COSC 3340: Introduction to Theory of Computation

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COSC 3340 Introduction to Theory of Computation

• University of Houston
• Dr. Verma
• Lecture 11

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Push Down Automaton (PDA)

• Language Acceptor Model for CFLs
• It is an NFA with a stack.

Finite State control
Input
Accept/Reject
Stack
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PDA (contd.)

• In one move the PDA can
• change state,
• consume a symbol from the input tape or ignore it
• pop a symbol from the stack or ignore it
• push a symbol onto the stack or not
• A string is accepted provided the machine when
  started in the start state consumes the string
  and reaches a final state.

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PDA (contd.)

• If PDA in state q can consume u, pop x from
  stack, change state to p, and push w on stack we
  show it as

u, x ? w
q0
q1
u, x w In JFLAP
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Example of a PDA

• PDA L anbn n ? 0

Push S to the stack in the beginning and then
pop it at the end before accepting.
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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JFLAP Simulation
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Definition of PDA

• Formally, a PDA M (K, ?, ?, ?, s, F), where
• K -- finite set of states
• ? -- is the input alphabet
• ? -- is the tape alphabet
• s ? K -- is the start state
• F ? K -- is the set of final states
• ? ? (K X ?? X ??) X (K X ??)

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Definition of L(M)

• Define ? as
• (1) ?(q, ?, ?) (q, ?, ?) ? (p, ?, ?) ((q,
  ?, ?), (p, ?)) ? ?
• (2) ?(q, uv, xy) U ?(p, v, wy) ((q, u, x),
  (p, w)) ? ?
• i.e., first compute ? for all successor
  configurations and then take the union of all
  those sets
• M accepts w if (f, ?, x) in ?(s, w, ?)
• Alternative if (f, ?, ? ) in ?(s, w, ?) we
  use
• L(M) w ? ? M accepts w

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Example

• What is L(M)?

Push S to the stack in the beginning and then
pop it at the end before accepting.