In computer science, a grammar is informally called a recursive grammar if it contains production rules that are recursive, meaning that expanding a non-terminal according to these rules can eventually lead to a string that includes the same non-terminal again. Otherwise it is called a non-recursive grammar.1
For example, a grammar for a context-free language is left recursive if there exists a non-terminal symbol A that can be put through the production rules to produce a string with A (as the leftmost symbol).23 All types of grammars in the Chomsky hierarchy can be recursive and it is recursion that allows the production of infinite sets of words.1
Properties
A non-recursive grammar can produce only a finite language; and each finite language can be produced by a non-recursive grammar.1 For example, a straight-line grammar produces just a single word.
A recursive context-free grammar that contains no useless rules necessarily produces an infinite language. This property forms the basis for an algorithm that can test efficiently whether a context-free grammar produces a finite or infinite language.4
References
References
- Nederhof, Mark-Jan; Satta, Giorgio (2002), "Parsing Non-recursive Context-free Grammars", Proceedings of the 40th Annual Meeting on Association for Computational Linguistics (ACL '02), Stroudsburg, PA, USA: Association for Computational Linguistics, pp. 112–119, doi:10.3115/1073083.1073104.
- Notes on Formal Language Theory and Parsing Archived 2017-08-28 at the Wayback Machine, James Power, Department of Computer Science National University of Ireland, Maynooth Maynooth, Co. Kildare, Ireland.
- Moore, Robert C. (2000), "Removing Left Recursion from Context-free Grammars", Proceedings of the 1st North American Chapter of the Association for Computational Linguistics Conference (NAACL 2000), Stroudsburg, PA, USA: Association for Computational Linguistics, pp. 249–255.
- Fleck, Arthur Charles (2001), Formal Models of Computation: The Ultimate Limits of Computing, AMAST series in computing, vol. 7, World Scientific, Theorem 6.3.1, p. 309, ISBN 9789810245009.