Abstract. An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if K(X ↾n) ≤ + K(0n) for all n, where K is the prefix-free complexity and ≤ + denotes inequality modulo a constant. In other words, if the information in any initial segment of it is merely the information in a sequence of 0s of the same length. We study the gap between the trivial complexity K(0n) and the complexity of a non-trivial sequence, i.e. the functions f such that (⋆) K(X ↾n) ≤ + K(0 n) + f(n) for all n for a non-trivial (in terms of initial segment complexity) sequence X. We show that given any ∆0 2 unbounded non-decreasing function f there exist uncountably many sequences X which satisfy (⋆). On the other hand there exists a ∆0 3 unbounded non-decreasing function f which does not satisfy (⋆) for any X with non-trivial initial segment complexity. This improves the bound ∆0 4 that was known from [CM06]. Finally we give some applications of these results. 1
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