real number computations have largely been unusable in practice because of their very bad performance, especially in comparison to floating point arithmetic implemented in hardware. This performance problem is to a very large extent due to the type-2 nature of the computable analysis frameworks usually employed. This problem can be overcome by employing a type-1 approach. This paper presents such an approach and deals with properties of it that have not been well studied before, namely the introduction of complexity measures for type-1 representations of real functions and ways to define intensional functions, i.e. functions that may return different real numbers for the same real argument given in different representations. 1

Abstract. We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative measure for the approximation. The domain-theoretic model of computational geometry provides such an object directed theory, which supports two such quantitative measures, one based on the Hausdorff metric and one on the Lebesgue measure. With respect to a new data type for the Euclidean space, given by its non-empty compact and convex subsets, we show that the convex hull, Voronoi diagram and Delaunay triangulation are Hausdorff and Lebesgue computable.