View source for Abstract family of languages

In [[computer science]], in particular in the field of [[formal language]] theory,
an '''abstract family of languages''' is an abstract mathematical notion generalizing characteristics common to the [[regular language]]s, the [[context-free language]]s and the [[recursively enumerable language]]s, and other families of formal languages studied in the scientific literature.

==Formal definitions==

A ''formal language'' is a set {{mvar|L}} for which there exists a finite set of abstract symbols {{math|&Sigma;}} such that <math>L \subseteq\Sigma^*</math>, where * is the [[Kleene star]] operation.

A ''family of languages'' is an ordered pair <math>(\Sigma,\Lambda)</math>, where
# {{math|&Sigma;}} is an infinite set of symbols;
# {{math|&Lambda;}} is a set of formal languages;
# For each {{mvar|L}} in {{math|&Lambda;}} there exists a finite subset <math>\Sigma_1 \subseteq \Sigma</math> such that <math>L \subseteq \Sigma_1^*</math>; and
# {{math|''L'' ≠ Ø}} for some {{mvar|L}} in {{math|&Lambda;}}.

A ''trio'' is a family of languages [[Closure (mathematics)|closed]] under [[homomorphism]]s that do not introduce the empty word, inverse [[homomorphism]]s, and intersections with a regular language.

A ''full trio,'' also called a  ''[[cone (formal languages)|cone]],'' is a trio closed under arbitrary homomorphism.

A ''(full) semi-AFL'' is a (full) trio  closed under [[Union (set theory)|union]].

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

==Some families of languages==
The following are some simple results from the study of abstract families of languages.<ref name="SpringerAFL">{{SpringerEOM| title=Abstract family of languages | id=Abstract_family_of_languages | oldid=18934 | first=A. | last=Mateescu | first2=A. | last2=Salomaa }}</ref>

Within the [[Chomsky hierarchy]], the regular languages, the context-free languages, and the recursively enumerable languages are all full AFLs. However, the [[Context-sensitive language|context sensitive languages]] and the [[recursive language]]s 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.<ref name="SpringerOps">{{SpringerEOM| title=AFL operations | id=AFL_operations | oldid=13097 | first=Gh. | last=Păun }}</ref>

==Origins==

[[Seymour Ginsburg]] of the [[University of Southern California]] and [[Sheila Greibach]] of [[Harvard University]] presented the first AFL theory paper at the IEEE Eighth Annual [[Symposium on Switching and Automata Theory]] in 1967.<ref>{{harvtxt|Ginsburg|Greibach|1967}}</ref>

==Notes==
<references/>

==References==
* {{cite conference
  | first1 = Seymour 
  | last1 = Ginsburg
  | first2 = Sheila 
  | last2= Greibach
  | title=Abstract Families of Languages
  | book-title = Conference Record of 1967 Eighth Annual Symposium on Switching and Automata Theory, 18–20 October 1967, Austin, Texas, USA
  | year = 1967
  | pages= 128–139
  |publisher = IEEE
}}
*Seymour Ginsburg, ''Algebraic and automata theoretic properties of formal languages'', North-Holland, 1975, {{ISBN|0-7204-2506-9}}.
* John E. Hopcroft and Jeffrey D. Ullman, ''[[Introduction to Automata Theory, Languages, and Computation]]'', Addison-Wesley Publishing, Reading Massachusetts, 1979. {{ISBN|0-201-02988-X}}. Chapter 11: Closure properties of families of languages.
* {{cite book |last1=Mateescu | first1=Alexandru |last2=Salomaa|first2=Arto |editor1-first=Grzegorz| editor1-last=Rozenberg|editor2-first=Arto| editor2-last=Salomaa |title=Handbook of Formal Languages. Volume I: Word, language, grammar |publisher=Springer-Verlag |year=1997 |pages=175–252 |chapter=Chapter 4: Aspects of Classical Language Theory |isbn=3-540-61486-9}}

[[Category:Formal languages]]
[[Category:Applied mathematics]]