Article · Wikipedia archive · Last revised Jun 14, 2026

Indexed language

Indexed languages are a class of formal languages discovered by Alfred Aho; they are described by indexed grammars and can be recognized by nested stack automata.

Last revised
Jun 14, 2026
Read time
≈ 3 min
Length
599 w
Citations
22
Source
Wikipedia ↗

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 Vijayashanker (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 indexed grammars. 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:

{ a n b n c n d n | n ≥ 1 } {\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n\geq 1\}} 3
{ a n b m c n d m | m , n ≥ 0 } {\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:

{ a 2 n | n ≥ 0 } {\displaystyle \{a^{2^{n}}|n\geq 0\}} 2
{ w w w | w ∈ { a , b } + } {\displaystyle \{www|w\in \{a,b\}^{+}\}} 3

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

{ ( a b n ) n | n ≥ 0 } {\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

Hayashi14 generalized the pumping lemma to indexed grammars. Conversely, Gilman7 gives a "shrinking lemma" for indexed languages.

See also

See also

References

References

  1. 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. 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. 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. 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). 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.
External links