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 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:

${\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n\geq 1\}}$ ^[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]

• 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] gives a "shrinking lemma" for indexed languages.

See also

• Chomsky hierarchy

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

• "NLP in Prolog" chapter on indexed grammars and languages