On the entropy of regular languages

Ceccherini‐Silberstein, Tullio; Machì, Antonio; Scarabotti, Fabio

doi:10.1016/s0304-3975(03)00094-x

Cited by 23 publications

(46 citation statements)

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“…In this note we close this gap by showing the following. The strategy (used in [4,5,6] and that we shall make use of here) for proving that every L in a given class L of languages is growth-sensitive is the following.…”

Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

“…This inequality was originally proved by Scarabotti [18]where some ideas of Gromov were used and the usual Perron-Frobenius theory was avoided-was then extended to oriented graphs with specified initial and terminal vertices in [4] where it was applied to the (regular) ergodic case, and it is now generalized for essential ergodicity.…”

Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

“…In the Appendix, for the sake of completeness and the convenience of the reader, the proof (from [4]) of the entropic inequality in the ergodic case is presented.…”

Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

“…linear/right linear) grammar. If this grammar is also unambiguous, we say that L is an ergodic, unambiguous language of the respective type.It is well known that ergodic regular languages of exponential growth are growthsensitive (see for instance [4]). This statement also has several analogues in different setups and terminologies, for instance in the context of Symbolic Dynamics [16, Cor.…”

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confidence: 99%

“…This statement also has several analogues in different setups and terminologies, for instance in the context of Symbolic Dynamics [16, Cor. 4.4.9], [3] and Asymptotic Group Theory [10].Recently it was shown in [5,6] The strategy (used in [4,5,6] and that we shall make use of here) for proving that every L in a given class L of languages is growth-sensitive is the following.Step 1. Consider the set (Σ 2 ) * of all words over Σ 2 .…”

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Growth and ergodicity of context-free languages II: The linear case

Ceccherini‐Silberstein

2006

Trans. Amer. Math. Soc.

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Abstract. A language L over a finite alphabet Σ is called growth-sensitive if forbidding any non-empty set F of subwords yields a sub-language L F whose exponential growth rate is smaller than that of L. Say that a context-free grammar (and associated language) is ergodic if its dependency di-graph is strongly connected. It is known that regular and unambiguous non-linear context-free languages which are ergodic are growth-sensitive. In this note it is shown that ergodic unambiguous linear languags are growth-sensitive, closing the gap that remained open. IntroductionLet L be a language over the alphabet Σ, that is, a subset of the free monoid Σ * of all finite words over Σ. We write ε for the empty word and Σ + = Σ * \ {ε}. For a word w ∈ Σ * , its length (number of letters) is denoted by |w|. The growth rate of L ⊆ Σ * is the numberAlso recall that L has exponential growth if γ(L) > 1 and it has sub-exponential growth otherwise, namely, if γ(L) = 1. A language L over Σ is growth-sensitive iffor any non-empty F ⊂ Σ * consisting of subwords of elements of L, where L F = {w ∈ L : no v ∈ F is a subword of w}.A context-free grammar is a quadruple C = (V, Σ, P, S), where V is a finite set of variables, disjoint from the finite alphabet Σ, the variable S is the start symbol, and P ⊂ V × (V ∪ Σ) * is a finite set of production rules. We write T u or (T u) ∈ P if (T, u) ∈ P. For v, w ∈ (V ∪ Σ) * , we write v =⇒ w if v = v 1 T v 2 and w = v 1 uv 2 , whereReceived by the editors November 4, 2004.2000 Mathematics Subject Classification. Primary 68Q45; Secondary 05A16, 05C20, 68Q42. Key words and phrases. Context-free grammar, linear language, dependency di-graph, ergodicity, ambiguity, growth, higher block languages, automaton, bilateral automaton. 605License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TULLIO CECCHERINI-SILBERSTEINA context-free language is a language generated by a context-free grammar. If in addition the production rules T u are such that u contains at most one variable in V, then the grammar and the corresponding language are termed linear. In particular, if u ∈ Σ * ∪ Σ * V (resp. Σ * ∪ VΣ * ), then the language is left (resp. right) linear. In both cases, language and grammar are also called regular.A general context-free grammar C is called unambiguous if for any w ∈ L(C) there is a unique rightmost derivation S * =⇒ w. A context-free language is unambiguous if it is generated by some unambiguous grammar. Note that there are context-free languages that cannot be generated by unambiguous grammars: these are called inherently ambiguous languages. In our setting, the languageis linear and inherently ambiguous (using Ogden's iteration lemma [17] (see also Chapter 6 in [9]) it can be deduced that one always has two different derivations for the words of the form a n b n c n ). Thus there exist inherently ambiguous linear languages.We shall always assume to have a reduced grammar C, that is, each variable is used in some rightmost derivation of a wor...

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Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

“…In the Appendix, for the sake of completeness and the convenience of the reader, the proof (from [4]) of the entropic inequality in the ergodic case is presented.…”

Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

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confidence: 99%

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Growth and ergodicity of context-free languages II: The linear case

Ceccherini‐Silberstein

2006

Trans. Amer. Math. Soc.

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A Framework for Estimating Simplicity of Automatically Discovered Process Models Based on Structural and Behavioral Characteristics

Kalenkova

Polyvyanyy

Rosa

2020

Lecture Notes in Computer Science

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A plethora of algorithms for automatically discovering process models from event logs has emerged. The discovered models are used for analysis and come with a graphical flowchart-like representation that supports their comprehension by analysts. According to the Occam's Razor principle, a model should encode the process behavior with as few constructs as possible, that is, it should not be overcomplicated without necessity. The simpler the graphical representation, the easier the described behavior can be understood by a stakeholder. Conversely, and intuitively, a complex representation should be harder to understand. Although various conformance checking techniques that relate the behavior of discovered models to the behavior recorded in event logs have been proposed, there are no methods for evaluating whether this behavior is represented in the simplest possible way. Existing techniques for measuring the simplicity of discovered models focus on their structural characteristics such as size or density, and ignore the behavior these models encoded. In this paper, we present a conceptual framework that can be instantiated into a concrete approach for estimating the simplicity of a model, considering the behavior the model describes, thus allowing a more holistic analysis. The reported evaluation over real-life event logs for several instantiations of the framework demonstrates its feasibility in practice.

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A Spectrum of Entropy-Based Precision and Recall Measurements Between Partially Matching Designed and Observed Processes

Kalenkova

Polyvyanyy

2020

Service-Oriented Computing

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On the entropy of regular languages

Cited by 23 publications

References 7 publications

Growth and ergodicity of context-free languages II: The linear case

Growth and ergodicity of context-free languages II: The linear case

A Framework for Estimating Simplicity of Automatically Discovered Process Models Based on Structural and Behavioral Characteristics

A Spectrum of Entropy-Based Precision and Recall Measurements Between Partially Matching Designed and Observed Processes

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