Browsing by Author "Brattka, Vasco"

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    Algorithmic randomness on computable metric spaces and hyperspaces
    (2012) Birch, Thomas; Brattka, Vasco
    In this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-empty compact subsets of a computable metric space, the study of which was first proposed by Barmpalias et al. [16] at the University of Florida in their work on the random closed subsets of the Cantor space. Much work has been done in the study of random sets with authors such as Diamondstone and Kjos-Hanssen [17] continuing the Florida approach, whilst others such as Axon [18] and Cenzer and Broadhead [19] have been studying the use of capacities to define hyperspace measures for use in randomness tests. Lastly in section 6.4 we shall be looking at the work done by Hertling and Weihrauch [13] on universal randomness tests in effective topological measure spaces and relate their results to randomness on computable metric measure spaces and in particular to the randomness of compact sets in the hyperspace of non-empty compact subsets of computable metric spaces.
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    Plottable Real Number Functions and the Computable Graph Theorem
    (2008) Brattka, Vasco
    The Graph Theorem of classical recursion theory states that a total function on the natural numbers is computable if and only if its graph is recursive. It is known that this result can be generalized to real number functions where it has an important practical interpretation: the total computable real number functions are precisely those which can be effectively plotted with any given resolution. We generalize the Graph Theorem to appropriate partial real number functions and even further to functions defined on certain computable metric spaces. Besides the nonuniform version of the Graph Theorem which logically relates computability properties of the function and computability properties of its graph, we also discuss the uniform version: given a program of a function, can we algorithmically derive a description of its graph? And, vice versa, given a description of the graph, can we derive a program of the function? While the passage from functions to graphs is always computable, the inverse direction fr...