353 research outputs found
If not empty, NP — P is topologically large
AbstractIn the classical Cantor topology or in the superset topology, NP and, consequently, classes included in NP are meagre. However, in a natural combination of the two topologies, we prove that NP — P, if not empty, is a second category class, while NP-complete sets form a first category class. These results are extended to different levels in the polynomial hierarchy and to the low and high hierarchies. P-immune sets in NP, NP-simple sets, P-bi-immune sets and NP-effectively simple sets are all second category (if not empty). It is shown that if C is any of the above second category classes, then for all B∈NP there exists an A∈C such that A is arbitrarily close to B infinitely often
Absolutely No Free Lunches!
This paper is concerned with learners who aim to learn patterns in infinite
binary sequences: shown longer and longer initial segments of a binary
sequence, they either attempt to predict whether the next bit will be a 0 or
will be a 1 or they issue forecast probabilities for these events. Several
variants of this problem are considered. In each case, a no-free-lunch result
of the following form is established: the problem of learning is a formidably
difficult one, in that no matter what method is pursued, failure is
incomparably more common that success; and difficult choices must be faced in
choosing a method of learning, since no approach dominates all others in its
range of success. In the simplest case, the comparison of the set of situations
in which a method fails and the set of situations in which it succeeds is a
matter of cardinality (countable vs. uncountable); in other cases, it is a
topological matter (meagre vs. co-meagre) or a hybrid computational-topological
matter (effectively meagre vs. effectively co-meagre)
Turning Borel sets into clopen sets effectively
We present the effective version of the theorem about turning Borel sets in
Polish spaces into clopen sets while preserving the Borel structure of the
underlying space. We show that under some conditions the emerging parameters
can be chosen in a hyperarithmetical way and using this we prove a uniformity
result
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