CSc 318
Final Examination
Saturday
16 December 2000
8 AM
>>>>>>>>>>>SUGGESTED ANSWERS<<<<<<<<<<<<<<<<<<<<<<<
1.
Consider the set of DFA’s which can be constructed over some alphabet
Σ
.
Prove
that equivalence of DFA’s is an equivalence relation on this set.
I show that the relation is reflexive, symmetric, and transitive.
First, a DFA is clearly equivalent to itself, because if M is the DFA, L(M)=L(M).
Second, if M is equivalent to M’ then L(M)=L(M’)==>L(M’)=L(M)==> M’ is
equivalent to M.
Third, if M is equivalent to M’ and M’ equivalent to M” then
L(M)=L(M’)=L(M”) ==> M is equivalent to M”.
2.
All semester you have been asked to write out definitions I have given you.
Now I
ask you to create the definition of what is called a “lazy NFA.”
All of the FA’s we
have defined this semester “chew up”
ε
or single symbols of an alphabet at each
move.
A lazy NFA “chews up” strings, that is, moving from one state to another
consumes a string.
Provide a formal definition for a lazy NFA, a formal definition of
acceptance of a string by a lazy NFA, and a formal definition of
the language of
a
lazy NFA.
A lazy NFA, M, is a 5-tuple, M=(Q,
Σ
, s, F,
∆
),
where Q is a finite set of states,
Σ
is an alphabet, s
∈
Q is the start state, F
⊆
Q is a set of final states, and
∆⊆
Qx
Σ
*xQ is a transition relation.
A string w is accepted by M if we may write
w=u
1
u
2
… u
n
and find states a
1
a
2
… a
n
such that a
n
∈
F, (s, a
1
, u
1
), (u
i
, a
i+1
, u
i+1
)
∈∆
for i=1, 2, …, n-1.
The language of M is the set of strings accepted by M.