128 research outputs found
Induced and higher-dimensional stable independence
We provide several crucial technical extensions of the theory of stable
independence notions in accessible categories. In particular, we describe
circumstances under which a stable independence notion can be transferred from
a subcategory to a category as a whole, and examine a number of applications to
categories of groups and modules, extending results of [MAa]. We prove, too,
that under the hypotheses of [LRV], a stable independence notion immediately
yields higher-dimensional independence as in [SV]
PAC Learning, VC Dimension, and the Arithmetic Hierarchy
We compute that the index set of PAC-learnable concept classes is
-complete within the set of indices for all concept classes of
a reasonable form. All concept classes considered are computable enumerations
of computable classes, in a sense made precise here. This family of
concept classes is sufficient to cover all standard examples, and also has the
property that PAC learnability is equivalent to finite VC dimension
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