3. CONTEXTFREE GRAMMARS

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3. CONTEXT-FREE GRAMMARS
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• 3.1 Introduction
• An approach generating the string of the
  language.
• A sentence of the language a syntactically
  correct string.
• Example
• The alphabet of our language is the set. a,
  the, Ahmad, Ali, burger, car, drivers, eats,
  slowly, frequently, big, brown
• The element of the alphabet are called the
  terminal symbols of the language.
• 2 rules of generating a sentence, e.g. var
  ltsentencegt
• ltsentencegt ? ltnoun phrasegtltverb-phrasegt
• ltsentencegt ? ltnoun phrasegtltverb-phrasegtltdirect-obj
  ect-phrasegt
• ltnoun phrasegt ? ltproper noungt
• ltnoun phrasegt ? ltdeterminergtlt common-noungt
• ltproper noungt ? Ahmad
• ltproper noungt ? Ali
• lt common-noungt ? car
• lt common-noungt ? burger

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• ltdeterminergt ? a
• ltdeterminergt ? the
• ltverb-phrasegt ? ltverbgtltadverbgt
• ltverb-phrasegt ? ltverbgt
• ltverbgt ? drivers
• ltverbgt ? eats
• ltadverbgt ? slowly
• ltadverbgt ? frequently

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• The sentence Ahmad drives frequently is
  generated by the following transformations
• Derivation Rule Applied
• ltsentencegt ? ltnoun phrasegt ltverb-phrasegt 1
• ? ltproper-noungt ltverb-phrasegt 3
• ? Ahmad ltverb-phrasegt 6
• ? Ahmad ltverbgtltadverbgt 11
• ? Ahmad drives ltadverbgt 13
• ? Ahmad drives frequently 16
• To complete the set of rules, the transformation
  for ltdirect-object-phrasegt
• must be given.
• e.g. To include any number of adjectives,
  including, preceding the direct object as
  shown in the following strings.

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• Ali eats burger
• Ali eats a big burger
• Ali eats a big juicy burger
• Ali eats a big brown juicy burger
• Ali eats a big big brown juicy burger
• Rules of grammar must be cup able of generating
  strings of arbitrary length.The use of recursive
  definition.
• ltadjective-listgt ? ltadjectivegt ltadjective-listgt
• ltadjective-listgt ? ?
• ltadjectivegt ? big
• ltadjectivegt ? juicy
• ltadjectivegt ? brown
• ltdirect-object-phrasegt ?ltadjective-listgtltcommongt
• ltdirect-object-phrasegt ?ltdeterminergtltadjective-lis
  tgtltcommongt

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• Exercise
• Write the sequence of rule applications for this
  sentence
• Ali eats a big juicy burger.
• modification
• 3 ltnoun phrasegt ? ltadjective-listgt
  ltproper-noungt
• 4 ltnoun phrasegt ? ltdeterminergt
  ltadjective-listgtltcommon noungt
• With this modification, the string big Ali eats
  frequently can be derived from the variable
  ltsentencegt.

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• 3.1 Context-Free Grammars and Languages
• Formal system of generating the strings of a
  language.
• Definition 3.1.1
• A context-free grammar is a quadruple ( V,S,P,S)
• where - V is a finite set of variables
• - S (the alphabet) is a finite set of
  terminal symbols.
• - P is a finite set of rules.
• - S is a distinguished element of V called
  the start symbol.
• The set V and S are assumed to be disjoint.
• A rule, called production is an element of the
  set VX (V ? S).
• The rule A,w is usually written A ? w
• A rule, A ? x, is called a null or lambda rule.

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• A ? w
• produces the string uwr
• Sentence uAr
• uAr ? uwr u prefix ,
• v suffix define the context in
  which variable A
  occurs.
• free ? general applicability of the rules.
• A string w is derivable from V if there is a
  finite seq. of rule applications that transform v
  to w. i. e. If a seq. of
• i.e v ?wi ?w2 ?. wn w
• transformation can be constructed from the rules
  of the grammar.
• v ?w

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• Definition 3.1.2
• Let G (V,S,P,S) be a context-free grammar
• and V ? ( V ? S ).
• The set of string derivable from v is defined
  recursively as follows
• Basis V is derivable from v.
• Recursive step If u xAy is derivable from v
  and A ?w ?P,
• then xwy is derivable from v.
• Closure Precisely those strings constructed
  from v by finitely many applications of
  the recursive step are derivable from v.
• Some rotations
• ? definition of rule
• ? application ( transforms one string to
  another)
• ? derivability utilizing one or more rule
  application.
• n? derivation length from one to other
• G? derivation utilizes the rule of grammar G

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• Definition 3.1.3
• Let G (V,S,P,S) be a context-free grammar.
• A string w ? (V? S) is a sentential form of G if
  there is s derivatives S?w in G.
• A string w ? S is a sentence of G if there is a
  derivation S ?w in G
• The language of G, demoted L (G), is the set w
  ? S S ?w
• Sentential forms-string derivable from the start
  symbols of the grammar.
• The sentences are the sentential form that
  contain only terminal symbols.
• The language of grammar is the set of sentences
  generated by the grammar.
• A set of strings over an alphabet S is called a
  context-free language if there is a context-free
  grammar that generate it.

