1. NFAs, DFAs, and regular expressions
The Chinese University of Hong Kong
Fall 2010
CSCI 3130: Automata theory and formal languages
NFAs, DFAs, and regular expressions
Andrej Bogdanov

2. Three ways of doing it L = {x Î S*: x ends in 01} S = {0, 1} (0+1)*011
0, 1 q0 q1 q2 qe 1 q0 q1
q00
q10
q01
q11
(0+1)*01
regular expression
NFA
DFA

3. They are all the same
NFA
regular expression
DFA
regular languages

4. Road map  
regular expression
NFA
NFA without e
DFA

5. Examples: regular expression → NFAq0 q1
R1 = 0
R2 = 01 1 q0 q1 q2

6. Examples: regular expression → NFAq1 q2 q0 q6 
R3 = q3 q4 q5 1 
q0’
q1’
R4 = (0 + 01)*
NFA3

7. Regular expressions → NFA
In general, how do we convert a regular expression to an NFA?
A regular expression over S is an expression formed using the following rules:
The symbol Æ is a regular expression
The symbol e is a regular expression
For every a  S, the symbol a is a regular expression
If R and S are regular expressions, so are R+S, RS and R*.

8. General method Æ e regular expr NFA q0 q0 q0 q1 a a  S RS q0 q1 
NFAR
NFAS

9. General method continued
regular expr
NFA
NFAR q0 q1 
R + S
NFAS q0 q1  R*
NFAR

10. Road map   
regular expression
NFA
NFA without e
DFA

11. Road map   
regular expression
NFA
NFA without e
DFA

12. Road map
regular expression
NFA
NFA without e
DFA
2-state GNFA
GNFA

13. Generalized NFAs
A generalized NFA is an NFA whose transitions are labeled by regular expressions, like moreover
It has exactly one accept state, different from its start state
No arrows come into the start state
No arrows go out of the accept state
0*1
e+10*
0*11 q0 q1 q2 01

14. Converting a DFA to a GNFA
regular expression
NFA
NFA without e
DFA
2-state GNFA
GNFA q1 q3 q5 q0 qf e

15.    Conversion example1 q0 e q3 1 q1 q2 
It has exactly one accept state, different from its start state
No arrows come into the start state
No arrows go out of the accept state  

16. GNFA state reduction
regular expression
NFA
NFA without e
DFA
2-state GNFA
GNFA
We will eliminate every state but the start and accept states

17. State elimination e+10* 0*11 0*1 q0 q1 q2 01 q0 q2 01

18. State elimination – general method
To eliminate state qk, for every pair of states (qi, qj) remember to do this even when qi = qj! R2 R1 R3
Replace qi qk qj R4
R1R2*R3 + R4 by qi qj

19.       Road map NFA without e NFA regular expression DFA GNFA
2-state GNFA
GNFA q0 q1 R
A 2-state GNFA is the same
as a regular expression R!

20. Conversion example Eliminate q1: Eliminate q2: Check: 0*1(00*1+1)* = 11 e 1 e q0 q1 q2 q3
00*1+1
0*1 e
Eliminate q1: q0 q2 q3
0*1(00*1+1)*
Eliminate q2: q0 q3 q1 q2 1 ?
Check:
0*1(00*1+1)* =

21. Check your answer! 0*1(00*1+1)* 0*1(0*1)* 011001000101q1 q2 1
All strings that end in 1
(0 + 1)*1
0*1(00*1+1)*
Always ends in 1 =
Does every string that ends in 1 have this form?
0*1(0*1)*
Yes!

22. Design Analyze Convert What you need to know regular expression NFA
DFA
regular languages
Design
Analyze
Convert