31 research outputs found
On the strength of the finite intersection principle
We study the logical content of several maximality principles related to the
finite intersection principle (F\IP) in set theory. Classically, these are
all equivalent to the axiom of choice, but in the context of reverse
mathematics their strengths vary: some are equivalent to \ACA over \RCA,
while others are strictly weaker, and incomparable with \WKL. We show that
there is a computable instance of F\IP all of whose solutions have
hyperimmune degree, and that every computable instance has a solution in every
nonzero c.e.\ degree. In terms of other weak principles previously studied in
the literature, the former result translates to F\IP implying the omitting
partial types principle (). We also show that, modulo
induction, F\IP lies strictly below the atomic model theorem
().Comment: This paper corresponds to section 3 of arXiv:1009.3242, "Reverse
mathematics and equivalents of the axiom of choice", which has been
abbreviated and divided into two pieces for publicatio
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
Ramsey’s theorem and products in the Weihrauch degrees
10.3233/com-180203Computability9285-11
Degree Spectra and Computable Dimensions in Algebraic Structures
Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of e#ort, and lacks generality. Another method is to code the original structure into a structure in the given class in a way that is e#ective enough to preserve the property in which we are interested. In this paper, we show how to transfer a # Partially supported by an Alfred P. Sloan Doctoral Dissertation Fellowship. Current address: Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago IL 60637, U.S.A.. ## Partially supported by NSF Grants DMS-9503503, DMS-9802843, and INT-9602579