On one-sided versus two-sided classification
نویسنده
چکیده
One-sided classiiers are computable devices which read the characteristic function of a set and output a sequence of guesses which converges to 1 ii the set on the input belongs to the given class. Such a classiier is two-sided if the sequence of its output in addition converges to 0 on sets not belonging to the class. The present work obtains the below mentioned results for one-sided classes (= 0 2 classes) w.r.t. four areas: Turing complexity, 1-reductions, index sets and measure. There are one-sided classes which are not two-sided. This can have two reasons: (1) the class has only high Turing complexity. Then there are some oracles which allow to construct noncomputable two-sided classiiers. (2) The class is diicult because of some topological constraints and then there are also no nonrecursive two-sided classiiers. For case (1), several results are obtained to localize the Turing complexity of certain types of one-sided sets. The concepts of 1-reduction, 1-completeness and simple sets is transferred to one-sided classes: There are 1-complete classes and simple classes, but no class is at the same time 1-complete and simple. The one-sided classes have a natural numbering. Most of the common index sets relative to this numbering have the high complexity 1 1 : the index sets of the class f0; 1g 1 , the index set of the equality problem and the index set of all two-sided classes. On the other side the index set of the empty class has complexity 0 2 ; 0 2 and 0 2 are the least complexities any non-trivial index set can have. Any one-sided class is measurable. It is shown that a one-sided class has effective measure 0 if it has measure 0, but that there are one-sided classes having measure 1 without having measure 1 eeectively. The measure of a two-sided class can be computed in the limit.
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عنوان ژورنال:
- Arch. Math. Log.
دوره 40 شماره
صفحات -
تاریخ انتشار 2001