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Automata theory and formal language

Introduction to formal language

May 1, 2017

1 - Introduction to formal language

The theory of formal languages {}{}was born within the domain of linguistics as a means of understanding the syntactic regularities of natural languages. In computer science, formal languages {}{}are used inter alia as a basis for defining programming languages. Language theory is both a mathematical, computer and linguistic branch, a formal language is a set of symbol strings with a set of rules that are specific to it. The alphabet of a formal language is the set of symbols or letters ... from which the chains of language can be formed; Often it is necessary to be finite. The strings formed from this alphabet are called words and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is often defined by means of a formal grammar such as a regular grammar or a grammar out of context, also called its formation rule. The domain of formal language theory mainly studies the purely syntactic aspects of these languages, that is, their internal structures.

2 - Alphabets, Words and Languages

begin{example} $Sigma ={0,1,2,...}=% %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion $ is an alphabet , $484,275,107,...$ are words end{example} begin{notation} The set of words on $Sigma $ will be noted $Sigma ^{ast }$ end{notation} begin{notation} The word that does not contain any letter is called, the empty word denoted $wedge $ or $varepsilon .$ end{notation} begin{remark} We are interested only in the form of a word which may be arbitrary and we omit the meaning of the word example: avhbrtys, jorbqhegd, ... are words. end{remark}
subsection{Formal Language} begin{definition} A formal language is set of words ( ie a subset of $Sigma ^{ast }$ $)$ end{definition} begin{example} ${$ $CRMEF,Oujda,$ $Hay,$ $AlMassira$ $}$ is language. end{example} begin{example} ${Lambda ,a,aa,aaa,...}={a^{n}/nin %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion }$ is infinite language generated by a finite alphabet $Sigma ={a}.$ end{example}

3 - Operations on words and languages

section{Operations on words and languages} subsection{Words concatenation} begin{definition} The concatenation of two words $x$ and $y$ is the word noed by $x.y $ or simply by $xy$ defined by : newline $x.y=xy$ ie if $x=x_{1}x_{2}...x_{r}$ and $y=y_{1}y_{2}...y_{s}$ then $x.y=x=x_{1}x_{2}...x_{r}$ $y_{1}y_{2}...y_{s}$ end{definition} begin{example} If $ x=langu$ and $y=age$ then $x.y=language$ end{example}
subsection{Concatenation of langauges} begin{definition} Let E and F be two languages of lthe alphabet $Sigma .$ The concatenation of $E$ and $F$ is the language defined by : newline $E.F={x.y/(x,y)in Etimes F}$ end{definition} begin{example} $E={u,v}$ , $F={uv,vu}$ $E.F={uuv,uvu,vuv,vvu}$ end{example} $E.F$ Will be noted later by $EF$ begin{definition} A language is said to be closed for the law "." If it is stable for concatenation ( i.e $forall x,yin E$ we have: $x.yin E$ end{definition} begin{example} ${x^{n}/nin %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion }$ is closed language $ $but the language ${x^{n}y^{m}/(n,m)in %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion times %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion $ $}$ is not closed. end{example}

begin{definition} Let $E$ be a language, the smallest closed language for the concatenation containing $E$ and ${wedge }$ Is called the closure of $E$ and is noted by $E^{ast }.$ $E^{ast }$ is also called Kleene closure or Kleene star. end{definition} begin{remark} $E^{ast }=underset{kgeq 0}{cup E^{k}}$ ( where $E^{k}$ is the concatenation $E.E...E$ ) end{remark} begin{example} Let $E$ be a language and $ x$ a word of $E$ then we have:newline $1)$ $ {x}^{ast }={x^{n}/nin %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion }$newline $2)$ ${xx}^{ast }={x^{2n}/nin %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion }$newline $3)$ ${xx,xxx}^{ast }={x^{n}/ngeq 2}cup {Lambda }$newline $4)$ ${0,1}^{ast }$ Is the set of all natural integers containing only the numbers $0$ or $1.$newline $5)$ $varnothing ^{ast }={Lambda }$ end{example}

