Languages and Finite Automata

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CSC312 Automata Theory Lecture 26 Chapter
12 by Cohen Context Free Grammars
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• Compiler
• A compiler is program that converts a high level
  language code into its equivalent assembly
  language. Grammar
• Grammar is a set of rules by which a valid
  sentence in a language is constructed.
• Parsing the sentence
• Parsing is the process of analyzing a text, made
  of a sequence of tokens (e.g. words), to
  determine its grammatical structure with respect
  to a given formal grammar.

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• Semantics
• The grammatical rules which involve the meaning
  of words are called Semantics e.g. in English
  language, the sentence Buildings sing make no
  sense.
• Syntactics
• The grammatical rules that dont involve the
  meaning of the words but the structure of the
  words.
• Context Free Grammar (CFG) general definition
• A grammar or language based on rules that
  describe a change in the string without reference
  to elements not in the string.
• The concept of CFG was introduce by the linguist
  Noam Chomsky in 1956.

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CFG Terminology

• Terminals
• The symbols that cannot be replaced by anything
  are called terminals.
• Non-Terminals
• The symbols that must be replaced by other things
  are called non-terminals. e.g.
• variable expr
• Derivation
• The sequence of application of the rules that
  produces the finished string of terminal from the
  starting symbol is called a derivation.
• Productions
• The grammatical rules are often called
  productions.

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• Context Free Grammar (CFG) technical definition
• A CFG is a collection of three things
• An alphabet ? of letters called terminal, from
  which strings or words of the language are
  formed.
• A set of symbols called non-terminals, one of
  which is the symbol S, standing for start here.
• A finite set of productions of the form
• One non-terminal ? finite string of terminals and
  /or non-terminals

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• Note
• The terminal are designated by small letters,
  while the non-terminals are designated by cpital
  letters.
• There is at least one production that has the
  non-terminal S as its left side.
• Example
• S ? aA bX
• A ? bA
• X ? cX
• The sign ? means can be replaced by and
• the sign ? means develop into

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• Context Free Language (CFL)
• The language generated by CFG is called context
  Free Language (CFL).
• Note CFG can generate all regular languages and
  some non-regular languages, but not all the
  non-regular languages.
• Examples

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• Ambiguous CFG
• If a CFG can generate a specific string in more
  than one different ways (i.e. more than one
  leftmost derivations or parse trees) then it is
  called ambiguous CFG.
• Note
• A CFL is inherently ambiguous if all the CFGs
  generating the language are ambiguous.
• Some programming language have ambiguougs
  grammars e.g. C-language.
• Examples of CFG

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• Note
• We can generate CFG for each and every regular
  language. However, there are some nonregular
  languages for which we can construct CFG and for
  some nonregualr language we cannot construct CFG.
• It may be noted that the productions
• S ? SS?
• always define the language which is closed w.r.t
  concatenation i.e. the language expressed by RE
  of type r

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• Note
• iii) Null string ? can not be terminal, because
  it is not part of alphabet. It terminates
  non-terminals.
• iv) The string where non-terminal remain on right
  side is called working string.
• v) If a word has ? at the end, then we write
  after removing it, otherwise we place ? sign.
• Examples