Universal computably enumerable sets and initial segment prefix-free complexity
Information and Computation
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An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if the prefix-free Kolmogorov complexity of each initial segment of X is the same as the complexity of the sequence of 0s of the same length, up to a constant. We study the gap between the minimum complexity K(0 n ) and the initial segment complexity of a nontrivial sequence, and in particular the nondecreasing unbounded functions f such that 驴 for a nontrivial sequence X, where K denotes the prefix-free complexity. Our first result is that there exists a $\varDelta^{0}_{3}$ unbounded nondecreasing function f which does not have this property. It is known that such functions cannot be $\varDelta^{0}_{2}$ hence this is an optimal bound on their arithmetical complexity. Moreover it improves the bound $\varDelta^{0}_{4}$ that was known from Csima and Montalbán (Proc. Amer. Math. Soc. 134(5):1499---1502, 2006).Our second result is that if f is $\varDelta^{0}_{2}$ then there exists a non-empty $\varPi^{0}_{1}$ class of reals X with nontrivial prefix-free complexity which satisfy (驴). This implies that in this case there uncountably many nontrivial reals X satisfying (驴) in various well known classes from computability theory and algorithmic randomness; for example low for 驴, non-low for 驴 and computably dominated reals. A special case of this result was independently obtained by Bienvenu, Merkle and Nies (STACS, pp. 452---463, 2011).