azfat2.ppt
TRANSCRIPT
Nondeterminism
Example from last time
0
1
…
…
……
q
q
q
q
q
q
q
q
q
q
q
0
1
0
1
0
1
1
1
1
1
0
• Construct a DFA over alphabet {0, 1} that accepts those strings that end in 101
• Sketch of answer:
Would be easier if…
• Suppose we could guess when the string we are reading has only 3 symbols left
• Then we could simply look for the sequence 101and accept if we see it
qdie
1 0 13 symbols left
This is not a DFA!
Nondeterminism
• Nondeterminism is the ability to make guesses, which we can later verify
• Informal nondeterministic algorithm for strings that end in 101:
1. Guess if you are approaching end of input2. If guess is yes, look for 101 and accept if you
see it3. If guess is no, read one more symbol and go
to step 1
Nondeterministic finite automaton
• This is a kind of automaton that allows you to make guesses
• Each state can have zero, one, or more transitions out labeled by the same symbol
1 0 1
0, 1
q q q q
Semantics of guessing
1 0 1
0, 1
q q q q
• State q0 has two transitions labeled 1
• Upon reading 1, we have the choice of staying in q0 or moving to q1
Semantics of guessing
1 0 1
0, 1
q q q q
• State q1 has no transition labeled 1
• Upon reading 1 in q1, we die; upon reading 0, we continue to q2
Semantics of guessing
1 0 1
0, 1
q q q q
• State q3 has no transition going out
• Upon reading anything in q3, we die
Meaning of automaton
1 0 1
0, 1
q q q q
Guess if you are 3 symbols away from
end of input
If so, guess you will see the
pattern 101
Check that you are at the end of input
Example
• Construct an NFA over alphabet {0, 1} that accepts those strings that contain the pattern 001
Example
• Construct an NFA over alphabet {0, 1} that accepts those strings that contain the pattern 001 somewhere
• Answer
0 0 1
0, 1
q q q q
0, 1
Example
• Construct an NFA that accepts all strings with an even number of 0s or an odd number of 1s
r0 r1
1 10
0even number of 0s
s0 s1
0 01
1odd number of 1s
q0
-transitionscan be taken for free (without reading input)
Formal definition
• A nondeterministic finite automaton (NFA) is a 5-tuple (Q, , , q0, F) where– Q is a finite set of states– is an alphabet– : Q × (∪) → subsets of Q is a transition
function
– q0 Q is the initial state
– F Q is a set of accepting states (or final states).
• Differences from DFA:– transition function can go into several states– It allows -transitions
Example
alphabet = {0, 1}states Q = {q0, q1, q2}initial state q0
accepting states F = {q2}
state
s
inputs
0 1q0
q1
q2
{q1}
table oftransition function
{q0, q1}
q0
, 1 0
0q1 q2
{q1}{q2}
Language of an NFA
The language of an NFA is the set of all strings for which there is some path that, starting from q0, leads to an accepting state as the string is read left to right (and -transitions are taken for free).• Example
, 00, 001, 101 are accepted, but 11, 0110 are not
q0
, 1 0
0q1 q2
Examples
q0
, 1
0q1 q2
q0
, 1
0q1 q2
1
2
00
0
01
2
3
4
5
q0
, 1
0q1 q2
01
2 3
4
q0
, 1
0q1 q2
01
2
3
54
or
or
Examples
q0
, 1
0q1 q2
15
001
0
q0
, 1
0q1 q2
01
4
q0
, 1
0q1 q2
01
2
3
4
1012
3
11
q0
, 1
0q1 q2
01
2
STOP
STOP
NFAs are as powerful as DFAs
• Obviously, an NFA can do everything a DFA can do
• But can it do more?
NFAs are as powerful as DFAs
• Obviously, an NFA can do everything a DFA can do
• But can it do more?
• Theorem
If a language L is accepted by some NFA, thenit is also accepted by some DFA.
NO!