1. COSC 3340: Introduction to Theory of Computation
University of Houston
Dr. Verma
Lecture 4
Lecture 4
UofH - COSC Dr. Verma

2. Formal definition of NFA acceptance
Define *(q, w) as a set of states: p ε *(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}
Lecture 4
UofH - COSC Dr. Verma

3. NFA acceptance (contd.)
w is accepted by NFA M iff *(q0, w)  F is nonempty.
L(M) = {w in * | w is accepted by M}.
Lecture 4
UofH - COSC Dr. Verma

4. NFA vs. DFA Is NFA more powerful than DFA? Theorem: Proof Idea:
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.)
Lecture 4
UofH - COSC Dr. Verma

5. 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.
Lecture 4
UofH - COSC Dr. Verma

6. Example: Equivalent DFA construction
NFA
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7. Equivalent DFA construction (contd.)
Lecture 4
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8. How to handle epsilon transitions?
Define e-closure of state q as *(q, ).
notation: e-closure(q).
Example:
Lecture 4
UofH - COSC Dr. Verma

9. 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.
Lecture 4
UofH - COSC Dr. Verma

10. Example: Handling epsilon transitions.
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UofH - COSC Dr. Verma

11. DFA = ?
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12. 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 >= 0 and each wi in L}.
Examples: if L = {a(2n+1) | n >= 0}. L' = {b(2n) | n > = 0}.
LL' = ?
Ans: LL' = {a(2n+1) b(2m) | n, m > = 0}
L* = ?
Ans: {an | n >= 0}
U, ., * are called regular operations.
Lecture 4
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13. Closure properties of regular languages.
Previously we saw closure under  and .
New: Regular languages are closed under
Concatenation
Kleene star
Complement.
Lecture 4
UofH - COSC Dr. Verma

14. L = {w in {a,b}* | w has even a’s }
Examples
L = {w in {a,b}* | w has even a’s }
Lecture 4
UofH - COSC Dr. Verma

15. Examples L' = {w in {a,b}* | w has at least one b} Lecture 4
UofH - COSC Dr. Verma

16. Construction for LL'
L’’ = (K,,,s,F) K = K1  K2 s = s1 F = F2  = 1  2  F1 X {e} X {s2}
Lecture 4
UofH - COSC Dr. Verma

17. L* and L'* L*
M = (K, , , s, F) K = {s}  K1 F = {s}  F1  = 1  F1 X {e} X {s1}  {(s, e, s1)}
Given M1 = (K1, , 1, s1, F1)
L’*
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UofH - COSC Dr. Verma

18. Complement of L and L' Complement of L Complement of L’ Lecture 4
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19. General Construction for Complement
DFA M = (K, , δ, s, F) K = K1 s = s1 F = K - F1 δ = δ1
L(M) = Complement of L(M1)
DFA M1 = (K1, , δ1, s1, F1)
Exercise: Will this construction work for NFAs?
Explain your answer.
Lecture 4
UofH - COSC Dr. Verma