From Wikipedia, the free encyclopediaIncomputer science, in the
area offormal language theory, frequent use is made of a variety
ofstring functions; however, the notation used is different from
that used oncomputer programming, and some commonly used functions
in the theoretical realm are rarely used when programming. This
article defines some of these basic terms.Contents[hide] 1Strings
and languages 2Alphabet of a string 3String substitution 4String
homomorphism 5String projection 6Right quotient 7Syntactic relation
8Right cancellation 9Prefixes 10See also 11Notes
12ReferencesStrings and languages[edit]A string is a finite
sequence of characters. Theempty stringis denoted by. The
concatenation of two stringandis denoted by, or shorter by.
Concatenating with the empty string makes no difference:.
Concatenation of strings is associative:.For example,.Alanguageis a
finite or infinite set of strings. Besides the usual set operations
like union, intersection etc., concatenation can be applied to
languages: if bothandare languages, their concatenationis defined
as the set of concatenations of any string fromand any string from,
formally. Again, the concatenation dotis often omitted for
shortness.The languageconsisting of just the empty string is to be
distinguished from the empty language. Concatenating any language
with the former doesn't make any change:, while concatenating with
the latter always yields the empty language:. Concatenation of
languages is associative:.For example, abbreviating, the set of all
three-digit decimal numbers is obtained as. The set of all decimal
numbers of arbitrary length is an example for an infinite
language.Alphabet of a string[edit]Thealphabet of a stringis the
set of all of the characters that occur in a particular string.
Ifsis a string, itsalphabetis denoted by
Thealphabet of a languageis the set of all characters that occur
in any string of, formally:.For example, the setis the alphabet of
the string, and theaboveis the alphabet of theabovelanguageas well
as of the language of all decimal numbers.String
substitution[edit]LetLbe alanguage, and let be its alphabet.
Astring substitutionor simply asubstitutionis a mappingfthat maps
letters in to languages (possibly in a different alphabet). Thus,
for example, given a lettera , one hasf(a)=LawhereLa *is some
language whose alphabet is . This mapping may be extended to
strings asf()=for theempty string, andf(sa)=f(s)f(a)for stringsL.
String substitutions may be extended to entire languages as[1]
Regular languagesare closed under string substitution. That is,
if each letter of a regular language is substituted by another
regular language, the result is still a regular
language.[2]Similarly,context-free languagesare closed under string
substitution.[3][note 1]A simple example is the conversionfuc(.) to
upper case, which may be defined e.g. as follows:lettermapped to
languageremark
xfuc(x)
a{ A }map lower-case char to corresponding upper-case char
A{ A }map upper-case char to itself
{ SS }no upper-case char available, map to two-char string
0{ }map digit to empty string
!{ }forbid punctuation, map to empty language
...similar for other chars
For the extension offucto strings, we have e.g. fuc(Strae) = {S}
{T} {R} {A} {SS} {E} = {STRASSE}, fuc(u2) = {U} {} = {U}, and
fuc(Go!) = {G} {O} {} = {}.For the extension offucto languages, we
have e.g. fuc({ Strae, u2, Go! }) = { STRASSE } { U } { } = {
STRASSE, U }.Another example is the conversion of anEBCDIC-encoded
string toASCII.String homomorphism[edit]Astring homomorphism(often
referred to simply as ahomomorphisminformal language theory) is a
string substitution such that each letter is replaced by a single
string. That is,f(a)=s, wheresis a string, for each lettera.[note
2][4]String homomorphisms aremonoid morphismson thefree monoid,
preserving thebinary operationofstring concatenation. Given a
languageL, the setf(L) is called thehomomorphic imageofL.
Theinverse homomorphic imageof a stringsis defined asf1(s) =
{w|f(w)=s}while the inverse homomorphic image of a languageLis
defined asf1(L) = {s|f(s) L}In general,f(f1(L)) L, while one does
havef(f1(L)) LandLf1(f(L))for any languageL.The class of regular
languages is closed under homomorphisms and inverse
homomorphisms.[5]Similarly, the context-free languages are closed
under homomorphisms[note 3]and inverse homomorphisms.[6]A string
homomorphism is said to be -free (or e-free) iff(a) for allain the
alphabet . Simple single-lettersubstitution ciphersare examples of
(-free) string homomorphisms.An example string homomorphismguccan
also be obtained by defining similar to theabovesubstitution:guc(a)
= A, ...,guc(0) = , but lettinggucundefined on punctuation chars.
