Indexed language
Indexed languages are a class of formal languages discovered by Alfred Aho;^{[1]} they are described by indexed grammars and can be recognized by nested stack automata.^{[2]}
Indexed languages are a proper subset of context-sensitive languages.^{[1]} They qualify as an abstract family of languages (furthermore a full AFL) and hence satisfy many closure properties. However, they are not closed under intersection or complement.^{[1]}
The class of indexed languages has practical importance in natural language processing as a computationally affordable generalization of context-free languages, since indexed grammars can describe many of the nonlocal constraints occurring in natural languages.
Gerald Gazdar (1988)^{[3]} and Vijay-Shanker (1987)^{[4]} introduced a mildly context-sensitive language class now known as linear indexed grammars (LIG).^{[5]} Linear indexed grammars have additional restrictions relative to IG. LIGs are weakly equivalent (generate the same language class) as tree adjoining grammars.^{[6]}
Examples
The following languages are indexed, but are not context-free:
- ^{[3]}
- ^{[2]}
These two languages are also indexed, but are not even mildly context sensitive under Gazdar's characterization:
- ^{[2]}
- ^{[3]}
On the other hand, the following language is not indexed:^{[7]}
Properties
Hopcroft and Ullman tend to consider indexed languages as a "natural" class, since they are generated by several formalisms, such as:^{[8]}
- Aho's indexed grammars^{[1]}
- Aho's one-way nested stack automata^{[10]}
- Fischer's macro grammars^{[11]}
- Greibach's automata with stacks of stacks^{[12]}
- Maibaum's algebraic characterization^{[13]}
Hayashi^{[14]} generalized the pumping lemma to indexed grammars. Conversely, Gilman^{[7]}^{[15]} gives a "shrinking lemma" for indexed languages.
See also
References
- 1 2 3 4 Aho, Alfred (1968). "Indexed grammars—an extension of context-free grammars". Journal of the ACM. 15 (4): 647–671. doi:10.1145/321479.321488.
- 1 2 3 Partee, Barbara; Alice ter Meulen; Robert E. Wall (1990). Mathematical Methods in Linguistics. Kluwer Academic Publishers. pp. 536–542. ISBN 978-90-277-2245-4.
- 1 2 3 Gazdar, Gerald (1988). "Applicability of Indexed Grammars to Natural Languages". In U. Reyle and C. Rohrer. Natural Language Parsing and Linguistic Theories. pp. 69–94.
- ↑ http://search.proquest.com/docview/303610666
- ↑ Laura Kallmeyer (2010). Parsing Beyond Context-Free Grammars. Springer Science & Business Media. p. 31. ISBN 978-3-642-14846-0.
- ↑ Laura Kallmeyer (16 August 2010). Parsing Beyond Context-Free Grammars. Springer Science & Business Media. p. 32. ISBN 978-3-642-14846-0.
- 1 2 Gilman, Robert H. (1996). "A Shrinking Lemma for Indexed Languages" (PDF). Theoretical Computer Science. 163 (1–2): 277–281. doi:10.1016/0304-3975(96)00244-7.
- ↑ Introduction to automata theory, languages, and computation,^{[9]}Bibliographic notes, p.394-395
- ↑ Hopcroft, John; Jeffrey Ullman (1979). Introduction to automata theory, languages, and computation. Addison-Wesley. p. 390. ISBN 0-201-02988-X.
- ↑ Alfred Aho (1969). "Nested Stack Automata". Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529.
- ↑ Michael J. Fischer (1968). "Grammars with Macro-Like Productions". Proc. 9th Ann. IEEE Symp. on Switching and Automata Theory (SWAT). pp. 131–142.
- ↑ Sheila A. Greibach (1970). "Full AFL's and Nested Iterated Substitution" (PDF). Information and Control. 16 (1): 7–35. doi:10.1016/s0019-9958(70)80039-0.
- ↑ T.S.E. Maibaum (1974). "A Generalized Approach to Formal Languages" (PDF). J. Computer and System Sciences. 8 (3): 409–439. doi:10.1016/s0022-0000(74)80031-0.
- ↑ T. Hayashi (1973). "On Derivation Trees of Indexed Grammars - An Extension of the uvxyz Theorem". Publication of the Research Institute for Mathematical Sciences. Research Institute for Mathematical Sciences. 9 (1): 61–92. doi:10.2977/prims/1195192738.
- ↑ Robert H. Gilman (Sep 1995). "A Shrinking Lemma for Indexed Languages".