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− | In addition to the usual set-theoretic operations on formal languages, one carries out: multiplication (or direct multiplication, or concatenation): | + | In addition to the usual set-theoretic operations on formal languages, one carries out: multiplication (or direct multiplication, or [[concatenation]]): |
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Revision as of 21:39, 12 January 2016
in mathematical linguistics
An arbitrary set of chains (that is, words, cf. Word) over some (finite or infinite) alphabet (sometimes also called a dictionary), that is, a set of expressions of the form
, where
; the number
, usually denoted by
, is the length of the chain
. One also considers the empty chain, denoted by
; one sets
. One often speaks of a language over the alphabet
, omitting the word "formal" . In mathematical linguistics and the theory of automata (cf. Automata, theory of) one considers various effective ways of specifying a formal language, principally by means of formal grammars (cf. Grammar, formal) and automata of various types, which can, in the majority of cases, be described as modifications of non-deterministic Turing machines (cf. Turing machine), often multi-tape, with some restrictions on the ways the machine works on each tape.
Operations on formal languages.
In addition to the usual set-theoretic operations on formal languages, one carries out: multiplication (or direct multiplication, or concatenation):
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left division:
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right division is defined by analogy with left division; iteration:
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where denotes
and
(in particular, the set of all chains in
is
); truncated iteration:
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and substitution: If is a language over the finite alphabet
and
are arbitrary languages, then
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if each of the languages ,
, consists of one chain
, a substitution is called a homomorphism; if all the
are non-empty, one speaks of an unabbreviated homomorphism. If the language
consists of one chain
, then instead of
,
, etc., one writes
,
, etc., respectively.
A variety of languages is an ordered pair (or
, if
is taken for granted), where
is an infinite alphabet and
is a class of languages such that: 1) for every
there is a finite alphabet
such that
; 2)
for some
; and 3)
is closed relative to union, multiplication, intersection with regular sets, truncated iteration, unabbreviated homomorphisms, and inverses of arbitrary homomorphisms. A variety that is closed relative to arbitrary homomorphisms is called complete.
References
[1] | A.V. Gladkii, "Formal grammars and languages" , Moscow (1973) (In Russian) |
[2] | S. Ginsburg, S. Greibach, Y. Hopcroft, "Studies in abstract families of languages" Mem. Amer. Math. Soc. , 87 (1969) pp. 1–32 |
Comments
For more details see Formal languages and automata.
Formal language. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_language&oldid=37513