4 - Combinatorics on words

section{Combinatorics of words} subsection{Length of words, prefixes and suffixes} begin{definition} The length of a word $x$ denoted $|x|$ , is the number of symbols that compose it. Example if $x=arbre$ alors $ |x|=5$ end{definition} begin{notation} Let $x$ be a word of $Sigma ^{ast }$ and $sigma $ be a symbol of $Sigma , $ the number of occurrences of $sigma $ in $x$ is denoted by $ |x|_{sigma }.$ end{notation} begin{example} $|arbre|_{r}=2,$ $|arbre|_{a}=1$ , $|abbcb|_{b}=3$ end{example} begin{definition} Let $x=x_{1}x_{2}...x_{r}$ be a word of $Sigma ^{ast }$ $ $the words $% :wedge ,$ $x_{1},$ $x_{1}x_{2},...,x_{1}x_{2}...x_{r}$ [ respectively $% wedge ,$ $x_{r},$ $x_{r-1}x_{r},$ $x_{r-2}x_{r-1}x_{r}$ ] Are called the prefixes of $x$ [ respectively the suffixes of $x]$ end{definition}
subsection{Parikh function's} Let $Sigma $ a finite and ordered alphabet, And can therefore be considered as a n-tuple $Sigma =(sigma _{1},sigma _{2},...,sigma _{n})$ the map $% psi :sum^{ast }rightarrow %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion ^{n}$ ,$xlongmapsto (|x|_{sigma _{1}},|x|_{sigma _{2}},...,|x|_{sigma _{n}})$ is called textbf{Parikh function's} begin{remark} For $n>1$ the parikh function's is not injective. end{remark} begin{notation} for any word $x=x_{1}x_{2}...x_{r}$ we put $% widetilde{x}=x_{r}x_{r-1}...x_{1}$ , and for any language $E$ we put : $% widetilde{E}={widetilde{x}/xin E}$ , $widetilde{x}$ is called the mirror image of $x$, and $widetilde{E}$ the mirror image of $E.$ end{notation} begin{definition} We call palindrome, every word $x$, verifying : $widetilde{x}=x$. end{definition} begin{example} laval, radar are palindromes. end{example}

5 - Regular Language

section{Regular Language } Let L be a language over alphabet $Sigma $ We say that L is regular or rational if L can be obtained recursively by: 1) from the only basic languages:newline $-$ language $emptyset $newline $-$ language ${wedge }$ ( Language formed only by the textbf{empty} symbol$)$newline $-$ Language of the form ${a}$ where $ain Sigma $newline $2)$By using only the following operations :newline $-$ Unions operationsnewline $-$ Concatenationnewline $-$ Kleene's star begin{example} The language ${x^{n}/nin %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion }$ regular end{example} begin{example} The language $L$ of the words of even length is rational since $L=(Sigma ^{2})^{ast }$ end{example} begin{remark} Every finite language is regular end{remark} begin{proposition} If $L$ is a regular language then $L^{ast }$ is also. end{proposition} begin{proposition} If $L_{1}$ $et L_{2}$ are regular then $L_{1}cdot L_{2}$ , $L_{1}cup L_{2}$ and $L_{1}cap L_{2}$ are regular. end{proposition}

6 - Residual Language

section{Residual Language} begin{definition} Let $L$ be a language over alphabet $Sigma $ and $u$ a word of $Sigma ^{ast }$, we get new language by putting $u^{-1}L=L/u={vin Sigma ^{ast }/uvin L}.$ $L/u$ is called the residual language of the language $L$ with respect to word $u.$ end{definition} begin{example} $L={uuv,uw,vv,uu}$ alors $L/u={uv,w,u}$ end{example} begin{theorem} (Myhill Nerode theorem's) A language is recognizable if and only if it has only a finite number of residuals. end{theorem}

7 - Topology on formal language

section{Topology on formal language} subsection{Metric over language} For any language $L$ on an alphabet $Sigma $ we define a distance $% d:Ltimes Llongrightarrow %TCIMACRO{U{211d} }% %BeginExpansion mathbb{R} %EndExpansion _{+}$ $(u,v)longmapsto 2^{|uwedge v|}$ where $uwedge v$ is the longest common prefix, moreover this distance is ultrametric $d(u,w)leq max (d(u,v),d(v,w))$ subsection{Topology inductive limit on Kleene closure $X^{ast }$} We define a topology on $X^{(k)}=X.X...X$ (The language on the alphabet $X$ formed of the words of length $k$) by conventioning that a part $O$ of $% X^{(k)}$ is open, if it is, any union of the sets of the form : $% O_{1}.O_{2}...O_{k}$ ( Concatenation of $O_{i}$ ) where $O_{1},...,O_{k}$ are opens subsets of $X.$ Then we equip the closing Kleene$X^{ast }$ of the inductive limit topology associated at the family of canonical injections $i_{k}:X^{(k)}hookrightarrow X^{ast }$ begin{remark} $X$ can be considered as a topological subspace of $X^{(k)}$ and consequently a topological subspace of $X^{ast } $by means of continuous injections $i:Xhookrightarrow X^{(k)}$ $ xlongmapsto x.varpi ...varpi $ where $varpi $ is the empty word. end{remark} begin{remark} Every language $L$ on the alphabet $X$ is considered as a topological subspace of $X^{ast }$ for the induced topology. end{remark}

Younes Derfoufi
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