3,846 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
Density Evolution and Functional Threshold for the Noisy Min-Sum Decoder
This paper investigates the behavior of the Min-Sum decoder running on noisy
devices. The aim is to evaluate the robustness of the decoder in the presence
of computation noise, e.g. due to faulty logic in the processing units, which
represents a new source of errors that may occur during the decoding process.
To this end, we first introduce probabilistic models for the arithmetic and
logic units of the the finite-precision Min-Sum decoder, and then carry out the
density evolution analysis of the noisy Min-Sum decoder. We show that in some
particular cases, the noise introduced by the device can help the Min-Sum
decoder to escape from fixed points attractors, and may actually result in an
increased correction capacity with respect to the noiseless decoder. We also
reveal the existence of a specific threshold phenomenon, referred to as
functional threshold. The behavior of the noisy decoder is demonstrated in the
asymptotic limit of the code-length -- by using "noisy" density evolution
equations -- and it is also verified in the finite-length case by Monte-Carlo
simulation.Comment: 46 pages (draft version); extended version of the paper with same
title, submitted to IEEE Transactions on Communication
Properties of Persistent Mutual Information and Emergence
The persistent mutual information (PMI) is a complexity measure for
stochastic processes. It is related to well-known complexity measures like
excess entropy or statistical complexity. Essentially it is a variation of the
excess entropy so that it can be interpreted as a specific measure of system
internal memory. The PMI was first introduced in 2010 by Ball, Diakonova and
MacKay as a measure for (strong) emergence. In this paper we define the PMI
mathematically and investigate the relation to excess entropy and statistical
complexity. In particular we prove that the excess entropy is an upper bound of
the PMI. Furthermore we show some properties of the PMI and calculate it
explicitly for some example processes. We also discuss to what extend it is a
measure for emergence and compare it with alternative approaches used to
formalize emergence.Comment: 45 pages excerpt of Diploma-Thesi
On Infinite Words Determined by Indexed Languages
We characterize the infinite words determined by indexed languages. An
infinite language determines an infinite word if every string in
is a prefix of . If is regular or context-free, it is known
that must be ultimately periodic. We show that if is an indexed
language, then is a morphic word, i.e., can be generated by
iterating a morphism under a coding. Since the other direction, that every
morphic word is determined by some indexed language, also holds, this implies
that the infinite words determined by indexed languages are exactly the morphic
words. To obtain this result, we prove a new pumping lemma for the indexed
languages, which may be of independent interest.Comment: Full version of paper accepted for publication at MFCS 201
An upper bound on asymptotic repetitive threshold of balanced sequences via colouring of the Fibonacci sequence
We colour the Fibonacci sequence by suitable constant gap sequences to
provide an upper bound on the asymptotic repetitive threshold of -ary
balanced sequences. The bound is attained for and and we
conjecture that it happens for infinitely many even 's.
Our bound reveals an essential difference in behavior of the repetitive
threshold and the asymptotic repetitive threshold of balanced sequences. The
repetitive threshold of -ary balanced sequences is known to be at least
for each . In contrast, our bound implies that the
asymptotic repetitive threshold of -ary balanced sequences is at most
for each , where is the golden mean.Comment: arXiv admin note: text overlap with arXiv:2112.0285
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