COSC%203340:%20Introduction%20to%20Theory%20of%20Computation

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COSC 3340 Introduction to Theory of Computation

• University of Houston
• Dr. Verma
• Lecture 4

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Formal definition of NFA acceptance

• Define ?(q, w) as a set of states p e ?(q, w)
  if there is a directed path from q to p labeled
  w
• Example consider NFA of Lecture 3
• ?(q0, 1) ?
• Ans q0, q1
• ?(q0, 11) ?
• Ans q0, q1, q2

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NFA acceptance (contd.)

• w is accepted by NFA M iff ?(q0, w) ? F is
  nonempty.
• L(M) w in ? w is accepted by M.

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NFA vs. DFA

• Is NFA more powerful than DFA?
• Ans No.
• Theorem
• For every NFA M there is an equivalent DFA M'
• Proof Idea
• NFA is in a set of states at any point during
  reading a string.
• DFA will use a lot of states to keep track of
  this.
• Important Assumption
• No transition labeled by epsilon.
• (Will get rid of this assumption later.)

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Equivalent DFA construction.

• NFA M (Q, ?, ?, s, F)
• DFA M' (Q', ?, ?, s', F') where
• Q' 2Q
• s' s
• F' P P ? F is nonempty
• ?(p1, p2, pm, ?) ?(p1, ?) ? ?(p2, ?) ? ...
  ? ?(pm, ?)
• i.e. find all the states that can be reached on ?
  from all the NFA states in a DFA state.

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Example Equivalent DFA construction
NFA
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Equivalent DFA construction (contd.)
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How to handle epsilon transitions?

• Define e-closure of state q as ?(q, ?).
• notation e-closure(q).
• Example

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Handling epsilon transitions (contd.)

• Extend e-closure to sets of states by
• e-closure(s1, ... , sm) e-closure(s1) ? ... ?
  e-closure(sm)
• Now let
• s' e-closure(s).
• and,
• ?(p1,..., pm, ?) e-closure(?(p1, ?)) ? ... ?
  e-closure(?(pm, ?))
• to complete construction of DFA.

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Example Handling epsilon transitions.
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DFA ?
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Language Operations

• Concatenation. Notation L?L' or just LL'
• L ? L' uv u in L, v in L'.
• Kleene Star. Notation L
• L w in ? w w1...wk for some k gt 0 and
  each wi in L.
• Examples if L a(2n1) n gt 0. L' b(2n)
  n gt 0.
• LL' ?
• Ans LL' a(2n1) b(2m) n, m gt 0
• L ?
• Ans an n gt 0
• U, ., are called regular operations.

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Closure properties of regular languages.

• Previously we saw closure under ? and ?.
• New Regular languages are closed under
• Concatenation
• Kleene star
• Complement.

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Examples
L w in a,b w has even as
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Examples
L' w in a,b w has at least one b
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Construction for L?L'
L (K,?,?,s,F)K K1 ? K2s s1F F2?
?1 ? ?2 ? F1 X e X s2
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L and L'
L
M (K, ?, ?, s, F)K s ? K1F s ? F1?
?1 ? F1 X e X s1 ? (s, e, s1) Given M1
(K1, ?, ?1, s1, F1)
L
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Complement of L and L'
Complement of L
Complement of L
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General Construction for Complement
DFA M (K, ?, d, s, F)K K1s s1F K -
F1d d1 L(M) Complement of L(M1) DFA M1
(K1, ?, d1, s1, F1) Exercise Will this
construction work for NFAs?
Explain your answer.