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.

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.

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) and Vijay-Shanker (1987) introduced a mildly context-sensitive language class now known as linear indexed grammars (LIG). Linear indexed grammars have additional restrictions relative to IG. LIGs are weakly equivalent (generate the same language class) as tree adjoining grammars.

Examples

The following languages are indexed, but are not context-free:

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

${\displaystyle \{a^{n}b^{m}c^{n}d^{m}|m,n\geq 0\}}$

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

${\displaystyle \{a^{2^{n}}|n\geq 0\}}$

${\displaystyle \{www|w\in \{a,b\}^{+}\}}$

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

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

Properties

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

• 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

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

See also

• Chomsky hierarchy

References

External links

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

1. Indexed grammar
2. Index of language articles
3. Search engine indexing
4. Indexed file
5. Index
6. Web server directory index
7. ISAM
8. Pumping lemma
9. Language Proficiency Index
10. Indexicality
11. Controlled vocabulary
12. Index of philosophy of language articles
13. TIOBE index
14. Zero-based numbering
15. List of language families
16. Sparse matrix
17. Lua (programming language)
18. List of programming languages
19. Nested stack automaton
20. Context-sensitive language