Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata

Abstract:We investigate weighted automata with discounting and their behaviours over semirings and finitely generated graded monoids. We characterize the discounted behaviours of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product. This extends a classical result of Kleene-Schützenberger. Here we show that the very special case of Schützenberger's result for free monoids over singleton alphabets suffices to deduce our generalization.

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“…[Sak87]. For background on the theory of formal power series, we refer the reader to [Con71,SS78,KS86,BR88,Kui97,Sak03].…”

Section: Introductionmentioning

confidence: 99%

Weighted automata with discounting

Droste

Sakarovitch

Vogler

2008

Information Processing Letters

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Section: Moreover If C Is Unambiguous/(essentially) Ergodic Then Almentioning

confidence: 99%

“…productions of the form T U , where T, U ∈ V, and ε-free, then d T (w) < ∞ always. We remark that there is a simple algorithm that transforms a (reduced) grammar into an equivalent (reduced) grammar without chain rules (see, for instance [11], Section 4.3, or [15]). Thus, with this terminology, the grammar is unambiguous if and only if d S (w) = 1 for all w ∈ L(C).…”

Section: Weighted-growth-sensitivity Of Linear Languagesmentioning

confidence: 99%

“…productions of the form A B, where A, B ∈ V. Again there is a simple algorithm that transforms a reduced grammar C into an equivalent reduced grammar C without chain rules; see e.g. [11, Section 4.3] or [15, Corollary 5.3].Note that these transformations preserve unambiguity and (essential) ergodicity, namely C is unambiguous (resp. (essentially) ergodic) if the original grammar C is such.…”

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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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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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“…This was the starting point for a large amount of work on formal power series, cf. [23,15,1,14] for surveys. Special cases of automata with multiplicities are networks with capacities (weights) and have been also investigated in operations research for algebraic optimization problems, cf.…”

Section: Introductionmentioning

confidence: 99%

On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables

Droste

Gastin

2007

Theory Comput Syst

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Abstract. Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non-commuting, then the product of any two aperiodic series is again aperiodic. We also show that if K is idempotent and the matrix monoids over K have a Burnside property (satisfied, e.g. by the tropical semiring), then the aperiodic and the star-free series coincide. This generalizes a classical result of Schützenberger (1961) for aperiodic regular languages and subsumes a result of Guaiana, Restivo and Salemi (1992) on aperiodic trace languages.

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Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata

Cited by 126 publications

References 90 publications

Weighted automata with discounting

Weighted automata with discounting

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

On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables

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