58,179 research outputs found
On the Growth Rates of Complexity of Threshold Languages
Threshold languages, which are the (k/(k-1))+-free languages over k-letter alphabets with k ≥, are the minimal infinite power-free languages according to Dejean's conjecture, which is now proved for all alphabets. We study the growth properties of these languages. On the base of obtained structural properties and computer-assisted studies we conjecture that the growth rate of complexity of the threshold language over k letters tends to a constant α̌ ≈ 1.242 as k tends to infinity. © 2010 EDP Sciences.The authors heartly thank the referees for their valuable comments and remarks
On Pansiot Words Avoiding 3-Repetitions
The recently confirmed Dejean's conjecture about the threshold between
avoidable and unavoidable powers of words gave rise to interesting and
challenging problems on the structure and growth of threshold words. Over any
finite alphabet with k >= 5 letters, Pansiot words avoiding 3-repetitions form
a regular language, which is a rather small superset of the set of all
threshold words. Using cylindric and 2-dimensional words, we prove that, as k
approaches infinity, the growth rates of complexity for these regular languages
tend to the growth rate of complexity of some ternary 2-dimensional language.
The numerical estimate of this growth rate is about 1.2421.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Diversity, competition, extinction: the ecophysics of language change
As early indicated by Charles Darwin, languages behave and change very much
like living species. They display high diversity, differentiate in space and
time, emerge and disappear. A large body of literature has explored the role of
information exchanges and communicative constraints in groups of agents under
selective scenarios. These models have been very helpful in providing a
rationale on how complex forms of communication emerge under evolutionary
pressures. However, other patterns of large-scale organization can be described
using mathematical methods ignoring communicative traits. These approaches
consider shorter time scales and have been developed by exploiting both
theoretical ecology and statistical physics methods. The models are reviewed
here and include extinction, invasion, origination, spatial organization,
coexistence and diversity as key concepts and are very simple in their defining
rules. Such simplicity is used in order to catch the most fundamental laws of
organization and those universal ingredients responsible for qualitative
traits. The similarities between observed and predicted patterns indicate that
an ecological theory of language is emerging, supporting (on a quantitative
basis) its ecological nature, although key differences are also present. Here
we critically review some recent advances lying and outline their implications
and limitations as well as open problems for future research.Comment: 17 Pages. A review on current models from statistical Physics and
Theoretical Ecology applied to study language dynamic
Statistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death
We analyze the dynamic properties of 10^7 words recorded in English, Spanish
and Hebrew over the period 1800--2008 in order to gain insight into the
coevolution of language and culture. We report language independent patterns
useful as benchmarks for theoretical models of language evolution. A
significantly decreasing (increasing) trend in the birth (death) rate of words
indicates a recent shift in the selection laws governing word use. For new
words, we observe a peak in the growth-rate fluctuations around 40 years after
introduction, consistent with the typical entry time into standard dictionaries
and the human generational timescale. Pronounced changes in the dynamics of
language during periods of war shows that word correlations, occurring across
time and between words, are largely influenced by coevolutionary social,
technological, and political factors. We quantify cultural memory by analyzing
the long-term correlations in the use of individual words using detrended
fluctuation analysis.Comment: Version 1: 31 pages, 17 figures, 3 tables. Version 2 is streamlined,
eliminates substantial material and incorporates referee comments: 19 pages,
14 figures, 3 table
Languages cool as they expand: Allometric scaling and the decreasing need for new words
We analyze the occurrence frequencies of over 15 million words recorded in millions of books published during the past two centuries in seven different languages. For all languages and chronological subsets of the data we confirm that two scaling regimes characterize the word frequency distributions, with only the more common words obeying the classic Zipf law. Using corpora of unprecedented size, we test the allometric scaling relation between the corpus size and the vocabulary size of growing languages to demonstrate a decreasing marginal need for new words, a feature that is likely related to the underlying correlations between words. We calculate the annual growth fluctuations of word use which has a decreasing trend as the corpus size increases, indicating a slowdown in linguistic evolution following language expansion. This ‘‘cooling pattern’’ forms the basis of a third statistical regularity, which unlike the Zipf and the Heaps law, is dynamical in nature
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
Self-replication and evolution of DNA crystals
Is it possible to create a simple physical system that is capable of replicating itself? Can such a system evolve interesting behaviors, thus allowing it to adapt to a wide range of environments? This paper presents a design for such a replicator constructed exclusively from synthetic DNA. The basis for the replicator is crystal growth: information is stored in the spatial arrangement of monomers and copied from layer to layer by templating. Replication is achieved by fragmentation of crystals, which produces new crystals that carry the same information. Crystal replication avoids intrinsic problems associated with template-directed mechanisms for replication of one-dimensional polymers. A key innovation of our work is that by using programmable DNA tiles as the crystal monomers, we can design crystal growth processes that apply interesting selective pressures to the evolving sequences. While evolution requires that copying occur with high accuracy, we show how to adapt error-correction techniques from algorithmic self-assembly to lower the replication error rate as much as is required
- …