Grammar

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Grammar
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Application of a production ?A????? in a
derivation step ?i ? ?i1
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Formal grammars (1/3)

• Example Let G1 have N A, B, C, T a, b,
  c and the set of productions
• ? ? A CB ? BC
• A ? aABC bB ? bb
• A ? abC bC ? bc
• cC ? cc
• The reader should convince himself that the word
  akbkck is in L(G1) for all k ? 1 and that only
  these words are in L(G1). That is,
• L(G1) akbkck k ? 1.

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Formal grammars (2/3)

• Example Grammar G2 is a modification of G1
• G2 ? ? A CB ? BC
• A ? aABC bB ? bb
• A ? abC bC ? b
• The reader may verify that L(G2) akbk k ?
  1. Note that the last rule, bC ? b, erases all
  the C's from the derivation, and that only this
  production removes the nonterminal C from
  sentential forms.

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Formal grammars (3/3)

• Example A simpler grammar that generates akbk
  k ? 1 is the grammar G3
• G3 ? ? S
• S ? aSb
• S ? ab
• A derivation of a3b3 is
• ? ? S ? aSb ? aaSbb ? aaabbb
• The reader may verify that L(G3) akbk k ?
  1.

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Type Format of Productions Remarks
0 fA?? f? ? Unrestricted Substitution Rules
1 fA?? f? ?, ??? ??? Context Sensitive Context Free Right Linear Left Linear
2 A ??, ??? ??? Context Sensitive Context Free Right Linear Left Linear
3 A?aB A?a ??? A?Ba A ?a ??? Context Sensitive Context Free Right Linear Left Linear
Contracting
Noncon- tracting
Regular
The four types of formal grammars
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Context-Sensitive Grammars(Type1)
Unrestricted Grammars(Type0)

• Definition A context-sensitive grammar G
  (N,T,P,?) is a formal grammar in which all
  productions are of the form
• fA??f??, ?? ?
• The grammar may also contain the production ?
  ??, if G is a context-sensitive (type1) grammar,
  then L(G) is a context-sensitive (type1) language.

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Context-Free Grammars (Type2)

• Definition A context-free grammar G(N,T,P,?)
  is a formal grammar in which all productions are
  of the form
• A??
• The grammar may also contain the production ?
  ??. If G is a context-free (type2) grammar, then
  L(G) is a context-free (type2) language.

A?N?? ??(N?T)-?
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Regular Grammars (Type3) (1/2)

• Definition A production of the form
• A?aB or A?a
• is called a right linear production. A
  production of the form
• A?Ba or A?a
• is a left linear production. A formal grammar is
  right linear if it contains only right linear
  productions, and is left linear if it contains
  only left linear production ? ??. Left and right
  linear grammars are also known as regular
  grammars. If G is a regular (type3) grammar, then
  L(G) is a regular (type3) language.

A?N?? B?N a?T
A?N?? B?N a?T
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Regular Grammars (Type3) (2/2)

• Example A left linear grammar G1 and a right
  linear grammar G2 have productions as follows
• G1 G2
• The reader may verify that
• L(G1) (10)11(01)L(G2)

? ? 1B ? ? 1 A ? 1B B ? 0A A ? 1
? ? B1 ? ? 1 A ? B1 B ? A0 A ? 1
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Ambiguity (1/2)

• Example Consider the context-free grammar
• G ? ? S
• S ? SS
• S ? ab
• We see that the derivations correspond to
  different tree diagrams. The grammar G is
  ambiguous with respect to the sentence ababab if
  the tree diagrams were used as the basis for
  assigning meaning to the derived string, mistaken
  interpretation could result.

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Ambiguity (2/2)

• Definition A context-free grammar is ambiguous
  if and only if it generates some sentence by two
  or more distinct leftmost derivations.