Abstract
The termF-cardinality of ℱ (=F-card(ℱ)) is introduced whereF: ℝn → ℝn is a partial function and ℱ is a set of partial functionsf: ℝn → ℝn. TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf ε ℱ can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs Ω(n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ℝn → ℝn and a natural numberk, it is almost always possible to construct a set ℱ such that itsF-cardinality has the valuek; in addition, ℱ can be required to be closed under composition of functionsf,g ε ℱ. Moreover, ifF is continuous, then ℱ consists of continuous functions.
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Huckenbeck, U. Constructing sets of functions which have a givenF-cardinality. Math. Systems Theory 25, 3–22 (1992). https://doi.org/10.1007/BF01368781
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DOI: https://doi.org/10.1007/BF01368781