2,073 research outputs found
Random strings and tt-degrees of Turing complete C.E. sets
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular,
sets of random strings. It is known that the set of random strings with respect
to any universal prefix-free machine is Turing complete, but that truth-table
completeness depends on the choice of universal machine. We show that for such
sets of random strings, any finite set of their truth-table degrees do not meet
to the degree~0, even within the c.e. truth-table degrees, but when taking the
meet over all such truth-table degrees, the infinite meet is indeed~0. The
latter result proves a conjecture of Allender, Friedman and Gasarch. We also
show that there are two Turing complete c.e. sets whose truth-table degrees
form a minimal pair.Comment: 25 page
Almost isomorphism for countable state Markov shifts
Countable state Markov shifts are a natural generalization of the well-known
subshifts of finite type. They are the subject of current research both for
their own sake and as models for smooth dynamical systems. In this paper, we
investigate their almost isomorphism and entropy conjugacy and obtain a
complete classification for the especially important class of strongly positive
recurrent Markov shifts. This gives a complete classification up to entropy
conjugacy of the natural extensions of smooth entropy expanding maps, including
all smooth interval maps with non-zero topological entropy
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