In formal language theory, a noncontracting grammar is in Kuroda normal form if all production rules are of the form:
- AB → CD or
- A → BC or
- A → B or
- A → 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 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 → CD is replaced by four context-sensitive rules AB → AZ, AZ → WZ, WZ → WD and WD → 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:
- AB → CD or
- A → BC or
- A → a or
- A → 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, 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 → 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 or
- A → BC or
- A → a
For every context-sensitive grammar, there exists a weakly equivalent one-sided normal form.
See also
References
- ^ Masami Ito; Yūji Kobayashi; Kunitaka Shoji (2010). Automata, Formal Languages and Algebraic Systems: Proceedings of AFLAS 2008, Kyoto, Japan, 20-22 September 2008. World Scientific. p. 182. ISBN 978-981-4317-60-3.
- ^ Mateescu, Alexandru; Salomaa, Arto (1997). "Chapter 4: Aspects of Classical Language Theory". In Rozenberg, Grzegorz; Salomaa, Arto (eds.). Handbook of Formal Languages. Volume I: Word, language, grammar. Springer-Verlag. p. 190. ISBN 978-3-540-61486-9.
- Willem J. M. Levelt (2008). An Introduction to the Theory of Formal Languages and Automata. John Benjamins Publishing. pp. 126–127. ISBN 978-90-272-3250-2.
- ^ Alexander Meduna (2000). Automata and Languages: Theory and Applications. Springer Science & Business Media. p. 722. ISBN 978-1-85233-074-3.
- Alexander Meduna (2000). Automata and Languages: Theory and Applications. Springer Science & Business Media. p. 728. ISBN 978-1-85233-074-3.
Further reading
- Sige-Yuki Kuroda (June 1964). "Classes of languages and linear-bounded automata". Information and Control. 7 (2): 207–223. doi:10.1016/S0019-9958(64)90120-2.
- G. Révész, "Comment on the paper 'Error detection in formal languages,'" Journal of Computer and System Sciences, vol. 8, no. 2, pp. 238–242, Apr. 1974. doi:10.1016/S0022-0000(74)80057-7 (Révész' trick)
- Penttonen, Martti (Aug 1974). "One-sided and two-sided context in formal grammars". Information and Control. 25 (4): 371–392. doi:10.1016/S0019-9958(74)91049-3.
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