String operations: Difference between revisions

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The concatenation of two string <math>s</math> and <math>t</math> is denoted by <math>s \cdot t</math>, or shorter by <math>s t</math>.

Concatenating with the empty string makes no difference: <math>s \cdot \varepsilon = s = \varepsilon \cdot s</math>.

Concatenation of strings is [[associative]]: <math>s \cdot (t \cdot u) = (s \cdot t) \cdot u</math>.

For example, <math>(\langle b \rangle \cdot \langle l \rangle) \cdot (\varepsilon \cdot \langle ah \rangle) = \langle bl \rangle \cdot \langle ah \rangle = \langle blah \rangle</math>.

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Similarly, [[context-free language]]s are closed under string substitution.<ref>Hopcroft, Ullman (1979), Sect.6.2, Theorem 6.2, p.131</ref><ref group="note">Although every regular language is also context-free, the previous theorem is not implied by the current one, since the former yields a shaper result for regular languages.</ref>

A simple example is the conversion ''f''_uc(.) to upper caseuppercase, which may be defined e.g. as follows:

{| class="wikitable"

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! ''x'' !! ''f''_uc(''x'') !!

|-

| ‹''a''› || { ‹''A''› } || map lower-caselowercase char to corresponding upper-caseuppercase char

|-

| ‹''A''› || { ‹''A''› } || map upper-caseuppercase char to itself

|-

| ‹''ß''› || { ‹''SS''› } || no upper-caseuppercase char available, map to two-char string

|-

| ‹0› || { ε } || map digit to empty string

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==String homomorphism==

A '''string homomorphism''' (often referred to simply as a [[Homomorphism#Homomorphisms and e-free homomorphisms in formalFormal language theory|homomorphism]] in [[formal language theory]]) is a string substitution such that each character is replaced by a single string. That is, <math>f(a)=s</math>, where <math>s</math> is a string, for each character <math>a</math>.<ref group="note" name="singleton sets">Strictly formally, a homomorphism yields a language consisting of just one string, i.e. <math>f(a) = \{s\}</math>.</ref><ref>Hopcroft, Ullman (1979), Sect.3.2, p.60-61</ref>

String homomorphisms are [[monoid morphism]]s on the [[free monoid]], preserving the empty string and the [[binary operation]] of [[string concatenation]]. Given a language <math>L</math>, the set <math>f(L)</math> is called the '''homomorphic image''' of <math>L</math>. The '''inverse homomorphic image''' of a string <math>s</math> is defined as

<math>f^{-1}(s) = \{ w |\mid f(w) = s \}</math>

while the inverse homomorphic image of a language <math>L</math> is defined as

<math>f^{-1}(L) = \{ s |\mid f(s) \in L \}</math>

In general, <math>f(f^{-1}(L)) \neq L</math>, while one does have

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A string homomorphism is said to be ε-free (or e-free) if <math>f(a) \neq \varepsilon</math> for all ''a'' in the alphabet <math>\Sigma</math>. Simple single-letter [[substitution cipher]]s are examples of (ε-free) string homomorphisms.

An example string homomorphism ''g''_uc can also be obtained by defining similar to the [[#String_substitution|above]] substitution: ''g''_uc(‹a›) = ‹A›, ..., ''g''_uc(‹0›) = ε, but letting ''g''_uc be undefined on punctuation chars.

<!---omitted, since sloppy notation, anyway, see above---

Besides this restriction of its input ___domain, ''g''_uc differs from ''f''_uc by returning strings,<ref group=note name="singleton sets"/> while the latter returned singleton sets of strings.

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:<math>\pi_\Sigma (L)=\{\pi_\Sigma(s)\ \vert\ s\in L \}</math>{{citation needed|date=August 2017}}

== Right and left quotient ==

The '''right quotient''' of a character ''a'' from a string ''s'' is the truncation of the character ''a'' in the string ''s'', from the right hand side. It is denoted as <math>s/a</math>. If the string does not have ''a'' on the right hand side, the result is the empty string. Thus:

<!---This definition deviates from Hopcroft.Ullman.1979, as remarked below. I guess the former doesn't have widespread use, if it has a source at all.--->

: <math>(sa)/ b = \begin{cases}

:<math>(sa)/ b = \begin{cases}

s & \mbox{if } a=b \\

\varepsilon & \mbox{if } a \ne b

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The quotient of the empty string may be taken:

: <math>\varepsilon / a = \varepsilon</math>

:<math>\varepsilon / a = \varepsilon</math>

Similarly, given a subset <math>S\subset M</math> of a monoid <math>M</math>, one may define the quotient subset as

: <math>S/a=\{s\in M\ \vert\ sa\in S\}</math>

'''Left quotients''' may be defined similarly, with operations taking place on the left of a string.{{citation needed|date=August 2017}}

:<math>S/a=\{s\in M\ \vert\ sa\in S\}</math>

Left quotients may be defined similarly, with operations taking place on the left of a string.{{citation needed|date=August 2017}}

Hopcroft and Ullman (1979) define the quotient ''L''₁/''L''₂ of the languages ''L''₁ and ''L''₂ over the same alphabet as {{nowrap|1=''L''₁/''L''₂ = {{mset| ''s'' |{{!}} ∃''t''∈''L''₂. ''st''∈''L''₁ }}}}.<ref>Hopcroft, Ullman (1979), Sect.3.2, p.62</ref>

This is not a generalization of the above definition, since, for a string ''s'' and distinct characters ''a'', ''b'', Hopcroft's and Ullman's definition implies {{nowrap||{{mset|''sa''}} / {{mset|''b''}}}} yielding {{mset|}}, rather than {{mset| ε }}.

The left quotient (when definieddefined similar to Hopcroft and Ullman 1979) of a singleton language ''L''₁ and an arbitrary language ''L''₂ is known as [[Brzozowski derivative]]; if ''L''₂ is represented by a [[regular expression]], so can be the left quotient.<ref>{{cite journal| author=Janusz A. Brzozowski| authorlink=Janusz Brzozowski (computer scientist)|title=Derivatives of Regular Expressions| journal=J ACM| year=1964| volume=11| issue=4| pages=481–494| doi=10.1145/321239.321249| s2cid=14126942| doi-access=free}}</ref>

==Syntactic relation==

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:<math>\{S/m\ \vert\ m\in M\}</math>

is finite. In the case that ''M'' is the monoid of words over some alphabet, ''S'' is then a [[regular language]], that is, a language that can be recognized by a [[finite -state automaton]]. This is discussed in greater detail in the article on [[syntactic monoid]]s.{{citation needed|date=August 2017}}

==Right cancellation==

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== References ==

* {{cite book | first1=John E. | last1=Hopcroft | first2=Jeffrey D. | last2=Ullman | title=Introduction to Automata Theory, Languages and Computation | publisher=Addison-Wesley Publishing | ___location=Reading, Massachusetts | year=1979 | isbn=978-0-201-02988-X8 | zbl=0426.68001 | url-access=registration | url=https://archive.org/details/introductiontoau00hopc }} ''(See chapter 3.)''

{{reflist}}

{{Strings}}

[[Category:Formal languages]]