56 research outputs found
Depth, Highness and DNR degrees
We study Bennett deep sequences in the context of recursion theory; in
particular we investigate the notions of O(1)-deepK, O(1)-deepC , order-deep K
and order-deep C sequences. Our main results are that Martin-Loef random sets
are not order-deepC , that every many-one degree contains a set which is not
O(1)-deepC , that O(1)-deepC sets and order-deepK sets have high or DNR Turing
degree and that no K-trival set is O(1)-deepK.Comment: journal version, dmtc
Depth, Highness and DNR Degrees
A sequence is Bennett deep [5] if every recursive approximation of the
Kolmogorov complexity of its initial segments from above satisfies that the difference
between the approximation and the actual value of the Kolmogorov complexity of
the initial segments dominates every constant function. We study for different lower
bounds r of this difference between approximation and actual value of the initial segment
complexity, which properties the corresponding r(n)-deep sets have. We prove
that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller
choices of r, i.e., r is any recursive order function, we show that depth implies either
highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order
depth already implies highness. As a corollary, we obtain that weakly-useful sets are
either high or DNR. We prove that not all deep sets are high by constructing a low
order-deep set.
Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that
if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition,
one obtains a notion which no longer satisfies the slow growth law (which
stipulates that no shallow set truth-table computes a deep set); however, under this
notion, random sets are not deep (at the unbounded recursive order magnitude). We
improve Bennett's result that recursive sets are shallow by proving all K-trivial sets
are shallow; our result is close to optimal.
For Bennett's depth, the magnitude of compression improvement has to be achieved
almost everywhere on the set. Bennett observed that relaxing to infinitely often is
meaningless because every recursive set is infinitely often deep. We propose an alternative
infinitely often depth notion that doesn't suffer this limitation (called i.o.
depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude
εn, and construct a π01- class where every member is an i.o. deep set of magnitude
εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep
of constant magnitude, and that every nonrecursive many-one degree contains such
a set
Depth, Highness and DNR Degrees
A sequence is Bennett deep [5] if every recursive approximation of the
Kolmogorov complexity of its initial segments from above satisfies that the difference
between the approximation and the actual value of the Kolmogorov complexity of
the initial segments dominates every constant function. We study for different lower
bounds r of this difference between approximation and actual value of the initial segment
complexity, which properties the corresponding r(n)-deep sets have. We prove
that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller
choices of r, i.e., r is any recursive order function, we show that depth implies either
highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order
depth already implies highness. As a corollary, we obtain that weakly-useful sets are
either high or DNR. We prove that not all deep sets are high by constructing a low
order-deep set.
Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that
if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition,
one obtains a notion which no longer satisfies the slow growth law (which
stipulates that no shallow set truth-table computes a deep set); however, under this
notion, random sets are not deep (at the unbounded recursive order magnitude). We
improve Bennett's result that recursive sets are shallow by proving all K-trivial sets
are shallow; our result is close to optimal.
For Bennett's depth, the magnitude of compression improvement has to be achieved
almost everywhere on the set. Bennett observed that relaxing to infinitely often is
meaningless because every recursive set is infinitely often deep. We propose an alternative
infinitely often depth notion that doesn't suffer this limitation (called i.o.
depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude
εn, and construct a π01- class where every member is an i.o. deep set of magnitude
εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep
of constant magnitude, and that every nonrecursive many-one degree contains such
a set
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
Maine inquirer: Vol.3, No. 22 - March 13, 1827
https://digitalmaine.com/skowhegan_history_house_newspapers/1013/thumbnail.jp
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