COSC 341 Lecture 11 Pushdown Automata

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COSC 341 Lecture 11 Pushdown Automata

• Context-free grammars A ? bBcC
• Regular languages are context-free, but some
  context-free languages are not regular
• Palindrome has the context-free grammar
• S ? aSa ? bSb ? a ? b ? ?.
• anbn n 0 has the CFG
• S ? aSb ? ?.
• We can design machines to accept the context-free
  languages.
• But there are languages that are not context-free.

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Pushdown automata (PDA)

• PDA NFA stack for memory
• PDAs may be non-deterministic - those that are
  deterministic accept only some of the
  context-free languages.
• Need finite set Q of states, input alphabet ?, qo
  in Q, F ? Q, a stack alphabet ?, and a
  transition function ? Q?(???)?(???)
  ??(Q?(???)).
• What does it mean if
• qn, B ? ?(qm, a, A)?
• It means that if the state is qm and the reading
  head sees input letter a while A is at the top of
  the stack, then the machine should
• change the state from qm to qn
• process input letter a
• pop A off the stack
• push B onto the stack.

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The role of ?

• What does qk, ? ? ?(qm, a, A) mean? Change
  from qm to qk, process a, pop A, and dont push
  anything onto the stack.
• What does qk, B ? ?(qm, a, ?) mean? Change
  from qm to qk, process a, dont pop anything from
  the stack, and push B on.
• What does qk, B ? ?(qm, ?, A) mean? Change
  from qm to qk, dont process anything from input
  string (and so dont move reading head to next
  square on right), pop A off and push B onto
  stack.
• What does qk, ? ? ?(qm, ?, ?) mean? Just
  change from qm to qk....!

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Deterministic PDAs

• The string w is accepted if there is a
  computation from q0, w, ? that ends in qi, ?,
  ? with qi ? F.
• L anbn n ? o is accepted by PDA

Note this is a deterministic PDA (though not
total).

• Palindrome the set of strings over a, b,
  of the form wz where z is w spelled
  backward and w? a, b, e.g. abba,
  aabaabaa, ...

(Note no choices, so deterministic PDA.)
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Non-deterministic PDAs

• The set of strings over a, b of even length
  and spelled the same forwards and backwards.

(We need a non-deterministic transition from q0
to q1 because unlike Palindrome strings have no
middle marker telling machine when to change
state.)

• an n0 ? anbn n 0

b A/ ?
a ?/A
b A/ ?
? ?/?
? A/?
q2
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The pumping lemma for PDA

• Suppose L is context-free. There is a number k
  (which depends on L) such that if z ? L with
  length(z) k, then z can be written uvwxy, with
• length (vwx) k
• length(v) length(x) gt 0
• uvnwxny ? L for all n 0.
• Why? Let G be a grammar for L and let m be the
  maximum number of symbols on the right of a rule.
• A node in a parse tree has m children, so a
  parse tree of depth d produces a string of length
  md.
• Pick any string of length gt mj where G has j
  non-terminals. A parse tree for that string has
  depth gt j so some nonterminal A is used twice
  along the same path. Suppose the subtree
  generated by the first occurrence of A produces
  vAx. We may use A again to produce vvAxx from
  vAx! This pumps v and x, which are not both empty
  (if we assume the grammar to be in Chomsky Normal
  Form).

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Application
L anbncn ? n 0 is not context-free. Suppos
e L is context-free let k be the number in the
pumping lemma. Let z akbkck. We show that, no
matter what choices are made for u, v, w, x, and
y, the string uv2wx2y ? L. Fact every string in
L has form anbncn and so has only one substring
ab only one substring bc no substrings ac, or
ba, or ca, or cb. Either v and x are solid blocks
of one letter (or ?) or not. If they are solid
blocks, then one or two of a, b, c will be
increased in uv2wx2y while the remaining
letter(s) will not increase, so uv2wx2y ? L. If
they are not solid blocks, then uv2wx2y will have
more of the 2-letter substrings ab, ac, ba, bc,
ca, cb than it is supposed to.
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Properties of CF languages

• If L1, L2 are context-free, then L1?L2, L1L2 and
  L1.
• However, if L1 and L2 are context-free then it
  is not necessarily so that L1?L2 is context-free,
  nor that the complement of L1 is context-free.
  Counterexample
• L1 anbncm ? n, m 0
• L2 ambncn ? n, m 0
• L1 and L2 are both context-free with grammars
• G1 S?BC G2 S?AB
• B?aBb?? A?aA??
• C?cC?? B?bBc??
• But L1?L2 anbncn ? n 0 which we have just
  proven not to be context free!
• (If complement were context-free, then also
  intersection - complement of the union of
  complements - contradiction!)
• (Exception if L1 is context-free and L2 is
  regular, then L1?L2 is context-free. Proof
  omitted.)