Nondeterministic Finite State
Automata (Dr. Torng)
NFA’s
1
Change: d is a relation
• For an FSA M, d(q,a) results in one and
only one state for all states q and characters
a.
– That is, d is a function
• For an NFA M, d(q,a) can result in a set of
states
– That is, d is now a relation
– Next step is not determined (nondeterministic)
2
Example NFA
a,b
a,b
a
a
b
a
•Why is this only an NFA and not an FSA? Identify
as many reasons as you can.
3
Computing with NFA’s
• Configurations: same as they are for FSA’s
• Computations are different
– Initial configuration is identical
– However, there may be several next configurations or
there may be none.
• Computation is no longer a “path” but is now a “graph” (often
a tree) rooted at the initial configuration
– Definition of halting, accepting, and rejecting
configurations is identical
– Definition of acceptance must be modified
4
a,b
Computation
Graph (Tree)
a,b
a
a
b
a
Input string aaaaba
(1, aaaaba)
(1, aaaba)
(2, aaaba)
(1, aaba)
(2, aaba)
(3, aaba)
(1, aba)
(2, aba)
(3, aba)
(1, ba)
(2, ba)
(1, a)
(1, l)
crash
(3, ba)
(4, a)
(2, l)
(5, l)
5
a,b
Definition of ├
unchanged *
a
a
b
a
Input string aaaaba
(1, aaaaba)├ (1, aaaba)
(1, aaaaba)
(1, aaaba)
a,b
(1, aaaaba)├ (2, aaaba)
(2, aaaba)
(1, aaaaba)├3 (1, aba)
(1, aaba)
(2, aaba)
(1, aaaaba)├3 (3, aba)
(3, aaba)
(1, aaaaba)├* (2, aba)
(1, aba)
(2, aba)
(1, ba)
(2, ba)
(3, aba)
crash
(1, aaaaba)├* (3, aba)
(1, aaaaba)├* (1, l)
(3, ba)
(1, aaaaba)├* (5, l)
(1, a)
(1, l)
(4, a)
(2, l)
(5, l)
6
a,b
a
Acceptance and
Rejection
(1, aaba)
(1, aba)
(2, aba)
(1, ba)
(2, ba)
(1, a)
(1, l)
b
a
M accepts string x if one of the
configurations reached is an
accepting configuration
(2, aaaba)
(2, aaba)
a
Input string aaaaba
(1, aaaaba)
(1, aaaba)
a,b
(q0, x) ├* (f, l),f in A
(3, aaba)
(3, aba)
crash
(3, ba)
M rejects string x if all
configurations reached are
either not halting
configurations or are rejecting
configurations
(4, a)
(2, l)
(5, l)
7
Comparison
b
a
a
a
a,b
b
a
b
b
FSA
a,b
a,b
a
a
b
a
NFA
8
Defining L(M) and LNFA
• M accepts string x if one
of the configurations
reached is an accepting
configuration
• L(M)
– (q0, x) |-* (f, l),f in A
• LNFA
• M rejects string x if all
configurations reached are
either not halting
configurations or are
rejecting configurations
• N(M)
– Language L is in language class
LNFA iff
9
Comparing language classes
LFSA subset of LNFA
10
LFSA subset LNFA
• Let L be an arbitrary language in LFSA
• Let M be the FSA such that L(M) = L
– M exists by definition of L in LFSA
•
•
•
•
Construct an NFA M’ such that L(M’) = L
Argue L(M’) = L
There exists an NFA M’ such that L(M’) = L
L is in LNFA
– By definition of L in LNFA
11
Visualization
•Let L be an arbitrary language in LFSA
•Let M be an FSA such that L(M) = L
•M exists by definition of L in LFSA
•Construct NFA M’ from FSA M
•Argue L(M’) = L
•There exists an NFA M’ such that L(M’)
=L
•L is in LNFA
L
LFSA
M
FSA’s
L
LNFA
M’
NFA’s
12
Construction
• We need to make M into an NFA M’ such
that L(M’) = L(M)
• How do we accomplish this?
13