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@@ -7,32 +7,25 @@
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__Regular Languages__ are the simplest family of formal languages.
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Can be defined in different ways:
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- algebrically
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-- through generating grammars
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-- through recognizer machines
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+- through __generating grammars__
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+- through __recognizer machines__
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-__Regular Expression__ (*regexp*) is a string $r$ defined with the terminal alphabet $\Sigma$
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-containing some *metasymbols*
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-##### Possible cases:
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+__Regular Expression__ or *regexp*
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+is a string $r$ defined over the elements of a terminal alphabet $\Sigma$, using one of the following *metasymbols* $\;\cdot\;\cup\;\varnothing\;(\;)$
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+#### regexp cases:
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$r,s,t$ are regexp
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1. $r=\varnothing$
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2. $r=a, \;\;a\in \Sigma$
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3. $r=s\cup t\;\; \text{or}\;\; r=s|t$
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4. $r=s.t\;\; \text{or}\;\; r=st$
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5. $r=s^*$ <!--*-->
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-##### Precedence of operators
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-1. star *
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-2. concatenation .
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+#### Precedence of operators
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+1. star $*$
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+2. concatenation $\;\cdot$
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3. union $\cup$ or |
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+<!--*-->
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-##### Derived operators
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-- __Power__ $e^h=ee...e_{n-times}$
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-- __Repetition__ from k to n>k times $[a]_k^n=a^k\cup a^{k+1}\cup ...\cup a^n$
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-- __Optionality__ $[a]=(\varepsilon\cup a)$
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-- __Interval__ of an ordered set $(0...9)(a...z)(A...Z)$
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-
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-By admitting the use of set-theoretic operations __Intersection__, __Complement__ and __Difference__ we obtain the so called __Extended regular expressions__.
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-
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-##### Regular languages
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+#### Regular languages
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A language is __regular__ if it is generated by a *regular expression*
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@@ -44,23 +37,27 @@ $s \cup t \; \text{or} \; s \mid t$ | $L_s \cup L_t$
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$s.t \; \text{or} \; st$ | $L_s.L_t$
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$s^*$|${L_s}^*$
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-##### example
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+##### examples
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>The regexp $e$ that generates the language of integers, possibly signed
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-$e={(+\cup - \cup\varepsilon)dd}^*$ $L_e=\{+,-,\varepsilon\}\{d\}\{d\}^*$
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+$e={(+\cup - \cup\varepsilon)dd}^*\;\;\;L_e=\{+,-,\varepsilon\}\{d\}\{d\}^*$
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+>The language over the binary alphabet $\{a,b\}$, such that the number of $a$ in every string is odd and every string contains at least one $b$
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+$A_p=b^*({ab}^*{ab}^*)^*\;\;A_d=b^*{ab}^*({ab}^*{ab}^*)^*$
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+$L=A_pbA_d | A_dbA_p$
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+<!--*-->
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-#### Sets of languages
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-__Family of regular languages__ or __REG__:
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-Is the collection of all regular languages.
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+### Sets of languages
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+__Family of regular languages__ or $REG$:
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+Is the collection of all *regular languages*.
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-__Family of finite languages__ or __FIN__:
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-Is the collection of all languages of finite cardinality.
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+__Family of finite languages__ or $FIN$:
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+Is the collection of all *languages of finite cardinality*.
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Every __finite language__ is *regular* as it is the union of a finite number of finite strings.
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But family of __regular languages__ contains also languages of *infinite cardinality*.
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$$FIN\subset REG$$
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-#### Subexpression of a regexp
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+### Subexpression of a regexp
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1. Consider a *fully parenthesized* regexp
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$$ e = (a\cup (bb))^*(c^+ \cup (a \cup (bb)))$$
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2. Number every alphabetic symbol that occurs in the regexp
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@@ -70,11 +67,67 @@ $$ (a_1\cup (b_2b_3))^*\;\;\;\;c_4^+ \cup (a_5 \cup (b_6b_7))$$
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$$ a_1\cup (b_2b_3)\;\;\;\;\;\;c_4^+\;\;\;\;a_5 \cup (b_6b_7)$$
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$$ a_1\;\;\;\;b_2b_3\;\;\;\;\;\;c_4^\;\;\;\;a_5 \;\;(b_6b_7)$$
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$$ a_1\;\;\;\;b_2\;\;\;b_3\;\;\;\;\;\;c_4^\;\;\;\;a_5 \;\;\;b_6\;\;\;b_7$$
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-<<!--*-->
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-#### Different parenthesization
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-#### Derivation
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-#### Ambiguity
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-#### Closure of the $REG$ family
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+<!--*-->
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+### Different parenthesization
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+And hence different interpretations of a regexp
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+$$f=(a\cup bb)^*(c^+\cup a\cup bb)$$
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+$$f=(a\cup bb)^*(c^+\cup (a\cup bb))$$
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+$$f=(a\cup bb)^*((c^+\cup a)\cup bb)$$
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+<!--*-->
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+Choosing one of the *interpretations*,
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+I obtain a new regexp defining a smaller language that the original one.
