Bulletin of Symbolic Logic

Shift-complex sequences

Mushfeq Khan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A Martin-Löf random sequence is an infinite binary sequence with the property that every initial segment $\sigma$ has prefix-free Kolmogorov complexity $K(\sigma)$ at least $|\sigma| - c$, for some constant $c \in \omega$. Informally, initial segments of Martin-Löf randoms are highly complex in the sense that they are not compressible by more than a constant number of bits. However, all Martin-Löf randoms necessarily have contiguous substrings of arbitrarily low complexity. If we demand that all substrings of a sequence be uniformly complex, then we arrive at the notion of shift-complex sequences. In this paper, we collect some of the existing results on these sequences and contribute two new ones. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is zero. We strengthen this result by proving that the Martin-Löf random sequences that do not compute shift-complex sequences are exactly the incomplete ones, in other words, the ones that do not compute the halting problem. In order to do so, we make use of the characterization by Franklin and Ng of the class of incomplete Martin-Löf randoms via a notion of randomness called difference randomness. Turning to the power of shift-complex sequences as oracles, we show that there are shift-complex sequences that do not compute Martin-Löf random (or even Kurtz random) sequences.

Article information

Bull. Symbolic Logic, Volume 19, Issue 2 (2013), 199-215.

First available in Project Euclid: 16 May 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32] 03D32: Algorithmic randomness and dimension [See also 68Q30] 03D28: Other Turing degree structures

Kolmogorov complexity subshifts Martin-Löf randomness


Khan, Mushfeq. Shift-complex sequences. Bull. Symbolic Logic 19 (2013), no. 2, 199--215. doi:10.2178/bsl.1902020. https://projecteuclid.org/euclid.bsl/1368716900

Export citation