MC 306 Theory of Computation Tuesday, 102803

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MC 306 Theory of ComputationTuesday, 10/28/03

• Last time
• Normal forms
• Backus-Naur form
• Chomsky normal forms
• See handout for completed example
• Today
• Pushdown Automata

• Reading
• 3.3, 3.4
• Exercises
• 3.3 3.18, 3.20, 3.22-3.24
• 3.4 3.29, Also Construct a pda equivalent to
  grammar in 3.3, and show its accepting
  computation of the string in part a.
• Hand-in
• No new hand-in

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A machine to accept context-free languages

• One of our archetypical context-free languages is
  anbn n ? 0
• We can soup up an NFA to accept this language
  by adding a stack
• a, ? 1 on arrow means do the following in
  order (1) Read a from input string (2) Pop ?
  from stack (3) Push 1 onto stack.
• Must be able to both read and pop to use arrow.

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Definition of PDA

• A (nondeterministic) pushdown automaton is a
  6-tuple
• Q is a finite set of states
• ? is the input alphabet
• ? is the stack alphabet
• s0 ? Q is the start state
• F ?? Q is the set of final (favorable) states
• ? , the transition relation, which is a set of
  ordered pairs of the form ( (q1, w1, v1), (q2,
  v2) ),
• where q1 and q2 are states
• where w1 ? ? ? ? (we read the input symbol w1)
• and where v1 and v2 ? ? (we pop v1, then push
  v2)
• Hence ? is a subset of (Q x (? ? ?) x ?) x (Q
  x ?)
• What is 6-tuple for preceding example?

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Computations and acceptance

• A configuraton of a pda is a triple (q, w, v),
  where q is a state, w ? ? (current input
  string), and v ? ?? (current stack)
• We say that a pda accepts a string w if there is
  some computation of the machine beginning with
  the starting configuration (s0, w, ?) and
  yielding an accepting configuration (f, ?, ?),
  where f is a final state
• I.e., you must be able to read the whole string
  and empty out the stack, ending in a final state
• Note JFLAP does not require emptying the stack
  to accept.
• Look at our example in JFLAP (anbn.jfl)
• Modify to make it do right thing!

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Exercises

• 1) Design a PDA to accept the language
• ai bj ck i, j, k 0, and i j or j k
• 2) Design a PDA to accept the language
• w ? a, b Na(w) Nb(w), where Nx(w) means
  the number of xs in w.
• 3) Prove If L is any regular language, there is
  some PDA that accepts L.
• 4) Is the converse of (3) true? Why?

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Equivalence of CFGs and PDAs

• Just as NFAs and regular expressions describe the
  same class of languages, namely the regular
  languages
• So do CFGs and PDAs describe the same class of
  languages, namely the context-free languages
• Given a CFG, we show how to construct an
  equivalent PDA
• Given a PDA, we show how to construct an
  equivalent CFG.

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From CFG to PDA

• Given a CFG G, we construct a PDA M that
  simulates a derivation of a string
• Q s, f s start state, f only final state
• ?M ?Q ?M ?Q ? NTQ
• Transitions
• Transition from s to f Read nothing, pop
  nothing push starting nonterminal S onto stack
• Transitions from f to f, corresponding to rules
• For each rule A ? ?, read nothing, pop A, push ?
• Transitions from f to f, corresponding to ?
  symbols
• For each x ??, read x, pop x, push nothing.

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Example Convert CFG to PDA

• A CFG that accepts w wR over alphabet ? a, b
• S ? a S a b S b ?
• A derivation of aabbaa
• S ? aSa ? aaSaa ? aabSbaa ? aabbaa
• Construct the PDA as defined by algorithm.
• What does computation of PDA look like on string
  aabbaa?