Examples for inverse homomorphic images are guc1({ SSS }) = { sss,
s, s }, sinceguc(sss) =guc(s) =guc(s) = SSS, and guc1({ A, bb }) =
{ a }, sinceguc(a) = A, while bb cannot be reached byguc.For the
latter language,guc(guc1({ A, bb })) =guc({ a }) = { A } { A, bb }.
The homomorphismgucis not -free, since it maps e.g. 0 to .String
projection[edit]Ifsis a string, andis an alphabet, thestring
projectionofsis the string that results by removing all letters
which are not in. It is written as. It is formally defined by
removal of letters from the right hand side:
Heredenotes theempty string. The projection of a string is
essentially the same as aprojection in relational algebra.String
projection may be promoted to theprojection of a language. Given
aformal languageL, its projection is given by
Right quotient[edit]Theright quotientof a letterafrom a
stringsis the truncation of the letterain the strings, from the
right hand side. It is denoted as. If the string does not haveaon
the right hand side, the result is the empty string. Thus:
The quotient of the empty string may be taken:
Similarly, given a subsetof a monoid, one may define the
quotient subset as
Left quotients may be defined similarly, with operations taking
place on the left of a string.Syntactic relation[edit]The right
quotient of a subsetof a monoiddefines anequivalence relation,
called therightsyntactic relationofS. It is given by
The relation is clearly of finite index (has a finite number of
equivalence classes) if and only if the family right quotients is
finite; that is, if
is finite. In this case,Sis arecognizable language, that is, a
language that can be recognized by afinite state automaton. This is
discussed in greater detail in the article onsyntactic
monoids.Right cancellation[edit]Theright cancellationof a
letterafrom a stringsis the removal of the first occurrence of the
letterain the strings, starting from the right hand side. It is
denoted asand is recursively defined as
The empty string is always cancellable:
Clearly, right cancellation and projectioncommute:
Prefixes[edit]Theprefixes of a stringis the set of allprefixesto
a string, with respect to a given language:
here.Theprefix closure of a languageis
Example:
A language is calledprefix closedif.The prefix closure operator
isidempotent:
Theprefix relationis abinary relationsuch thatif and only if.
This relation is a particular example of aprefix order
From Wikipedia, the free encyclopediaIncomputer science, in the
area offormal language theory, frequent use is made of a variety
ofstring functions; however, the notation used is different from
that used oncomputer programming, and some commonly used functions
in the theoretical realm are rarely used when programming. This
article defines some of these basic terms.Contents[hide] 1Strings
and languages 2Alphabet of a string 3String substitution 4String
homomorphism 5String projection 6Right quotient 7Syntactic relation
8Right cancellation 9Prefixes 10See also 11Notes
12ReferencesStrings and languages[edit]A string is a finite
sequence of characters. Theempty stringis denoted by. The
concatenation of two stringandis denoted by, or shorter by.
Concatenating with the empty string makes no difference:.
Concatenation of strings is associative:.For example,.Alanguageis a
finite or infinite set of strings. Besides the usual set operations
like union, intersection etc., concatenation can be applied to
languages: if bothandare languages, their concatenationis defined
as the set of concatenations of any string fromand any string from,
formally. Again, the concatenation dotis often omitted for
shortness.The languageconsisting of just the empty string is to be
distinguished from the empty language. Concatenating any language
with the former doesn't make any change:, while concatenating with
the latter always yields the empty language:. Concatenation of
languages is associative:.For example, abbreviating, the set of all
three-digit decimal numbers is obtained as. The set of all decimal
numbers of arbitrary length is an example for an infinite
language.Alphabet of a string[edit]Thealphabet of a stringis the
set of all of the characters that occur in a particular string.
Ifsis a string, itsalphabetis denoted by
Thealphabet of a languageis the set of all characters that occur
in any string of, formally:.For example, the setis the alphabet of
the string, and theaboveis the alphabet of theabovelanguageas well
as of the language of all decimal numbers.String
substitution[edit]LetLbe alanguage, and let be its alphabet.
