Context-free Grammars

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Context-free Grammars
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• Recursive definition
• Grammar
• Derivation

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• What is the syntax of the description
  (meta-language)?
• External form of the grammar
• How do we associate a language with such a
  description?
• Semantics or use of the grammar
• Generators Grammars specify languages by
  generating strings in the language.
• Recognizers Grammars can be viewed as a notation
  for describing a family of recognition algorithms.

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CFG (V, ,P,S)

• V Finite set of variables/non-terminals
• Alphabet/Finite set of terminals
• P Finite set of rules/productions
• S Start symbol

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• a represents a context-free language because we
  can write a CFG for it.
• A-rule, null-rule,
  -rule.
• Context-freeness An A-rule can be applied
  whenever A occurs in a string, irrespective of
  the context (that is, non-terminals and terminals
  around A).
• Cf. context-sensitive grammar (declare-use)

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From CFG to Language

• One-step Derivation (Defn 3.1.2)
• w is derivable from v in CFG, if there is a
  finite sequence of rule applications such that

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• Let G(V, S, P, S) be a CFG.
• is a sentential form,
  if
• .
• is a sentence, if
  .
• The language of G,
• L(G)

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• Derivation of aabb
• Derivation tree

l
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Derivation Tree (abstracts derivation)
S
A
B
A
B
b
a
l
l
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Examples CFGs and CFLs
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Note that parentheses are part of the
meta-language.
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Left to right generation of string.
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Regular Grammar

• A regular grammar is a CFG in which each of the
  rules has the form
• (Right Linear Grammar)

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Equivalent Regular Grammar

• A regular grammar is a CFG in which each of the
  rules has the form
• (Left Linear Grammar)

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Derivations
Note that the above result does not hold for
sentential forms.