Catagory, Measure, Inductive Inference: A Triality Theorem and its Applications
Carl H. Smith
Abstract:
The famous SierpinskiErd¨os Duality Theorem states, informally, that any
theorem about effective measure 0 and/or first category sets is also true
when all occurrences of ``effective measure 0'' are replaced by ``first
category'' and vice versa. This powerful and nice result shows that
``measure'' and ``category'' are equally useful notions neither of which
can be preferred to the other one when making formal the intuitive notion
``almost all sets.'' Effective versions of measure and category are used
in recursive function theory and related areas, and resourcebounded
versions of the same notions are used in Theory of Computation. Again they
are dual in the same sense.
We show that in the world of recursive functions there is a third equipotent
notion dual to both measure and category. This new notion is related to
learnability (also known as inductive inference or identifiability). We use
the term ``triality'' to describe this threeparty duality.