16,190 research outputs found
Finitary reducibility on equivalence relations
We introduce the notion of finitary computable reducibility on equivalence
relations on the natural numbers. This is a weakening of the usual notion of
computable reducibility, and we show it to be distinct in several ways. In
particular, whereas no equivalence relation can be -complete under
computable reducibility, we show that, for every , there does exist a
natural equivalence relation which is -complete under finitary
reducibility. We also show that our hierarchy of finitary reducibilities does
not collapse, and illustrate how it sharpens certain known results. Along the
way, we present several new results which use computable reducibility to
establish the complexity of various naturally defined equivalence relations in
the arithmetical hierarchy
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
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