Recap: Nondeterministic Finite Automaton NFA

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Recap Nondeterministic Finite Automaton (NFA)

• A deterministic finite automaton (NFA) is a
  5-tuple (Q,?,?,s,F) where
• Q is a finite set of elements called states
• ? is a finite input alphabet
• s ? Q called the start state
• F ? Q called the favorable states

• ? (Q (? ? e)) ? ?(Q)

Constant!

a1
a2
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Acceptance of a Word in an NFA A

• Acceptance w is accepted by A if at least one
  sequence of transformations yields the empty word
  and end up in an accepting state

• The oracle explanation of NFAs there is an
  oracle that always makes the right choice

Oracle will find a path to an accepting state
that process all characters only if there is one
such path
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Pushdown Automaton

• A pushdown automaton is a 6-tuple (Q,?,?, ?,s,F)
  where
• Q is a finite set of elements called states
• ? is a finite input alphabet
• ? is a set of stack symbols
• s ? Q called the start state
• F ? Q called the favorable states

• ? (Q (? ? e) (? ? e) ) ? ?( Q (? ?
  e) )

Constant!

a1
a2
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Computation in Pushdown Automata
PA (Q,?,?, ?,s,F) where

• ?(Q (? ? e) (? ? e) ) ? ?( Q (? ?
  e) )

Let (q,?)) ? ?((q, ?,?)), it is compute as
follows

• if
• the automaton is in an state q
• the next character is ?
• The word ? is on top of the stack

• Then
• pop ? from top of the stack
• Jump to the state q
• Push the word ? on top of the stack

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String Accepted by Pushdown Automaton

• Given a pushdown automaton A, and a string w ?
  ?, we say that w is accepted by A if there is a
  sequence of computations that
• begin from the starting state, the word w, and
  the empty stack
• terminates in a favorable state and process all
  characters in w
• Note the stack does not have to be empty at the
  end
• The set of all words accepted by an automaton A
  form the language L(A) accepted or recognized by
  the automaton.

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Example
Pushdown automaton recognizing anbn n 0, 1,
2,
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Example
Pushdown automaton recognizing
L wwR w is in ?
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Homework for Friday 12.

• 2.3 (n), (o)
• 2.4 (c)
• 2.6 (b)
• 2.21
• Construct a pushdown automaton for the language
  of exercise 2.21