Kuroda normal form
In formal language theory, a grammar is in Kuroda normal form if all production rules are of the form:[1]
- 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.[1] Some sources omit the A → B pattern.[2]
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.[3]
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.[2]
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.[1]
There is a similar normal form for unrestricted grammars as well, which at least some authors call "Kuroda normal form" too:[4]
- 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.[2]
If the rule AB → CD is eliminated from the above, then one obtains context-free languages.[5] The Penttonen normal form (for unrestricted grammars) is a special case where A = C in the first rule above.[4] For context-sensitive grammars, the Penttonen normal form, also called the one-sided normal form (following Penttonen's own terminology) is just:[1][2]
- AB → AD or
- A → BC or
- A → a
As the name suggests, for every context-sensitive grammar, there exists a [weakly] equivalent one-sided/Penttonen normal form.[2]
See also
References
- 1 2 3 4 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.
- 1 2 3 4 5 Mateescu, Alexandru; Salomaa, Arto (1997). "Chapter 4: Aspects of Classical Language Theory". In Rozenberg, Grzegorz; Salomaa, Arto. Handbook of Formal Languages. Volume I: Word, language, grammar. Springer-Verlag. p. 190. ISBN 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 90-272-3250-4.
- 1 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" (PDF). 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" (PDF). Information and Control 25 (4): 371–392. doi:10.1016/S0019-9958(74)91049-3.
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