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@@ -10,8 +10,9 @@ Are composed by:
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- A __set of rules__ which can be applied a finite number of times
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A language can be defined by rules that after multiple applications allow to generates all and only the phrases in the language
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+### Context-free grammars
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-##### example:
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+##### example 1:
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__Language__
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$$L=\{uu^R|u\in \{a,b\}^*\}=\{\varepsilon,aa,bb,abba,baab,...,abbbba,...\}$$
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__Grammar rules__
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@@ -23,11 +24,87 @@ __Derivation chain__
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$$\text{frase}\Rightarrow a\;\text{frase}\;a$$
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Lista, frase are __non-terminal__ symbols
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##### example 2:
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-The metalanguage of regular expressions
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+__The metalanguage of regular expressions__
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+
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+__alphabet__ $\Sigma_{r.e.}=\{a,b,\cup,*,\emptyset,(,)\}$ <!--*-->
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__syntax__
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$1.\;espr \rightarrow\emptyset$
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+$2.\;espr \rightarrow a$
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+$3.\;espr \rightarrow b$
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+$4.\;espr \rightarrow (espr\cup espr)$
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+$5.\;espr \rightarrow (espr\;espr)$
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+$6.\;espr \rightarrow (espr)^*$
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+<!--*-->
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+### Backus Normal Form - BNF
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+
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+Is defined using four entities
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+- $V$ non-terminal alphabet
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+- $\Sigma$ terminal alphabet
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+- $P$ syntactic rules
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+- $S\in V$ is a particular non-terminal called *axiom*
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+
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+Each rule $P$ is a pair $(X,\alpha)$ of:
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+- $X\in V$ element of the non-terminal alphabet
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+- $\alpha\in(V\cup \Sigma)^*$ element of terminal or non-terminal alphabet
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+
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+Some examples are:
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+- $X\rightarrow\alpha$
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+- $X\rightarrow\alpha_1|\alpha_2|...|\alpha_n$
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+- $X\rightarrow\alpha_1\cup\alpha_2\cup...\cup\alpha_n$
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+<!--*-->
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+
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+### Reduction of the grammars
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+A grammar is in __reduced form__ or simply __reduced__ if both the following conditions hold:
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+- Every non-terminal A is reachable from the axiom $S\overset{*}{\Rightarrow}\alpha A\beta$
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+- Every non-terminal A generates a non-empty set of strings $L_A(G)\neq\emptyset$
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+<!--*-->
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+There exists an algorithm in two phases:
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+- The first phase identifies the non-terminal symbols that are undefined
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+- The second phase identifies those that ar unreachable
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+
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+__Phase 1__
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+
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+- Construct the complement set $DEF$ of the defined non-term symbols.
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+$$DEF = V\setminus UNDEF$$
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+- Initially $DEF$ is set to contain the non-terminal symbols that are expanded by terminal rules
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+$$DEF:=\{A|(A\rightarrow u)\in P, \text{with}\;u\in\Sigma^*\}$$ <!--*-->
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+- Then the following transformation is applied repeatedly, until it converges
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+$$DEF:=DEF\cup\{B|(B\rightarrow D_1D_2...D_n)\in P\}$$
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+Each symbol $D_i$ is either a terminal or belongs to $DEF$
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+
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+In every iteration of the first phase, two events are possible:
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+- New non-terminal symbols are identified and added to the set $DEF$
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+- the $DEF$ set does not change, which means that convergence has been reached, therefore the algorithm terminates.
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+
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+The non-terminal symbols that belong to the $UNDEF$ set can be eliminated along with the rules where they appear.
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+
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+__Phase 2__
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+
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+The identification of the non-terminal symbols that are reachable starting from the axiom $S$ can be reduced to the problem of the existence of a path in the graph associated with the following binary relation produce
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+
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+$A\overset{produce}{\rightarrow}B \;\;\text{if and only if} \;\;A\rightarrow\alpha B\beta \;\;\text{with}\;\;A\neq B\;\;\;\alpha,\beta\;\text{are any strings}$
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+
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+The non-terminal $C$ is reachable from the axiom $S$ if and only if
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+the graph of the binary relation *produce* contains a path from $S$ to $C$
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+
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+A __third property__ is often required for a grammar to be in the reduced form:
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+
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+The grammar G may not have __circular derivations__, which are inessential and moreover give raise to ambiguity
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+
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+#### example
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+
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+$$\{S\rightarrow a,\;A\rightarrow b\}\;\text{reduced to}\;\{S\rightarrow a\}$$
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+
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+### Recursive grammars
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+
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+To generate an __infinite language__, composed of an infinite number of strings,
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+it is necessary to allow derivations of *unbounded length*, this is called __recursive grammar__
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+
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+$$A\overset{n}{\Rightarrow}xAy\;\;n\ge 1\;\;\text{is recursive}$$
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-#### Context-free Grammars
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-Is a generalization of regexp
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+__Necessary and sufficient__ condition for a language $L(G)$ to be __infinite__:
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+>where $G$ is a grammar in the reduced form and does not have any circular derivations,
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+is that $G$ admits recursive derivations
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-###Reduction of the grammars
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+A grammar __does not contain__ recursion (or is recursion-free) if and only if
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+the __graph__ of the binary relation *produce* is __acyclic__
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