Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complexity and Avoidance   In this dissertation we examine the relationships between the several
hierarchies, including the complexity, $\mathrm{LUA}$ (Linearly Universal
Avoidance), and shift complexity hierarchies, with an eye towards quantitative
bounds on growth rates therein. We show that for suitable $f$ and $p$, there
are $q$ and $g$ such that $\mathrm{LUA}(q) \leq_\mathrm{s} \mathrm{COMPLEX}(f)$
and $\mathrm{COMPLEX}(g) \leq_\mathrm{s} \mathrm{LUA}(p)$, as well as quantify
the growth rates of $q$ and $g$. In the opposite direction, we show that for
certain sub-identical $f$ satisfying $\lim_{n \to \infty}{f(n)/n}=1$ there is a
$q$ such that $\mathrm{COMPLEX}(f) \leq_\mathrm{w} \mathrm{LUA}(q)$, and for
certain fast-growing $p$ there is a $g$ such that $\mathrm{LUA}(p)
\leq_\mathrm{s} \mathrm{COMPLEX}(g)$, as well as quantify the growth rates of
$q$ and $g$.
  Concerning shift complexity, explicit bounds are given on how slow-growing
$q$ must be for any member of $\rm{LUA}(q)$ to compute $\delta$-shift complex
sequences. Motivated by the complexity hierarchy, we generalize the notion of
shift complexity to consider sequences $X$ satisfying $\operatorname{KP}(\tau)
\geq f(|\tau|) - O(1)$ for all substrings $\tau$ of $X$ where $f$ is any order
function. We show that for sufficiently slow-growing $f$, $f$-shift complex
sequences can be uniformly computed by $g$-complex sequences, where $g$ grows
slightly faster than $f$.
  The structure of the $\mathrm{LUA}$ hierarchy is examined using bushy tree
forcing, with the main result being that for any order function $p$, there is a
slow-growing order function $q$ such that $\mathrm{LUA}(p)$ and
$\mathrm{LUA}(q)$ are weakly incomparable. Using this, we prove new results
about the filter of the weak degrees of deep nonempty $\Pi^0_1$ classes and the
connection between the shift complexity and $\mathrm{LUA}$ hierarchies.

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The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: eess