Astring substitutionor simply asubstitutionis a mappingfthat maps
letters in to languages (possibly in a different alphabet). Thus,
for example, given a lettera , one hasf(a)=LawhereLa *is some
language whose alphabet is . This mapping may be extended to
strings asf()=for theempty string, andf(sa)=f(s)f(a)for stringsL.
String substitutions may be extended to entire languages as[1]
Regular languagesare closed under string substitution. That is,
if each letter of a regular language is substituted by another
regular language, the result is still a regular
language.[2]Similarly,context-free languagesare closed under string
substitution.[3][note 1]A simple example is the conversionfuc(.) to
upper case, which may be defined e.g. as follows:lettermapped to
languageremark
xfuc(x)
a{ A }map lower-case char to corresponding upper-case char
A{ A }map upper-case char to itself
{ SS }no upper-case char available, map to two-char string
0{ }map digit to empty string
!{ }forbid punctuation, map to empty language
...similar for other chars
For the extension offucto strings, we have e.g. fuc(Strae) = {S}
{T} {R} {A} {SS} {E} = {STRASSE}, fuc(u2) = {U} {} = {U}, and
fuc(Go!) = {G} {O} {} = {}.For the extension offucto languages, we
have e.g. fuc({ Strae, u2, Go! }) = { STRASSE } { U } { } = {
STRASSE, U }.Another example is the conversion of anEBCDIC-encoded
string toASCII.String homomorphism[edit]Astring homomorphism(often
referred to simply as ahomomorphisminformal language theory) is a
string substitution such that each letter is replaced by a single
string. That is,f(a)=s, wheresis a string, for each lettera.[note
2][4]String homomorphisms aremonoid morphismson thefree monoid,
preserving thebinary operationofstring concatenation. Given a
languageL, the setf(L) is called thehomomorphic imageofL.
Theinverse homomorphic imageof a stringsis defined asf1(s) =
{w|f(w)=s}while the inverse homomorphic image of a languageLis
defined asf1(L) = {s|f(s) L}In general,f(f1(L)) L, while one does
havef(f1(L)) LandLf1(f(L))for any languageL.The class of regular
languages is closed under homomorphisms and inverse
homomorphisms.[5]Similarly, the context-free languages are closed
under homomorphisms[note 3]and inverse homomorphisms.[6]A string
homomorphism is said to be -free (or e-free) iff(a) for allain the
alphabet . Simple single-lettersubstitution ciphersare examples of
(-free) string homomorphisms.An example string homomorphismguccan
also be obtained by defining similar to theabovesubstitution:guc(a)
= A, ...,guc(0) = , but lettinggucundefined on punctuation chars.
Examples for inverse homomorphic images are guc1({ SSS }) = { sss,
s, s }, sinceguc(sss) =guc(s) =guc(s) = SSS, and guc1({ A, bb }) =
{ a }, sinceguc(a) = A, while bb cannot be reached byguc.For the
latter language,guc(guc1({ A, bb })) =guc({ a }) = { A } { A, bb }.
The homomorphismgucis not -free, since it maps e.g. 0 to .String
projection[edit]Ifsis a string, andis an alphabet, thestring
projectionofsis the string that results by removing all letters
which are not in. It is written as. It is formally defined by
removal of letters from the right hand side:
Heredenotes theempty string. The projection of a string is
essentially the same as aprojection in relational algebra.String
projection may be promoted to theprojection of a language. Given
aformal languageL, its projection is given by
Right quotient[edit]Theright quotientof a letterafrom a
stringsis the truncation of the letterain the strings, from the
right hand side. It is denoted as. If the string does not haveaon
the right hand side, the result is the empty string. Thus:
The quotient of the empty string may be taken:
Similarly, given a subsetof a monoid, one may define the
quotient subset as
Left quotients may be defined similarly, with operations taking
place on the left of a string.Syntactic relation[edit]The right
quotient of a subsetof a monoiddefines anequivalence relation,
called therightsyntactic relationofS. It is given by
The relation is clearly of finite index (has a finite number of
equivalence classes) if and only if the family right quotients is
finite; that is, if
is finite. In this case,Sis arecognizable language, that is, a
language that can be recognized by afinite state automaton. This is
discussed in greater detail in the article onsyntactic
monoids.Right cancellation[edit]Theright cancellationof a
letterafrom a stringsis the removal of the first occurrence of the
letterain the strings, starting from the right hand side. It is
denoted asand is recursively defined as
The empty string is always cancellable:
Clearly, right cancellation and projectioncommute:
Prefixes[edit]Theprefixes of a stringis the set of allprefixesto
a string, with respect to a given language:
here.Theprefix closure of a languageis
Example:
A language is calledprefix closedif.The prefix closure operator
isidempotent:
Theprefix relationis abinary relationsuch thatif and only if.
