Asymptotic entropy of random walks on regular languages over a finite alphabet

In this paper we consider a superclass of automaton grammars that can be represented in terms of paths on graphs. With this approach, we assume that vertices of graph are labeled by symbols of ﬁnite alphabet A . We will call such grammars graph-generated grammars or G-grammars. In contrast to the graph grammars that are used to describe graph structure transformations, G-grammars using a graphs as a means of representing formal languages. We will give an algorithm for constructing G-grammar which generate the language recognized by deterministic ﬁnite automaton. Moreover, we will show that the class of languages generated by G-grammars is a proper superset of regular languages.

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Asymptotic entropy of transformed random walks

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.90 ◽

2016 ◽

Vol 37 (5) ◽

pp. 1480-1491 ◽

Cited By ~ 1

Author(s):

BEHRANG FORGHANI

Keyword(s):

Random Walk ◽

Random Walks ◽

Ergodic Theory ◽

Stopping Time ◽

Stopping Times ◽

Discrete Groups ◽

Differential Entropy ◽

Random Walks On Groups ◽

Rate Of Escape ◽

Asymptotic Entropy

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

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ON PANSIOT WORDS AVOIDING 3-REPETITIONS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054112400631 ◽

2012 ◽

Vol 23 (08) ◽

pp. 1583-1594

Author(s):

IRINA A. GORBUNOVA ◽

ARSENY M. SHUR

Keyword(s):

Growth Rate ◽

Growth Rates ◽

Regular Language ◽

Numerical Estimate ◽

Regular Languages ◽

Finite Alphabet

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 ≈1.2421.

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Hausdorff dimension of the harmonic measure on trees

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798108180 ◽

1998 ◽

Vol 18 (3) ◽

pp. 631-660 ◽

Cited By ~ 15

Author(s):

VADIM A. KAIMANOVICH

Keyword(s):

Hausdorff Dimension ◽

Random Walks ◽

Large Class ◽

Harmonic Measure ◽

Random Environment ◽

Free Groups ◽

Equivalence Relations ◽

Markov Operators ◽

Rate Of Escape ◽

Asymptotic Entropy

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

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Random walks on regular languages and algebraic systems of generating functions

Algebraic Methods in Statistics and Probability - Contemporary Mathematics ◽

10.1090/conm/287/04787 ◽

2001 ◽

pp. 201-230 ◽

Cited By ~ 2

Author(s):

Steven P. Lalley

Keyword(s):

Random Walks ◽

Generating Functions ◽

Regular Languages ◽

Algebraic Systems

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Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3580 ◽

2008 ◽

Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽

Author(s):

Lorenz A. Gilch

Keyword(s):

Random Walks ◽

Finite Groups ◽

Transition Probabilities ◽

Finite Alphabet ◽

Free Products ◽

Rate Of Escape ◽

International Audience ◽

Special Case ◽

Current Word ◽

Natural Way

International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

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