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• Rule A ? u Av directly recursive
• This rule can generate any member of copies of
  the string u followed by an A and an equal number
  of Vs.
• A variable A is called recursive if there is
  derivation
• A ? u Av
• A w ? u Av , A is not in w ? indirectly
  recursive.
• Example Derivation of ababaa in G A grammar
  G that generates the language consisting of
  strings with a positive, even number of as.
• G ( V,S,P,S )
• V S,A
• S a,b
• P S ? AA
• A ? AAA bAAba

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• S ? AA S ? AA S ? AA
• ? aA ? AAA ? Aa
• ? aAAA ? aAAA ? AAAa
• ? abAAA ? abAAA ? AAbAa
• ? abaAA ? abaAA ? AAbaa
• ? ababAA ? ababAA ? AbAbaa
• ? ababaA ? ababaA ? Ababaa
• ? ababaa ? ababaa ? ababaa
• (a) (b) (c)
• left most left most right most
• S ? AA
• ? aA
• ? aAAA
• ? aAAa
• ? abAAa (d) left most right most.
• ? abAbAa
• ? ababAa
• ? ababaa

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• WEEK ? 3
• Definition 3.1.4
• Let G ( V,?,P,S ) be a context-free grammar and
  S G? w a derivation.
• The derivation tree, DT, of S G? w is an
  ordered tree that can be built interactively as
  follows
• Initialize DT with roots.
• If A ? x1x2 xn with xi (V ??) is the rule in
  the derivation applied to the string uAv, then
  add x1,x2, xn as the children of A in the tree.
• If A ?? is the rule in the derivation applied to
  the string uAv, then add ? as the only child of A
  in the tree.

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• Example 3.2.1
• Let G be the grammar given by the productions.
• S ? aSaaBa
• B?bBb
• Then L (G) anbman ngt0, mgt0
• Explanation
• The rule S?aSa recursively builds an equal
  number of as on each end of the string.
• The recursion is terminated by the application of
  the rule S?aBa, ensuring at least one leading and
  one trailing a.
• The recursive B rule then generates any number of
  bs. To remove the variable B from the string and
  obtain a sentence of the language, the rule B ?b
  must be applied, forcing the presence of at least
  one b.

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• Exercise
• L (G) anbmcmd2n n0, mgt0
• is given by the production
• S ? aSdd A
• A ? bAc bc
• give ? valid string
• example
• A string w is a palindrome if w wR , ?
  a,b
• write the rules of s to generate such string.
• S ? a b ?
• Consider the grammar
• S ? abscB ?
• B ? bB b,
• Explain what kind of string generated by that
  rules.

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• Example 3.2.6
• Let G1 and G2 be the grammars
• G1 S ?AB G2 S ?aS aB
• A ?aA a B
  ? bB ?
• B ? bB ?
• Both grammars generated the language
• a b
• Note a aa , non null string of as
• The A rules on G1 provide the standard method of
  generately a non null
• string of as use of ? rule to terminate the
  recursion allow the posibility of having no bs.
• Grammar G2 builds the same language in a
  left-to-right manner.

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• Exercise
• Write grammar G1 and G2 generate the strings
  over a,b that contain exactly two bs
• ? a baba8
• G1 S ? AbAbA G2 S ? as bA
• A ? aA ? A ? aA bc
• C ? aC ?
• (2) Modify the above grammar to generate with
  at least two bs.
• G1 S ? AbAbA G2 S ? as bA
• A ? aA bA ? A ? aA bc
• C ? aC bc ?

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A Context-Free Grammar For Pascal

• John Backus (1959) and Peter Naur (1963) used a
  system of rules to define the programming
  language ALGOL 60,
• Known as BNF (Backus-Naur form)
• BNF definition of Pascal
• ltprogramgt ? ltprogram headinggt ltprogram blockgt
• ltprogram blockgt ? ltblockgt
• ltprogram headinggt ? program ltidentifiergtltfile
  identifiergt , ltfile
  identifiergt)
• ltfile identifiergt ? ltidentifiergt
• ltidentifiergt ? ltlettergt ltletter or digitgt
• ltletter or digitgt ? ltlettergt ltdigitgt
• ltlettergt ? abcz
• ltdigitgt ? 012..9

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Example 3.6.1 The constant 5 is generated by the
derivation

• ltexpressiongt ? ltsimple expressiongt
• ? lttermgt
• ? ltfactorgt
• ? ltunsigned constantgt
• ? ltunsigned numbergt
• ? ltunsigned integergt
• ? ltdigitgt
• ? 5

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Example 3.6.2 The expression x 5

• ltexpressiongt ? ltsimple expressiongt
• ? ltsimple
  expressiongtltadding operatorgtlttermgt
• ? lttermgtltadding
  operatorgtlttermgt
• ? ltfactorgt gtltadding
  operatorgtlttermgt
• ? ltvariablegt gtltadding
  operatorgtlttermgt
• ? ltentire variablegt
  gtltadding operatorgtlttermgt
• ? ltvariable identifiergt
  gtltadding operatorgtlttermgt
• ? ltidentifiergt gtltadding operatorgtlttermgt
• ? ltlettergtltidentifier-tailgt
  gtltadding operatorgtlttermgt
• ? x ltidentifier-tailgtltaddin
  g operatorgtlttermgt
• ? x ltadding
  operatorgtlttermgt
• ? x lttermgt
• ? x 5