Complexity and Avoidance
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In this dissertation we examine the relationships between the several hierarchies, including the complexity, 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 LUA(q) ≤_sCOMPLEX(f) and COMPLEX(g) ≤_sLUA(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 →∞f(n)/n=1 there is a q such that COMPLEX(f) ≤_wLUA(q), and for certain fast-growing p there is a g such that LUA(p) ≤_sCOMPLEX(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 LUA(q) to compute δ-shift complex sequences. Motivated by the complexity hierarchy, we generalize the notion of shift complexity to consider sequences X satisfying KP(τ) ≥ f(|τ|) - O(1) for all substrings τ 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 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 LUA(p) and LUA(q) are weakly incomparable. Using this, we prove new results about the filter of the weak degrees of deep nonempty Π^0_1 classes and the connection between the shift complexity and LUA hierarchies.
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