Pushdown Automaton (PDA)

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Pushdown Automaton (PDA)

• A Pushdown Automaton is a nondeterministic finite
  state automaton (NFA) that permits
  e-transitions and a stack.

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Pushdown Automaton (PDA)
Q A finite set of states. S A finite set of
input symbols. G A finite stack alphabet. d The
transition function with input qi is a state in
Q. a is a symbol in S or a e (the empty
string). tm is a stack symbol, tm Î G. and
the output is a finite set of pairs qk the new
state. tn is the string of stack symbols that
replaces tm at the top of the stack. If tn
e, then the stack is popped. q0 The start
state. t0 Initially, the PDAs stack consists
this symbol and nothing else. F The set of
accepting states.
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PDA Example

• The language, Lwwr, is the even-length
  palindromes over alphabet 0,1.
• Lwwr is a Context-Free Language (CFL) generated
  by the grammar

One PDA for Lwwr is given on the following
slide...
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PDA for Lwwr
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A Graphical Notation for PDAs

1. The nodes correspond to the states of the PDA.
2. An arrow labeled Start indicates the unique start
   state.
3. Doubly circled states are accepting states.
4. Edges correspond to transitions in the PDA as
   follows
5. An edge labeled (ai, tm)/tn from state q to state
   p means that d(q, ai, tm) contains the pair (p,
   tn), perhaps among other pairs.

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Graphical Notation for PDA of Lwwr
(0, 0)/00 (0, 1)/01 (1, 0)/10 (1, 1)/11
(0,0)/e (1,1)/e
(0, t0)/0t0 (1, t0)/1t0
start
(e,t0) / t0
(e, t0) / t0 (e, 0) / 0 (e, 1) / 1
(EOF,t0) / t0
All possibilities that do not have explicit
edges, have implicit edges that go to an implicit
reject state.

• This is a nondeterministic machine.
• Think of the machine as following all possible
  paths.
• Kill a path if it leads to a reject state.
• If any path leads to an accept state, then the
  machine accepts.

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Exercise 1

• Design a PDA that recognizes legal sequences of
  if and else statements in a C program.
• In the PDA, let i stands for if and e
  stands for else.
• Hint There is a problem whenever the number of
  else statements in any prefix exceeds the
  number of if statements in that prefix.

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Exercise 2

• Design a PDA to accept the language