COSC 3340: Introduction to Theory of
Computation
University of Houston
Dr. Verma
Lecture 4
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Lecture 4
UofH - COSC 3340 - Dr. Verma
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
–

*(q0, 1) = ?
–

Ans: {q0, q1}
*(q0, 11) = ?
–
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Example: consider NFA of Lecture 3
Ans: {q0, q1, q2}
Lecture 4
UofH - COSC 3340 - Dr. Verma
NFA acceptance (contd.)
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
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 3340 - Dr. Verma
NFA vs. DFA

Is NFA more powerful than DFA?
–

Theorem:
–

For every NFA M there is an equivalent DFA M'
Proof Idea:
–
–

Ans: No.
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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Lecture 4
UofH - COSC 3340 - Dr. Verma
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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Lecture 4
UofH - COSC 3340 - Dr. Verma
Example: Equivalent DFA construction
NFA
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Lecture 4
UofH - COSC 3340 - Dr. Verma
Equivalent DFA construction (contd.)
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Lecture 4
UofH - COSC 3340 - Dr. Verma
How to handle epsilon transitions?

Define e-closure of state q as *(q, ).
–

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notation: e-closure(q).
Example:
Lecture 4
UofH - COSC 3340 - Dr. Verma
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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Lecture 4
UofH - COSC 3340 - Dr. Verma
Example: Handling epsilon transitions.
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Lecture 4
UofH - COSC 3340 - Dr. Verma
DFA = ?
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Lecture 4
UofH - COSC 3340 - Dr. Verma
Language Operations
1.
Concatenation. Notation: LL' or just LL'
–
2.
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.


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L  L' = {uv | u in L, v in L'}.
Lecture 4
UofH - COSC 3340 - Dr. Verma
Closure properties of regular
languages.


Previously we saw closure under  and .
New: Regular languages are closed under
–
–
–
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Concatenation
Kleene star
Complement.
Lecture 4
UofH - COSC 3340 - Dr. Verma
Examples
L = {w in {a,b}* | w has even a’s }
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Lecture 4
UofH - COSC 3340 - Dr. Verma
Examples
L' = {w in {a,b}* | w has at least one b}
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Lecture 4
UofH - COSC 3340 - Dr. Verma
Construction for LL'
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L’’ = (K,,,s,F)
K = K1  K2
s = s1
F = F2
 = 1  2  F1 X {e} X
{s2}
Lecture 4
UofH - COSC 3340 - Dr. Verma
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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Lecture 4
UofH - COSC 3340 - Dr. Verma
Complement of L and L'
Complement of L
Complement of L’
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Lecture 4
UofH - COSC 3340 - Dr. Verma
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.
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Lecture 4
UofH - COSC 3340 - Dr. Verma