Formal languages

Kuroda normal form

In formal language theory, a context-sensitive grammar is in Kuroda normal form if all production rules are of the form: AB → CD orA → BC orA → B orA → a where A, B, C and D are nonterminal symbols and a is a terminal symbol. Some sources omit the A → B pattern. It is named after Sige-Yuki Kuroda, who originally called it a linear bounded grammar—a terminology that was also used by a few other authors thereafter. Every grammar in Kuroda normal form is noncontracting, and therefore, generates a context-sensitive language. Conversely, every context-sensitive language which does not generate the empty string can be generated by a grammar in Kuroda normal form. A straightforward technique attributed to György Révész transforms a grammar in Kuroda's form to Chomsky's CSG: AB → CD is replaced by four context-sensitive rules AB → AZ, AZ → WZ, WZ → WD and WD → CD. This technique also proves that every noncontracting grammar is context-sensitive. There is a similar normal form for unrestricted grammars as well, which at least some authors call "Kuroda normal form" too: AB → CD orA → BC orA → a orA → ε where ε is the empty string. Every unrestricted grammar is weakly equivalent to one using only productions of this form. If the rule AB → CD is eliminated from the above, then one obtains context-free languages. The Penttonen normal form (for unrestricted grammars) is a special case where first rule above is AB → AD. Similarly, for context-sensitive grammars, the Penttonen normal form, also called the one-sided normal form (following Penttonen's own terminology) is: AB → AD orA → BC orA → a For every context-sensitive grammar, there exists a weakly equivalent one-sided normal form. (Wikipedia).

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Related pages

Context-sensitive language | Greibach normal form | Context-sensitive grammar | Noncontracting grammar | Chomsky normal form | Empty string | Unrestricted grammar | Backus–Naur form