# 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[citation needed] 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:

${\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n\geq 1\}}$ [3]
${\displaystyle \{a^{n}b^{m}c^{n}d^{m}|m,n\geq 0\}}$ [2]

These two languages are also indexed, but are not even mildly context sensitive under Gazdar's characterization:

${\displaystyle \{a^{2^{n}}|n\geq 0\}}$ [2]
${\displaystyle \{www|w\in \{a,b\}^{+}\}}$ [3]

On the other hand, the following language is not indexed:[7]

${\displaystyle \{(ab^{n})^{n}|n\geq 0\}}$

## Properties

Hopcroft and Ullman tend to consider indexed languages as a "natural" class, since they are generated by several formalisms, such as:[9]

Hayashi[14] generalized the pumping lemma to indexed grammars. Conversely, Gilman[7] gives a "shrinking lemma" for indexed languages.

## References

1. ^ a b c d Aho, Alfred (1968). "Indexed grammars—an extension of context-free grammars". Journal of the ACM. 15 (4): 647–671. doi:10.1145/321479.321488. S2CID 9539666.
2. ^ a b c Partee, Barbara; ter Meulen, Alice; Wall, Robert E. (1990). Mathematical Methods in Linguistics. Kluwer Academic Publishers. pp. 536–542. ISBN 978-90-277-2245-4.
3. ^ a b c Gazdar, Gerald (1988). "Applicability of Indexed Grammars to Natural Languages". In Reyle, U.; Rohrer, C. (eds.). Natural Language Parsing and Linguistic Theories. Studies in Linguistics and Philosophy. Vol. 35. Springer Netherlands. pp. 69–94. doi:10.1007/978-94-009-1337-0_3. ISBN 978-94-009-1337-0.
4. ^ Vijayashanker, K. (1987). A study of tree adjoining grammars (Thesis). ProQuest 303610666.
5. ^ Kallmeyer, Laura (2010). Parsing Beyond Context-Free Grammars. Springer. p. 31. ISBN 978-3-642-14846-0.
6. ^ Kallmeyer, Laura (16 August 2010). Parsing Beyond Context-Free Grammars. Springer. p. 32. ISBN 978-3-642-14846-0.
7. ^ a b Gilman, Robert H. (1996). "A Shrinking Lemma for Indexed Languages". Theoretical Computer Science. 163 (1–2): 277–281. arXiv:math/9509205. doi:10.1016/0304-3975(96)00244-7. S2CID 14479068.
8. ^ Hopcroft, John; Ullman, Jeffrey (1979). Introduction to automata theory, languages, and computation. Addison-Wesley. p. 390. ISBN 978-0-201-02988-8.
9. ^ Introduction to automata theory, languages, and computation,[8] Bibliographic notes, p.394-395
10. ^ Aho, Alfred V. (July 1969). "Nested Stack Automata". Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529. S2CID 685569.
11. ^ Fischer, Michael J. (October 1968). "Grammars with macro-like productions". 9th Annual Symposium on Switching and Automata Theory (Swat 1968). 9th Annual Symposium on Switching and Automata Theory (swat 1968). pp. 131–142. doi:10.1109/SWAT.1968.12.
12. ^ Greibach, Sheila A. (March 1970). "Full AFLs and nested iterated substitution". Information and Control. 16 (1): 7–35. doi:10.1016/s0019-9958(70)80039-0.
13. ^ Maibaum, T.S.E. (June 1974). "A generalized approach to formal languages". Journal of Computer and System Sciences. 8 (3): 409–439. doi:10.1016/s0022-0000(74)80031-0.
14. ^ Hayashi, Takeshi (1973). "On derivation trees of indexed grammars: an extension of the {$uvwxy$}-theorem". Publications of the Research Institute for Mathematical Sciences. 9 (1): 61–92. doi:10.2977/prims/1195192738.