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+
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+### Derivation
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+Between two regexp $e'$ and $e''$:
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+$$e'\Rightarrow e''$$
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+If the two regexp $e'$ and $e''$ can be factored as
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+$$e'=\alpha\beta\gamma \;\;\; e''=\alpha\delta\gamma$$
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+With
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+- $\beta$ subexpression of $e'$
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+- $\gamma$ subexpression of $e''$
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+- $\gamma$ is a *choice* of $\beta$
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+##### examples
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+- one step derivation
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+$$a^*\cup b^+\Rightarrow a^*\;\;\;a^*\cup b^+\Rightarrow b^+$$
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+- multiple derivation
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+$$a^*\cup b^+\Rightarrow a^*\Rightarrow\varepsilon\;\;\;a^*\cup b^+\overset{2}{\Rightarrow} \varepsilon$$
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+
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+### Equivalence
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+Two regexps are __equivalent__ if they define the same language
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+
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+The language defined by a *derived regexp* is contained in the language defined by the *deriving regexp*
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+
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+### Ambiguity
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+A string may be obtained by two derivations differing not only in the order of choices
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+$$(a\cup b)^*a(a\cup b)^*$$
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+$$(a\cup b)^*a(a\cup b)^* \Rightarrow (a\cup b)a(a\cup b)^*\Rightarrow aa(a\cup b)^*\Rightarrow aa\varepsilon\Rightarrow aa$$
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+$$(a\cup b)^*a(a\cup b)^* \Rightarrow \varepsilon a(a\cup b)^*\Rightarrow \varepsilon a(a\cup b)\Rightarrow \varepsilon aa\Rightarrow aa$$
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+<!--*-->
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+
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+A regexp is __ambiguous__ if the corresponding marked (with numbers on letters) regexp $f_\#$ generates two marked strings $x$ and $y$ such that, removing the indices, we obtain the same string (sufficient condition, not necessary)
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+##### examples
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+$$(aa|ba)^*a|b(aa|ba)^*\;\; \text{is ambiguous}$$
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+in fact the marking
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+$$(a_1a_2|b_3a_4)^*a_5|b_6(a_7a_8|b_9a_{10})^*$$
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+Generates the two strings
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+$$b_3a_4a_5\;\;\;b_6a_7a_8$$
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+That project ambiguously to the same phrase: $\;\;\;baa$
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+
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+### Other operators
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+__Basic operators:__ union, concatenation, star
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+
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+__Derived operators:__
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+- __power__ $e^h = \underset{h\;times}{ee...e}$
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+- __cross__ $e^+ = {ee}^*$
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+<!--*-->
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+- __Repetition__ from k to n>k times $[a]_k^n=a^k\cup a^{k+1}\cup ...\cup a^n$
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+
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+- __Optionality__ $[a]=(\varepsilon\cup a)$
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+- __Interval__ of an ordered set $(0...9)(a...z)(A...Z)$
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+
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+By admitting the use of set-theoretic operations __Intersection__, __Complement__ and __Difference__ we obtain the so called __Extended regular expressions__.
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+
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+### Closure of the $REG$ family to operations
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> A family of languages is said to be closed with respect to an operator $\theta$ if the result language belongs to the same family as the operand languages.
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@@ -92,9 +145,39 @@ Regular languages are also closed with respect to __mirroring__
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This means that combining two regular languages with one of the said operators, we obtain another regular language.
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-#####The family $REG$ of the regular languages is the smallest family of languages such that both the following properties hold:
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+##### The family $REG$ of the regular languages is the smallest family of languages such that both the following properties hold:
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- $REG$ contains all finite languages
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- $REG$ is closet w.r.t conatenation, union, and star.
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The $LIB$ family of the context-free languages is closed w.r.t. concatenation, union and star but is not the smallest of such family, in fact:
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$$REG \subset LIB$$
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+
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+### Linguistic Abstraction
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+Linguistic abstraction transforms the phrases of a real language to a simple form, called __abstract representation__
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+
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+Compiler design makers reference to the *abstract language* to process, not to the *effective language*
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+
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+Artificial languages (e.g. programming languages) contain few abstract structures, one of these is __lists__ which can be easily modeled by regexps
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+
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+## Effective and abstracts lists
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+
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+A __list__ is a sequence of *elements* with number not fixed in advance
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+An __element__ can be a __terminal__ (alphabetic symbol) or a __compound object__ (for instance, a list itself)
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+
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+##### example:
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+List with separators and start-marker of end-marker
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+$$ie(se)^*f\;\;\;i[e(se)^*]f$$
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+
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+### Substitution
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+Operation that replaces the terminal characters of the source language with the phrases of the destination languages
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+##### example
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+$$L\subset \Sigma^*\;\;\;L_b\subset\Delta^*$$
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+$$x\in L \;\;\;x=a_1a_2...a_n\;\;\text{and for some}\;\;\;a_i=b$$
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+Substituting the language $L_b$ to the letter b in the string x produces a language over the alphabet $(\Sigma\setminus \{b\})\cup \Delta$
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+$$\{y|y=y_1y_2...y_n\wedge (ifa_i\neq b\;\text{then}\;y_i=a_i\;\text{else}\;y_i\in L_b)\}$$
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+
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+### Nested lists (or precedence lists)
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+Lists may contain atomic objects as elements, but also other lists of lower level
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+##### example
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+>livello 1: begin istr1;istr2;...;istrn end
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+livello 2: STAMPA(var1,var2,...,varn)
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