1. Intro to Theory of Computation
LECTURE 4
Last time:
Equivalence of NFAs and DFAs
Closure properties of regular languages
Today:
Finish closure properties Equivalence of NFAs, DFAs and regular expressions CS
332
Sofya Raskhodnikova
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

2. Operations on languages
Regular
Operations on languages
Complement: A = { w | w  A }
Union: A  B = { w | w  A or w  B }
Union: A  B = { w | w  A or w  B }
Intersection: A  B = { w | w  A and w  B }
Reverse: AR = { w1 …wk | wk …w1  A }
Concatenation: A  B = { vw | v  A and w  B }
Concatenation: A  B = { vw | v  A and w  B }
Star: A* = { w1 …wk | k ≥ 0 and each wi  A }
Star: A* = { w1 …wk | k ≥ 0 and each wi  A }
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

3. Closure properties of the class of regular languages
THEOREM. The class of regular languages is closed under all 6 operations.
If A and B are regular, applying any of these operation yields a regular language.
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

4. Sofya Raskhodnikova; based on slides by Nick Hopper
Star operation
Star: A* = { w1 …wk | k ≥ 0 and each wi  A }
Theorem. If A is regular, A* is also regular. ε ε
L(M)=A ε
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

5. The class of regular languages is closed under
Regular operations
Union: A  B = { w | w  A or w  B }
Union: A  B = { w | w  A or w  B }
Concatenation: A  B = { vw | v  A and w  B }
Star: A* = { w1 …wk | k ≥ 0 and each wi  A }
Other operations
Complement: A = { w | w  A }
Intersection: A  B = { w | w  A and w  B }
Reverse: AR = { w1 …wk | wk …w1  A }
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

6. Sofya Raskhodnikova; based on slides by Nick Hopper
Regular expressions
In a regular expression, we can use
Constants: ε ,  , a set Σ, members of Σ.
Regular operations: *,  , ∪
Examples:
0*1*
(01)*
0*1*(ε 0)
L(R) = the language regular expression R describes
= { w | w has a run of 0s followed by a run of 1s}
= the set of all strings over the alphabet Σ={0,1}
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

7. Sofya Raskhodnikova; based on slides by Nick Hopper
Precedence
*  ∪
EXAMPLE
𝑹 𝟏 𝑹 𝟐 ∗ ∪ 𝑹 𝟑 𝑹 𝟒 =
(𝑹 𝟏 ( 𝑹 𝟐 ∗ ))∪( 𝑹 𝟑 𝑹 𝟒 )
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

8. Sofya Raskhodnikova; based on slides by Nick Hopper
Regular expressions: examples 1) 2) 3)
{ w | w has exactly one character 1, any # of 0s}
0*10*
{ w | w has length ≥ 3 and its 3rd symbol is 0 }
(01)(01) 0 (01)*
{ w | every odd position of w is a 1 }
(1(01))* (ε  1)
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

9. Regular expressions to NFAs
Theorem. Every regular expression has an equivalent NFA.
Proof: Induction on the length of regular expression R.
Base case: length 1
Inductive step: follows from closure of the class of regular languages under the regular operations.
R = ε
R =  
R = 
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

10. Exercise What should the induction hypothesis be?
Suppose some regular expression of length 𝑘 can be converted an NFA, for some 𝑘∈ℕ.
Suppose each regular expression of length 𝑘 can be converted an NFA, for some 𝑘∈ℕ.
Suppose each regular expression of length at most k can be converted an NFA, for some 𝑘∈ℕ.
None of the above.
{madam, dad, but, tub, book}

11. Regular expression to NFA
Transform (1(0  1))* to an NFA ε 1
1,0 ε
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

12. NFAs to regular expressions
Theorem. Every NFA has an equivalent regular expression.
Proof idea:
Transform NFA to a regular expression by removing states and relabeling the arrows with regular expressions.
01*0 1
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

13. Exercise
What is the regular expression that generates all strings that take this machine from 𝒒 𝒔 to 𝒒 𝒂 ?
𝑎 𝑏 ∗ 𝑐∪𝑑
𝑎 𝑏 ∗ 𝑐∩𝑑
𝑎𝑏𝑐∪𝑑
𝑎 𝑏 ∗ 𝑐𝑑 ∗
None of the above. 𝒃 𝒂 𝒄 qs q qa 𝒅
{madam, dad, but, tub, book}

14. Sofya Raskhodnikova; based on slides by Nick Hopper
Generalized NFAs
Each transition is labeled with a regular expression
Unique and distinct start and accept states
No transitions to the start state
No transitions from the accept state
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

15. G accepts w if ∃q0,q1,…,qk, w1,…,wk: wi is generated by R(qi-1,qi)
Generalized NFAs
a  b
a*b a qs q qa
G accepts w if ∃q0,q1,…,qk, w1,…,wk:
wi is generated by R(qi-1,qi)
w = w1w2…wk
q0=qs, qk=qa
R(qs,q) = a*b
R(qa,q) = Ø
R(q,qs) = Ø

16. NFA to GNFA NFA ε ε ε ε Add a new start state with no incoming arrows.
Make a unique accept state with no outgoing arrows.
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

17. GNF to regular expressionε ε ε
NFA
While machine has more than 2 states:
Pick an internal state, rip it out and relabel the arrows with regular expressions to account for the missing state.
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

18. q1 b a ε q2
a,b ε
(a*b)(ab)*
a*b q0 q3
R(q0,q3) = (a*b)(ab)*

19. Sofya Raskhodnikova; based on slides by Nick Hopperbb a q1 q2 a ε b ε b a ε q3
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

20. Sofya Raskhodnikova; based on slides by Nick Hopperbb a q1 q2
a  ba a ε b b ε a ε
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

21. Sofya Raskhodnikova; based on slides by Nick Hopper
bb  (a  ba)b*a bb
= R(q1,q1) a q1 q2
a  ba ε
b  (a  ba)b* b ε
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

22. Sofya Raskhodnikova; based on slides by Nick Hopper
bb  (a  ba)b*a
= R(q1,q1) q1 ε
b  (a  ba)b*
= R(q1,qa) qs qa
(bb  (a  ba)b*a)*(b  (a  ba)b*)
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

23. Sofya Raskhodnikova; based on slides by Nick Hopper
Conversion procedures
DFA
NFA
definition
Regular
Language
Regular
Expression
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

24. Sofya Raskhodnikova; based on slides by Nick Hopper
REGULAR OR NOT?
Design an NFA for the language:
{0𝑛1𝑛 | 0 < 𝑛≤ 2}
0𝑛1𝑛 0 < 𝑛≤ 𝑘}
{0𝑛1𝑛 | 𝑛 > 0}? n
(For R a regexp, R2 means RR, and Rn means RR…R)
8/24/2019
Sofya Raskhodnikova; based on slides by Nick Hopper