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Kontextsensitive Grammatik Beispiel Essay

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Definition:  Sei  = (, , , ) eine Grammatik, wobei die Menge der Variablen ist, die Menge der Terminal­zeichen, die Menge der Produktionen und das Startsymbol, ferner  =    das Gesamt­alphabet.

Die Grammatik heißt kontext­sensitiv, wenn jede Produktion von der Form

mit   ,    + und ,   * ist.

Als Ausnahme ist die Produktion   ε erlaubt, wobei dann nicht auf der rechten Seite einer Produktion vorkommen darf.

Eine Sprache heißt kontext­sensitiv, wenn es eine kontext­sensitive Grammatik gibt, die sie erzeugt.

Die Bezeichnung kontext­sensitiv (im Gegensatz zu kontextfrei) rührt daher, dass die Ersetzung    nur im Kontext ... zulässig ist; d.h. wenn irgendwo in einem Wort vorkommt, darf es nur dann durch ersetzt werden, wenn direkt links von das Teilwort und direkt rechts von das Teilwort steht.

 

Definition:  Eine kontext­sensitive Grammatik ist in Kuroda-Normalform, wenn jede Produktion von der Form

mit    und , , ,    ist.

Als Ausnahme ist die Produktion ε erlaubt, um das leere Wort ε zu erzeugen; dann darf aber nicht auf der rechten Seite einer Produktion vorkommen.

 

Satz:  Zu jeder kontext­sensitiven Sprache gibt es eine kontext­sensitive Grammatik in Kuroda-Normalform.

Eine kontextfreie Grammatik in Chomsky-Normalform ist offensichtlich in Kuroda-Normalform.

Definition:  Eine Grammatik heißt monoton, wenn für alle Produktionen gilt

||  ||

Bei einer Ersetzung von durch in einem Wort wird also niemals kürzer.

Als Ausnahme ist die Produktion ε erlaubt; dann darf aber nicht auf der rechten Seite einer Produktion vorkommen.

Satz:  Zu jeder kontext­sensitiven Sprache gibt es eine monotone Grammatik.

Eine kontext­sensitive Grammatik in Kuroda-Normalform ist offensichtlich monoton. Kontextfreie Grammatiken in Chomsky- und in Greibach-Normalform sowie rechts­lineare Grammatiken sind ebenfalls monoton.

Die Sprache  = { abc  |     } ist nicht kontextfrei. Dies lässt sich mit dem Pumping-Lemma für kontextfreie Sprachen zeigen. ist aber kontext­sensitiv.

Eine monotone Grammatik für ist die folgende:

ac  |  abc
cc
bbb

Eine Grammatik in Kuroda-Normalform für ist folgende:

 

Weiter mit: 

 

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