CSCI 2670 Introduction to Theory of Computing

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CSCI 2670Introduction to Theory of Computing
September 21, 2005
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Agenda

• Yesterday
• Pushdown automata
• Today
• Quiz
• More on pushdown automata
• Pumping lemma for CFGs

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Finite automata and PDA schematics
State control
FA
State control
PDA
Stack Infinite LIFO (last in first out) device
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Definition of pushdown automaton

• A pushdown automaton is a 6-tuple (Q,?,?,?,q0,F),
  where Q,?,?, and F are all finite sets, and
• Q is the set of states
• ? is the input alphabet
• ? is the stack alphabet
• ? Q ? ?e ? ?e ? P(Q ? ?e) is the transition
  function
• q0 ? Q is the start state, and
• F ? Q are the accept states.

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Strings accepted by a PDA

• Let w be a string in ? and P a PDA. w is in
  L(P) iff w can be written ww1w2wn, where each
  wi??e, and there exist r0,r1,,rn?Q and
  s0,s1,,sn?? satisfying the following
• r0q0 and s0e
• M starts in the start state with an empty stack
• (ri1,b)??(ri,wi1,a), where siat and si1bt
  for some a,b??e and t??
• M moves according to transition rules for the
  state, input and stack
• rm?F
• accept state occurs at input end

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A closer look at the transition rule

• (ri1,b)??(ri,wi1,a), where siat and si1bt
  for some a,b??e and t??
• The top symbol is
• Pushed if ae and b?e
• Popped if a?e and be
• Changed if a?e and b?e
• Unchanged if ae and be
• Symbols below the top of the stack may be
  considered, but not changed
• This is ts role

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Example

• Find ? for the PDA that accepts all strings in
  0,1 with the same number of 0s and 1s
• Need to keep track of equilibrium point with a
  on the stack
• If stack top is not , it contains the symbol
  currently dominating in the string

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Example

• Find ? for the PDA that accepts all strings in
  0,1 with the same number of 0s and 1s
• Push a symbol on the stack as its read if
• It matches the top of the stack, or
• The top of stack is
• Pop the symbol off the top of the stack if you
  read a 0 and the top of stack is 1 or vice versa

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Example
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Example
This PDA is equivalent to the one on the previous
slide
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Example

• Nested parentheses

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Equivalence of PDAs and CFGs

• Theorem A language is context free if and only
  if some pushdown automaton recognizes it
• Proved in two lemmas one for the if direction
  and one for the only if direction

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CFGs are recognized by PDAs

• Lemma If a language is context free, then some
  pushdown automaton recognizes it
• Proof idea Construct a PDA following CFG rules

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Constructing the PDA

• You can read any symbol in ? when that symbol is
  at the top of the stack
• Transitions of the form a,a?e
• The rules will be pushed onto the stack when a
  variable A is on top of the stack and there is a
  rule A?w, you pop A and push w
• You can go to the accept state only if the stack
  is empty