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 contextsensitive 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 contextfree languages, since indexed grammars can describe many of the nonlocal constraints occurring in natural languages.
Gerald Gazdar (1988)^{[3]} and VijayShanker (1987)^{[4]} introduced a mildly contextsensitive 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 contextfree:

\{a^n b^n c^n d^n n \geq 1 \} ^{[3]}

\{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 \geq 0 \} ^{[2]}

\{www  w \in \{a,b\}^+ \} ^{[3]}
On the other hand, the following language is not indexed:^{[7]}

\{(a b^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]}^{[15]} gives a "shrinking lemma" for indexed languages.
See also
References

^ ^{a} ^{b} ^{c} ^{d}

^ ^{}a ^{b} ^{c}

^ ^{}a ^{b} ^{c} 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 ContextFree Grammars. Springer Science & Business Media. p. 31.

^ Laura Kallmeyer (16 August 2010). Parsing Beyond ContextFree Grammars. Springer Science & Business Media. p. 32.

^ ^{}a ^{b} Gilman, Robert H. (1996). "A Shrinking Lemma for Indexed Languages". Theoretical Computer Science 163 (1–2): 277–281.

^

^ Introduction to automata theory, languages, and computation,^{[8]}Bibliographic notes, p.394395

^ Alfred Aho (1969). "Nested Stack Automata". Journal of the ACM 16 (3): 383–406.

^ Michael J. Fischer (1968). "Grammars with MacroLike 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". Information and Control 16 (1): 7–35.

^ T.S.E. Maibaum (1974). "A Generalized Approach to Formal Languages". J. Computer and System Sciences 8 (3): 409–439.

^ T. Hayashi (1973). Theorem"uvxyz"On Derivation Trees of Indexed Grammars  An Extension of the . Publication of the Research Institute for Mathematical Sciences (Research Institute for Mathematical Sciences) 9 (1): 61–92.

^ Robert H. Gilman (Sep 1995). "A Shrinking Lemma for Indexed Languages".
External links

"NLP in Prolog" chapter on indexed grammars and languages




Each category of languages, except those marked by a ^{*}, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.


This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.