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| In [[formal language theory]], a '''cone''' is a set of [[formal language]]s that has some desirable [[closure (mathematics)|closure]] properties enjoyed by some well-known sets of languages, in particular by the families of [[regular language]]s, [[context-free language]]s and the [[recursively enumerable language]]s.<ref>{{harvtxt|Ginsburg|Greibach|1967}}</ref> The concept of a cone is a more abstract notion that subsumes all of these families. A similar notion is the '''faithful cone''', having somewhat relaxed conditions. For example, the [[context-sensitive language]]s do not form a cone, but still have the required properties to form a faithful cone.
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| The terminology ''cone'' has a French origin. In the American oriented literature one usually speaks of a ''full trio''. The ''trio'' corresponds to the faithful cone.
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| ==Definition==
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| A cone is a non-empty family <math>\mathcal{S}</math> of languages such that, for any <math>L \in \mathcal{S}</math> over some alphabet <math>\Sigma</math>,
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| * if <math>h</math> is a [[homomorphism]] from <math>\Sigma^\ast</math> to some <math>\Delta^\ast</math>, the language <math>h(L)</math> is in <math>\mathcal{S}</math>;
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| * if <math>h</math> is a homomorphism from some <math>\Delta^\ast</math> to <math>\Sigma^\ast</math>, the language <math>h^{-1}(L)</math> is in <math>\mathcal{S}</math>;
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| * if <math>R</math> is any regular language over <math>\Sigma</math>, then <math>L\cap R</math> is in <math>\mathcal{S}</math>.
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| The family of all regular languages is contained in any cone.
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| If one restricts the definition to homomorphisms that do not introduce the empty word <math>\lambda</math> then one speaks of a ''faithful cone''; the inverse homomorphisms are not restricted. Within the [[Chomsky hierarchy]], the regular languages, the context-free languages, and the [[recursively enumerable language]]s are all cones, whereas the context-sensitive languages and the recursive languages are only faithful cones.
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| ==Relation to Transducers==
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| A [[finite state transducer]] is a finite state automaton that has both input and output. It defines a transduction <math>T</math>, mapping a language <math>L</math> over the input alphabet into another language <math>T(L)</math> over the output alphabet. Each of the cone operations (homomorphism, inverse homomorphism, intersection with a regular language) can be implemented using a finite state transducer. And, since finite state transducers are closed under composition, every sequence of cone operations can be performed by a finite state transducer.
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| Conversely, every finite state transduction <math>T</math> can be decomposed into cone operations. In fact, there exists a normal form for this decomposition,<ref>{{harvtxt|Nivat|1968}}</ref> which is commonly known as ''Nivat's Theorem'':<ref>cf. {{harvtxt|Mateescu|Salomaa|1997}}</ref>
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| Namely, each such <math>T</math> can be effectively decomposed as
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| <math>T(L) = g(h^{-1}(L) \cap R)</math>, where <math>g, h</math> are homomorphisms, and <math>R</math> is a regular language depending only on <math>T</math>.
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| Altogether, this means that a family of languages is a cone if it is closed under finite state transductions. This is a very powerful set of operations. For instance one easily writes a (nondeterministic) finite state transducer with alphabet <math>\{a,b\}</math> that removes every second <math>b</math> in words of even length (and does not change words otherwise). Since the context-free languages form a cone, they are closed under this exotic operation.
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| ==See also==
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| * [[Abstract family of languages]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite conference
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| | first1 = Seymour
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| | last1 = Ginsburg
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| | first2 = Sheila
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| | last2= Greibach
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| | title=Abstract Families of Languages
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| | booktitle = Conference Record of 1967 Eighth Annual Symposium on Switching and Automata Theory, 18–20 October 1967, Austin, Texas, USA
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| | year = 1967
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| | pages= 128–139
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| |publisher = IEEE
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| }}
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| * {{cite doi| 10.5802/aif.287}}
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| *[[Seymour Ginsburg]], ''Algebraic and automata theoretic properties of formal languages'', North-Holland, 1975, ISBN 0-7204-2506-9.
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| * John E. Hopcroft and Jeffrey D. Ullman, ''[[Introduction to Automata Theory, Languages, and Computation]]'', Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X. Chapter 11: Closure properties of families of languages.
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| * {{cite book |last1=Mateescu | first1=Alexandru |last2=Salomaa|first2=Arto |editor1-first=Grzegorz| editor1-last=Rozenberg|editor2-first=Arto| editor2-last=Salomaa |title=Handbook of Formal Languages. Volume I: Word, language, grammar |publisher=Springer-Verlag |year=1997 |pages=175–252 |chapter=Chapter 4: Aspects of Classical Language Theory |isbn=3-540-61486-9}}
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| ==External links==
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| *[http://eom.springer.de/T/t110060.htm Encyclopedia of mathematics: Trio], Springer.
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| [[Category:Formal languages]]
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