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| An '''[[Ordinal number|ω]]-language''' is a [[Set (mathematics)|set]] of infinite-length sequences of [[symbol (formal)|symbols]].
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| ==Formal definition==
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| Let Σ be a set of symbols (not necessarily finite). Following the standard definition from [[formal language]] theory, Σ<sup>*</sup> is the set of all ''finite'' words over Σ. Every finite word has a length, which is, obviously, a natural number. Given a word ''w'' of length ''n'', ''w'' can be viewed as a function from the set {0,1,...,''n''-1} → Σ. The infinite words, or ω-words, can likewise be viewed as functions from <math>\mathbb{N}</math> to Σ, with the value at ''i'' giving the symbol at position ''i''. The set of all infinite words over Σ is denoted Σ<sup>ω</sup>. The set of all finite ''and'' infinite words over Σ is sometimes written Σ<sup>∞</sup>.
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| Thus, an ω-language ''L'' over Σ is a [[subset]] of Σ<sup>ω</sup>.
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| ==Operations==
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| Some common operations defined on ω-languages are:
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| * ''Intersection and union''. Given ω-languages ''L'' and ''M'', both ''L'' ∪ ''M'' and ''L'' ∩ ''M'' are ω-languages.
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| * ''Left catenation''. Let ''L'' be an ω-language, and ''K'' be a language of finite words only. Then ''K'' can be catenated on the left ''only'' to ''L'' to yield the new ω-language ''KL''.
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| * ''Omega (infinite iteration)''. As the notation hints, the operation (<math>\cdot</math>)<sup>ω</sup> is the infinite version of the [[Kleene star]] operator on finite-length languages. Given a formal language ''L'', ''L''<sup>ω</sup> is the ω-language of all infinite sequence of words from ''L''; in the functional view, of all functions <math>\mathbb{N}</math>→''L''.
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| * ''Prefixes''. Let ''w'' be an ω-word. Then the formal language Pref(''w'') contains every ''finite'' [[Prefix (computer science)|prefix]] of ''w''.
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| * ''Limit''. Given a finite-length language ''L'', an ω-word ''w'' is in the ''limit'' of ''L'' if and only if Pref(''w'') ∩ ''L'' is an ''infinite'' set. In other words, for an arbitrarily large natural number ''n'', it is always possible to choose some word in ''L'', whose length is greater than ''n'', ''and'' which is a prefix of ''w''. The limit operation on ''L'' can be written ''L''<sup>δ</sup> or <math>\vec{L}</math>.
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| ==Distance between ω-words==
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| The set Σ<sup>ω</sup> can be made into a [[metric space]] by definition of the [[metric (mathematics)|metric]] d:Σ<sup>ω</sup> × Σ<sup>ω</sup> → '''R''' as:
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| : '''if''' ''w'' and ''v'' share any finite prefix, then d(''w'',''v'')= inf {2<sup>-|''x''|</sup> : ''x'' in Σ<sup>*</sup>, and ''x'' in both Pref(''w'') and Pref(''v'') }.
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| : '''otherwise''' d(''w'', ''v'')=1
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| where |''x''| is interpreted as "the length of ''x''" (number of symbols in ''x''), and '''inf''' is the [[infimum]] over sets of [[real number]]s. If ''w''=''v'', they have no longest finite prefix, and d(''w'',''v'')=0; it can be shown that d satisfies all the other necessary properties of a [[metric (mathematics)|metric]].
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| ==Important subclasses==
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| The most widely used subclass of the ω-languages is the set of [[omega-regular language|ω-regular languages]], which enjoy the useful property of being recognizable by [[Büchi automaton|Büchi automata]]; thus the [[decision problem]] of ω-regular language membership is decidable and fairly straightforward to compute.
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| ==Bibliography==
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| * Perrin, D. and Pin, J-E. "[http://www.liafa.jussieu.fr/~jep/Resumes/InfiniteWords.html Infinite Words Automata, Semigroups, Logic and Games]". Pure and Applied Mathematics Vol 141, Elsevier, 2004.
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| * Staiger, L. "[http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=CB21AA7D00A17BADAE650D2B342D7752?doi=10.1.1.48.4015&rep=rep1&type=pdf ω-Languages]". In G. Rozenberg and [[Arto Salomaa|A. Salomaa]], editors, ''Handbook of Formal Languages'', Volume 3, pages 339-387. Springer-Verlag, Berlin, 1997.
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| * Thomas, W. "Automata on Infinite Objects". In [[Jan van Leeuwen]], editor, ''Handbook of Theoretical Computer Science'', Volume B: Formal Models and Semantics, pages 133-192. Elsevier Science Publishers, Amsterdam, 1990.
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| [[Category:Theory of computation]]
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| [[Category:Formal languages]]
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