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**1 - 2**of**2**### Complexity in the Real world

, 2005

"... Whereas Turing Machines lay a solid foundation for computation of functions on countable sets, a lot of real-world calculations require real numbers. The question arises naturally whether there is a satisfying extension to functions on uncountable sets. This thesis states and discusses such a genera ..."

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Whereas Turing Machines lay a solid foundation for computation of functions on countable sets, a lot of real-world calculations require real numbers. The question arises naturally whether there is a satisfying extension to functions on uncountable sets. This thesis states and discusses such a generalization, based on previous research. It also discusses higher order functions, e.g. differentiation. In contrast to preceding works, however, the focus is on complexity – after computability, of course. By giving a different perspective on Weihrauch’s excellent definition of computability in the uncountable case, we show that this theory indeed admits a useful notion of complexity. Various examples are given to demonstrate the theory, including an application to distributions, also called generalized functions, as a form of ‘stress-test’.

### in a higher-type setting

"... We show that, in a fairly general setting including higher-types, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a non-deterministic prog ..."

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We show that, in a fairly general setting including higher-types, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a non-deterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semi-decidability of similar tests under the simultaneous presence of non-deterministic and probabilistic choice. Keywords: Non-deterministic and probabilistic computation, higher-type computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains. 1