753 research outputs found
The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length
Theories of classification distinguish classes with some good structure
theorem from those for which none is possible. Some classes (dense linear
orders, for instance) are non-classifiable in general, but are classifiable
when we consider only countable members. This paper explores such a notion for
classes of computable structures by working out a sequence of examples.
We follow recent work by Goncharov and Knight in using the degree of the
isomorphism problem for a class to distinguish classifiable classes from
non-classifiable. In this paper, we calculate the degree of the isomorphism
problem for Abelian -groups of bounded Ulm length. The result is a sequence
of classes whose isomorphism problems are cofinal in the hyperarithmetical
hierarchy. In the process, new back-and-forth relations on such groups are
calculated.Comment: 15 page
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
The Cardinality of an Oracle in Blum-Shub-Smale Computation
We examine the relation of BSS-reducibility on subsets of the real numbers.
The question was asked recently (and anonymously) whether it is possible for
the halting problem H in BSS-computation to be BSS-reducible to a countable
set. Intuitively, it seems that a countable set ought not to contain enough
information to decide membership in a reasonably complex (uncountable) set such
as H. We confirm this intuition, and prove a more general theorem linking the
cardinality of the oracle set to the cardinality, in a local sense, of the set
which it computes. We also mention other recent results on BSS-computation and
algebraic real numbers
Degeneration and orbits of tuples and subgroups in an Abelian group
A tuple (or subgroup) in a group is said to degenerate to another if the
latter is an endomorphic image of the former. In a countable reduced abelian
group, it is shown that if tuples (or finite subgroups) degenerate to each
other, then they lie in the same automorphism orbit. The proof is based on
techniques that were developed by Kaplansky and Mackey in order to give an
elegant proof of Ulm's theorem. Similar results hold for reduced countably
generated torsion modules over principal ideal domains. It is shown that the
depth and the description of atoms of the resulting poset of orbits of tuples
depend only on the Ulm invariants of the module in question (and not on the
underlying ring). A complete description of the poset of orbits of elements in
terms of the Ulm invariants of the module is given. The relationship between
this description of orbits and a very different-looking one obtained by Dutta
and Prasad for torsion modules of bounded order is explained.Comment: 13 pages, 1 figur
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