Topological Complexity with Continuous Operations

AbstractThe topological complexity of algorithms is studied in a general context in the first part and for zero-finding in the second part. In the first part thelevel of discontinuityof a functionfis introduced and it is proved that it is a lower bound for the total number of comparisons plus 1 in any algorithm computingfthat uses only continuous operations and comparisons. This lower bound is proved to be sharp if arbitrary continuous operations are allowed. Then there exists even a balanced optimal computation tree forf. In the second part we use these results in order to determine the topological complexity of zero-finding for continuous functionsfon the unit interval withf(0) ·f(1) < 0. It is proved that roughly log2log2ϵ−1comparisons are optimal during a computation in order to approximate a zero up to ϵ. This is true regardless of whether one allows arbitrary continuous operations or just function evaluations, the arithmetic operations {+, −, *, /}, and the absolute value. It is true also for the subclass of nondecreasing functions. But for the subclass of increasing functions the topological complexity drops to zero even for the smaller class of operations

Nonlinear Lebesque and Itô integration problems of high complexity

We analyze the complexity of nonlinear Lebesgue integration problems in the average case setting for continous functions with the Wiener measure and the complexity of approximating the Itô stochasitcal integral. Wasilkowski and Woźniakowski (1999) studied these problems, observed that their complexities are closely related, and showed that for certain classes of smooth functions with boundedness conditions on derivatives the complexity is proportional to ε¯¹. Here ε>0 is the desired precision with which the integral is to be approximated. They showed also that for certain natural function classes with weaker smoothness conditions the complexity ist most of order ε¯² and conjectured that this bound is sharp. We show that this conjecture is true

Continuity and Computability of Relations

The main theorem of the theory of effectivity ( cf. Kreitz and Weihrauch [KWl], [Wl]) states that in admissibly represented topological spaces a function is continuous iff it has a continuous representation. Hence continuity is a necessary condition for computability. We investigate an extended model of computability in order to compute relations. From another point of view these relations are nondeterministic operations or set-valued functions. We show that for a special class of topological spaces (including the complete separable metric ones) and for a certain notion of continuity for relations the main theorem can be extended too