14,464 research outputs found
On the time complexity of 2-tag systems and small universal Turing machines
We show that 2-tag systems efficiently simulate Turing machines. As a
corollary we find that the small universal Turing machines of Rogozhin, Minsky
and others simulate Turing machines in polynomial time. This is an exponential
improvement on the previously known simulation time overhead and improves a
forty year old result in the area of small universal Turing machines.Comment: Slightly expanded and updated from conference versio
Small Turing universal signal machines
This article aims at providing signal machines as small as possible able to
perform any computation (in the classical understanding). After presenting
signal machines, it is shown how to get universal ones from Turing machines,
cellular-automata and cyclic tag systems. Finally a halting universal signal
machine with 13 meta-signals and 21 collision rules is presented
A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences
Given the widespread use of lossless compression algorithms to approximate
algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression
algorithms fall short at characterizing patterns other than statistical ones
not different to entropy estimations, here we explore an alternative and
complementary approach. We study formal properties of a Levin-inspired measure
calculated from the output distribution of small Turing machines. We
introduce and justify finite approximations that have been used in some
applications as an alternative to lossless compression algorithms for
approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of
the relevant properties of both and and compare them to Levin's
Universal Distribution. We provide error estimations of with respect to
. Finally, we present an application to integer sequences from the Online
Encyclopedia of Integer Sequences which suggests that our AP-based measures may
characterize non-statistical patterns, and we report interesting correlations
with textual, function and program description lengths of the said sequences.Comment: As accepted by the journal Complexity (Wiley/Hindawi
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