Abstract family of languages

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An abstract family of languages is a grouping of formal languages such that the membership of a language in a given family is proven by its sharing of specific characteristics with the languages already known to be of that family. A family must contain at least one non-empty language, and while there may be no limit to the size of alphabets in the languages of a family, there must be a finite alphabet over which each of the languages in the family are defined.^[1]

Contents 1 Formal definitions 2 Some families of languages 3 Origins 4 References

[edit] Formal definitions

A formal language is a set L for which there exists a finite set of abstract symbols Σ such that , where * is the Kleene star operation.

A family of languages is an ordered pair (Σ,Λ), where

1. Σ is an infinite set of symbols;
2. Λ is a set of formal languages;
3. For each L in Λ there exists a finite subset Σ₁ ⊂ Σ such that L ⊆ ; and
4. L ≠ Ø for some L in Λ.

A trio is a family of languages closed under e-free homomorphism, inverse homomorphism, and intersection with regular language.

A full trio, also called a cone, is a trio closed under arbitrary homomorphism.

A (full) semi-AFL is a (full) trio closed under union.

A (full) AFL is a (full) semi-AFL closed under concatenation and the Kleene plus.

[edit] Some families of languages

The following are some simple results from the study of abstract families of languages.^[2][1]

Within the Chomsky hierarchy, the regular languages, the context-free languages, and the recursively enumerable languages are all full AFLs. However, the context sensitive languages and the recursive languages are AFLs, but not full AFLs because they are not closed under arbitrary homomorphisms.

The family of regular languages are contained within any cone (full trio). Other categories of abstract families are identifiable by closure under other operations such as shuffle, reversal, or substitution.^[3]

[edit] Origins

Seymour Ginsburg of the University of Southern California and Sheila Greibach of Harvard University first presented their AFL theory paper at the IEEE Eighth Annual Symposium on Switching and Automata Theory in 1967.^[4]

[edit] References

1. ^ ^{a b} Springer, "abstract family of languages"
2. ^ Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, ISBN 0 7204 2506 9.
3. ^ Springer, "AFL operations"
4. ^ Seymour Ginsburg, Sheila Greibach: "Abstract families of languages", Proc. Eighth Annual Symposium on Switching and Automata Theory, 1967.