MC 306 Theory of Computation Thursday, 101603

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MC 306 Theory of ComputationThursday, 10/16/03

• Last time
• Pumping Lemma
• Todays Class
• More on Pumping Lemma, non-regular languages
• New Chapter!
• Context-free Languages

• Reading 3.1
• Exercises p. 110 1, 2, 5
• New hand-in problems pp. 110-111 3, 8

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Regular languages

• So far Regular languages
• Described by regular expressions
• Accepted by finite automata
• Applications
• Pattern matching in searches
• Lexical analysis of programming languages
• Limitations Lots of reasonable languages are not
  regular
• Example Cant test for balanced parentheses
  using regular expressions or FAs
• Example Cant test for primality using FAs

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Context-free languages

• Context-free languages
• Described by context-free grammars
• Accepted by pushdown automata
• Properties and Applications
• Developed originally for study of human languages
• Used for describing and checking syntax of
  programming languages in compiler design
• Contain regular languages as a subclass
• Still does not describe all possible languages

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Context-free grammars a simple example

• The expressions to the right form a context-free
  grammar well call G1
• There are two types of symbols
• Variables (nonterminals) A, B
• Terminals (constants) 0, 1, 2
• Each line is substitution rule (or production)
• One variable is designated as the starting symbol
  (A in this example)
• A grammar generates strings by repeatedly
  applying rules
• This is called a derivation of the string 0002111

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Parse-trees

• Parse-trees can be used to represent derivations
  pictorially
• Root of the tree is the start symbol
• Internal nodes are variables
• Leaves are terminals
• What strings does G1 generate?

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A more complex example

• Some sentences generated by this grammar
• A girl sees
• The boy sees a flower
• The girl with a flower likes the boy
• Derive the first sentence

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Definition of Context-free grammar

• A context-free grammar is a quadruple (?, NT, R,
  S), where
• ? is the alphabet (of terminals)
• NT is the set of nonterminals (or variables)
• R is the set of rules, a subset of
• S ? NT is the starting symbol
• Thus a rule is a pair (A, v) (usually written A
  ? v, where A is a single variable, and v is any
  string composed of variables and/or terminals
  (such a string is called a sentential form)

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Derivations and the language of a grammar

• We apply a rule A?v to a sentential form w that
  contains the symbol A, by substituting v for A,
  to get a new sentential form w
• If we get w by applying a rule to w, we say that
  w yields w in one step, and we write w ? w.
• If we get w by applying a sequence of rules,
  starting with w, we say that w yields w, and we
  write w? w. We also say that w is derivable
  from w.
• The language of the grammar G, denoted L(G), is
  the set of strings over ? that can be derived
  from the start symbol S

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More examples

• Consider the grammar
• G S ? AB, A ? aA ?, B ? bB ?
• Give the elements of its quadruple
• Derive the string aab
• What is L(G)?
• Design a grammar whose language is all
  palindromes over ? a, b
• Design a grammar whose language is all balanced
  strings of parentheses