1. NFAs Sipser 1.2 (pages 47–54)

2. CS 311 Mount Holyoke College 2 Recall… Last time we showed that the class of regular languages is closed under: –Complement –Union –Intersection

3. CS 311 Mount Holyoke College 3 Concatenation operation Let A and B be languages. The concatenation of A and B is A ° B = {xy | x ∈ A and y ∈ B}. Example Let A = {w | w is a string of 0 s and 1 s containing an odd number of 1 s } Let B = {w | w is a string of 0 s and 1 s containing an even number of 1 s } What are A ° B ; B ° A ; A ° A ; B ° B ? –Are any of these languages regular?

4. CS 311 Mount Holyoke College 4 Concatenation Conjecture: The class of regular languages is closed under the concatenation operation. Proof idea:

5. CS 311 Mount Holyoke College 5 Hmm… A = {w | w is a string of 0 s and 1 s containing an odd number of 1 s } B = {w | w is a string of 0 s and 1 s containing an even number of 1 s } Create a machine by gluing… how?

6. CS 311 Mount Holyoke College 6 We need another approach ? ?

7. CS 311 Mount Holyoke College 7 The empty symbol ε Seems like it might work Let’s try running it…

8. Relaxing the rules Deterministic (DFA) Nondeterministic (NFA)

9. Running DFA vs NFA DFA computation (path)NFA computation (tree)

10. CS 311 Mount Holyoke College 10 An example computation What about 1011?

11. CS 311 Mount Holyoke College 11 Another example What language is being recognized? Hint: can you start listing strings accepted?

12. CS 311 Mount Holyoke College 12 Formally… A nondeterministic finite automaton (NFA) is a 5-tuple (Q, Σ, δ, q 0, F), where – Q is a finite set called the states – Σ is a finite set called the alphabet – δ: Q × Σε → P(Q) is the transition function – q 0 ∈ Q is the start state – F ⊆ Q is a set of accept states In-class exercise:

13. CS 311 Mount Holyoke College 13 NFA computation Let N=(Q, Σ, δ, q 0, F) be a NFA and let w be a string over the alphabet Σ Then N accepts w if – w can be written as w 1 w 2 w 3 …w m with each w i in Σε and –There exists a sequence of states s 0,s 1,s 2,…,s m exists in Q with the following conditions: 1.s 0 =q 0 2.s i+1 ∈ δ(s i,w i+1 ) for i = 0,…,m-1 3.s n ∈ F

14. CS 311 Mount Holyoke College 14 Nondeterminism makes life easier Let’s build an NFA that recognizes B = {w | w is a string over {a,b} that starts and ends with the same symbol} C = {w | w is a string over {0,1} that contains at least three 1’s } D = {w | w is a string over {0,1} that contains at least three consecutive 1’s }

15. CS 311 Mount Holyoke College 15 If at first you don’t succeed… … adjust your goal! We wanted to prove: The class of regular languages is closed under the concatenation operation.

16. CS 311 Mount Holyoke College 16 If at first you don’t succeed… … adjust your goal! Instead, let’s prove the theorem: The class of languages recognized by NFAs is closed under the concatenation operation.