8' Pushdown Automata

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8. Pushdown Automata Context-free Languages
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8.1 Pushdown Automata

• Theorem 7.5.1 says the language aibi i gt 0
  is not regular.
• Means it does not accepted by any finite
  automaton.
• To accept this language, a machine needs to
  record the processing of any finite number of
  as.
• So, PDA machine is introduced using stack
  operation

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Definition 8.1.1

• A PDA is sixtuple (Q, ?, ?, ?, q0, F),
• Q a finite states
• ? a finite set of input alphabet
• ? a finite set of stack alphabet
• q0 the start state
• F ? Q a set of final state, and
• ? a transition function from
• Q X (???) X (???) to
  subset Q X (???)
• PDA has 2 alphabets
• An input alphabet ? (input string)
• A stack alphabet ? (stored on the stack)

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• Example

R
L
A
a
a
b
b
Top
A
Input string, ?
B
B
Stack, ?
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• The computation of a PDA begins with machine in
• state q0,
• ?(qi, a,A) qj , B,qk, C
• means the transition function, ? lists all
  possible transitions for
• a given state, symbol, and stack top
  combination.
• ? Here, ? indicates 2 transitions are possible
  when the automaton is in the state qj , scanning
  an a with A on the top of the stack.

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• qj, B ? ?qi, a, A

Current stack top
New state
Current input symbol
New stack top
Current state
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• Causes the machine to
• Change the state from qi to qj
• Process the symbol, a (advance the tape head)
• Remove A from the top of the stack (pop the
  stack)
• Push B onto the stack

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State diagram of PDA
qi
qj
a A/B
?qi, a, A qj, B
A/B replacing A on the top by B
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? Transition

• A transition
• qj, B ? ?qi, a, ?
• will enter qj and add B to the top, whenever the
  machine is in state qi scanning an a.
• If the input position is ? , the transition doest
  not process any symbol.

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• qi, ? ? ?qi, ?, A
• ii) qi, A ? ?qi, ?, ?

qi
Pop symbol A without altering the state
? A/?
qi
qi
Push symbol A without altering the state
? ?/A
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• iii) qj, ? ? ?qi, a, ?
• That PDA is equivalent to a finite automaton
  transition
• Configuration of PDA
• qj, ?, ?

a ?/?
qi
qj
Machine state
The stack
Unprocessed input
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• qj, ?, ? ---- qj, ?, ?

M
Can be obtained from by a single transition of
PDA M
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The construction of PDA M to accept the language
ai bi , i? 0

• Start with input string , w and empty stack
• Process a, ? push A onto the stack
• Process b, ? pop the stack
• Match the number of as to bs
• M Q q0, q1
• ? a,b
• ? A
• F q0, q1

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Actions of M to process input string aabb

• q0, aabb, ?
• --- q0, abb, A
• --- q0, bb, AA
• --- q1, b, A
• --- q1, ?, ?
• ?(q0, a, ?) q0, A
• ?(q0, b, A) q1, ?
• ?(q1, b, A) q1, ?

a ?/A
b A/?
b A/?
q0
q1
Halt in an accepting with empty stack
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Definition 8.1.2

• Let M (Q, ?, ?, ?, q0, F be a PDA. A string
  is accepted by M if there is a computation
• q0, ?, ? --- qi, ?, ?
• where qi ? F, language L(M)
• Example 8.1.1
• The PDA M accept the language wcwR w ?
  a,b

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• M Q q0, q1 ?(q0, a,
  ?) q0, A
• ? a,b,c
  ?(q0, b, ?) q0, B
• ? A,B ?(q0,
  c, ?) q1, ?
• F q1
  ?(q1, a, A) q1, ?
• 
  ?(q1, b, B) q1, ?

b ?/B
a A/?
a ?/A
b B/?
c ? /?
q0
q1
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• The computation of M with input abcba is given
  below
• q0, abcba, ?
• --- q0, bcba, A
• --- q0, cba, BA
• --- q1, ba, BA
• --- q1, a, A
• --- q1, ?, ?
• A PDA is deterministic if there is at most one
  transition that is applicable for each
  combination of state, input symbol, and stack
  top.

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Example 8.1.2

• Suppose we want to construct a NDPA that accept
  the language
• L w ? a,b w contains equally may as and
  bs
• We must count the occurrences of as and bs.
  This can be done by pushing symbols on the stack
  when one input character is seen and removing
  them when the other is seen.

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Pushdown Automata and Contact-Free Language

• Every cfg is accepted by an extended PDA
• Let L is a contact-free language, and G is a
  garmmar in Greibach Normal Form (GNF)
• ? L(G) L
• The rules of G, accepts for
• S ? ?, have the form
• A ? aA1 A2 A3 An

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Greibach Normal Form

• A contact-free grammar G (V, ,P,S) is in in GNF
  if each rule has one of the following forms
• A ? aA1 A2 A3 An
• A ? a
• S ? ?
• Where a ? ?, and Ai ? V S for I1,2,3,,n

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• The GNF, G that accepts ai bi i gt 0 is used
  to illustrate the construction of an equivalent
  PDA
• G S ? aAB aB
• A ? aAB aB
• B ? b
• PDA has two states
• q0 start state
• q1 accepting state

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• S ? aA1 A2 A3 An
• Generates a transition that processes the
  terminal symbol a, pushes the variables aA1 A2 A3
  An onto the stack, and enter state q1
• - ? given as follows
• ?(q0, a, ?) q1, AB, (q1, B)
• ?(q1, a, A) q1, AB, (q1, B)
• ?(q1, b, B) q1, ?

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Computation of string aaabbb

• GNF
• S ? aAB
• ? aaABB
• ? aaaBBB
• ? aaabBB
• ? aaabbB
• ? aaabbb
• PDA
• q0, aaabbb, ?
• --- q1, aabbb, AB
• --- q1, abbb, ABB
• --- q1, bbb, BBB
• --- q1, bb, BB
• --- q1, b, B
• --- q1, ?, ?

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