This relation is a particular example of aprefix order
CmSc 365 Theory of Computation
Alphabets and Languages Definitions Operations on strings
Languages Operations on languages ProblemsLearning goalsExam-like
problems
1. Some definiions and propertiesAlphabet: A finite set of
symbols.E.G {a,b,c,x.y.z}. {0,1},{0,1,2,3,4,5,6,7,8,9}Stringover an
alphabet: A finite sequence of symbols from the alphabetE.G.:
thisisastring - over {a,b,c,,z}01011 - over {0,1}3786 - over
{0,1,2,3,4,5,6,7,8,9}A string with one symbol only = symbol
itselfEmpty string- no symbols, notation:eNote: we use the letters
a,b,c,, w,x,y,z both for naming strings and for writing instances
of strings.Usually for names of strings we use the last letters: w,
x,y,zThus x = abc means thatabcis a string and we call itx.Length
of a string- its length as a sequence (the number of symbols)ifw=
abcd, |w| = 4Ifw= classroom, |w| = 9We can match a position in a
string with the symbol there:Ifw= classroom,w(3) = a,w(4) = s,
andw(5) = sTo be able to distinguish between same symbols, we refer
to them as different occurrences of the same symbol.2. Operations
on stringsConcatenation:combines two strings by putting them one
after the other.E.Gx= abc,y= mnop, thenx y= abcmnop, or simplyxy=
abcmnopThe concatenation of the empty string with any other string
gives the string itself:xe=ex = xSubstring:Ifwis a string, thenvis
a substring ofwif there exist stringsxandysuch thatw = xvyxis
calledprefix, andyis calledsuffixofwThe i-th concatenation of a
string with itself is defined in the following way:w0=ewi+1= wi
wfor each i 0.Sow1= w, bang2= bangbangKleene star operationon
strings: Letwbe a string.w*is the set of strings obtainedby
applying any number of concatenations ofwwith itself, including the
empty string.Example:a* = { e, a, aa, aaa, aaaa, aaaaa, }Reversalof
a stringwdenotedwRis the string spelled backwardsFormal
definition:Ifwis a string of length 0, thenwR= w =eIfwis a string
of length n+1 > 0, thenw = uafor some a, andwR= a uR.3.
LanguagesIf is an alphabet, then * is the set of all strings over
.Language:any set of strings over an alphabet , i.e. any subset of
*.* is a countably infinite set. Its elements can be ordered in the
following way:a. The alphabet is a finite set, so we can order the
symbols in some way.b. The set * can be partitioned into disjoint
sets with respect to the length of the strings (there are infinite
number of strings, however each string has a finite length)c. For
eachk 0 first we enumerate all strings of lengthkbefore all strings
of lengthk+1. This means that we first order the strings of length
0 (this is the empty string), then strings of length 1, then of
length 2, etc.d. Strings of length k, denoted as nkare enumerated
lexicographically :ai1ai2aikprecedesaj1aj2ajkif for somem, 0 m k-1,
we have ih=jhfor h = 1,,mand im+1< jm+1. Note that if ih=jhmeans
that aihis the same as ajh.4. Operations on languagesLanguages are
sets, so the operations union, intersection and difference are
applicable. There are two operations specific for
languages:Concatenation of languagesConcatenation of languages is
defined in the following way:If L1and L2are languages, then L =
L1L2(or simply L1L2) is the set:L = {w*:w = xy , xL1, yL2}i.e. L
consists of all possible concatenations between strings in L1and
strings in L2.Concatenation of languages corresponds to the
Cartesian product of sets.Kleene starof a language L: the set of
all strings obtained by concatenating zero or more strings from L.
It is denoted by L*.If we consider as a language, then * would be
the Kleene star of that language.5. Problemsa. Is the set of all
possible meaningful English sentences countable?b. Is the set of
all possible meaningless English sentences countable?c. Define the
relation
