44 research outputs found
A statistical mechanical interpretation of instantaneous codes
In this paper we develop a statistical mechanical interpretation of the
noiseless source coding scheme based on an absolutely optimal instantaneous
code. The notions in statistical mechanics such as statistical mechanical
entropy, temperature, and thermal equilibrium are translated into the context
of noiseless source coding. Especially, it is discovered that the temperature 1
corresponds to the average codeword length of an instantaneous code in this
statistical mechanical interpretation of noiseless source coding scheme. This
correspondence is also verified by the investigation using box-counting
dimension. Using the notion of temperature and statistical mechanical
arguments, some information-theoretic relations can be derived in the manner
which appeals to intuition.Comment: 5 pages, Proceedings of the 2007 IEEE International Symposium on
Information Theory, pp.1906 - 1910, Nice, France, June 24 - 29, 200
The Tsallis entropy and the Shannon entropy of a universal probability
We study the properties of Tsallis entropy and Shannon entropy from the point
of view of algorithmic randomness. In algorithmic information theory, there are
two equivalent ways to define the program-size complexity K(s) of a given
finite binary string s. In the standard way, K(s) is defined as the length of
the shortest input string for the universal self-delimiting Turing machine to
output s. In the other way, the so-called universal probability m is introduced
first, and then K(s) is defined as -log_2 m(s) without reference to the concept
of program-size. In this paper, we investigate the properties of the Shannon
entropy, the power sum, and the Tsallis entropy of a universal probability by
means of the notion of program-size complexity. We determine the convergence or
divergence of each of these three quantities, and evaluate its degree of
randomness if it converges.Comment: 5 pages, to appear in the Proceedings of the 2008 IEEE International
Symposium on Information Theory, Toronto, ON, Canada, July 6 - 11, 200
Properties of optimal prefix-free machines as instantaneous codes
The optimal prefix-free machine U is a universal decoding algorithm used to
define the notion of program-size complexity H(s) for a finite binary string s.
Since the set of all halting inputs for U is chosen to form a prefix-free set,
the optimal prefix-free machine U can be regarded as an instantaneous code for
noiseless source coding scheme. In this paper, we investigate the properties of
optimal prefix-free machines as instantaneous codes. In particular, we
investigate the properties of the set U^{-1}(s) of codewords associated with a
symbol s. Namely, we investigate the number of codewords in U^{-1}(s) and the
distribution of codewords in U^{-1}(s) for each symbol s, using the toolkit of
algorithmic information theory.Comment: 5 pages, no figures, final manuscript to appear in the Proceedings of
the 2010 IEEE Information Theory Workshop, Dublin, Ireland, August 30 -
September 3, 201