In formal language theory, a noncontracting grammar is in Kuroda normal form if all production rules are of the form:
where A, B, C and D are nonterminal symbols and a is a terminal symbol. Some sources omit the A â 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 noncontracting grammar that does not generate the empty string can be converted to Kuroda normal form.
A straightforward technique attributed to György Révész transforms a grammar in Kuroda normal form to a context-sensitive grammar: AB â CD is replaced by four context-sensitive rules AB â AZ, AZ â WZ, WZ â WD and WD â CD. This proves that every noncontracting grammar generates a context-sensitive language.
There is a similar normal form for unrestricted grammars as well, which at least some authors call "Kuroda normal form" too:
where õ is the empty string. Every unrestricted grammar is weakly equivalent to one using only productions of this form.
If the rule AB â CD is eliminated from the above, one obtains context-free grammars in Chomsky Normal Form. The Penttonen normal form (for unrestricted grammars) is a special case where first rule above is AB â AD. Similarly, for context-sensitive grammars, the Penttonen normal form, also called the one-sided normal form (following Penttonen's own terminology) is:
For every context-sensitive grammar, there exists a weakly equivalent one-sided normal form.