Indexed languages are a proper subset of context-sensitive languages. 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. Gerald Gazdar has characterized the mildly context-sensitive languages in terms of linear indexed grammars.
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.
The following languages are indexed, but are not context-free:
These two languages are also indexed, but are not even mildly context sensitive under Gazdar's characterization:
On the other hand, the following language is not indexed:
- Aho's indexed grammars
- Aho's one-way nested stack automata
- Fischer's macro grammars
- Greibach's automata with stacks of stacks
- Maibaum's algebraic characterization
- Aho, Alfred (1968). "Indexed grammars—an extension of context-free grammars". Journal of the ACM 15 (4): 647–671. doi:10.1145/321479.321488.
- 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.
- 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.
- Gilman, Robert H. (1996). "A Shrinking Lemma for Indexed Languages". Theoretical Computer Science 163 (1–2): 277–281. doi:10.1016/0304-3975(96)00244-7.
- Hopcroft, John; Jeffrey Ullman (1979). Introduction to automata theory, languages, and computation. Addison-Wesley. p. 390. ISBN 0-201-02988-X.
- Introduction to automata theory, languages, and computation,Bibliographic notes, p.394-395
- Alfred Aho (1969). "Nested Stack Automata". Journal of the ACM 16 (3): 383–406.
- 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". 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). "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.
- Robert H. Gilman (Sep 1995). A Shrinking Lemma for Indexed Languages.
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