1. LING/C SC/PSYC 438/538
Lecture 17
Sandiway Fong

2. Today's Topic Review of ungraded homework Closure properties of FSA
Practice! Homework out on Thursday…

3. From last time Ungraded Homework 6
apply the set-of-states construction technique to the two machines on the ε- transition slide (repeated below)
self-check your answer:
verify in each case that the machine produced is deterministic and accurately simulates its ε-transition counterpart 1 a ε 2 3 b
> 1 a ε 2 3 b
>

4. Ungraded Homework 6 Review
Converting a NDFSA into a DFSA a b
Note: this machine with an ε-transition
is non-deterministic
> 1 2 3 ε
{1,3}
{2}
{3}
Note: this machine is
deterministic
> a b
[Powerpoint animation]

5. Ungraded Homework 6 Review
Converting a NDFSA into a DFSA a b
Note: this machine with an ε-transition
is non-deterministic
> 1 2 3 ε
{1,2}
{2}
{3}
Note: this machine is
deterministic
> a b b
[Powerpoint animation]

6. Regular Languages and FSA
Formal (constructive) set-theoretic definition of a regular language
Correspondence between REs and Regular Languages
concatenation (juxtaposition)
union (| also [ ])
Kleene closure (*) Note: x+ = xx*)
Note:
backreferences are memory devices and thus are too powerful
e.g. L = {ww} and prime number testing (see earlier lectures)

7. Regular Languages and FSA
Closure properties: i.e. do we still have a regular language afterward applying the operation?
Properties not necessarily preserved higher up: e.g. context-free grammars as we’ll see later

8. Regular Languages and FSA
Textbook gives one direction only
case by case:
Empty string
Empty set
Any character from the alphabet

9. Regular Languages and FSA
Concatenation:
Link final state of FSA1 to initial state of FSA2 using an empty transition
Note: empty transition ε can be deleted using the set of states construction

10. Regular Languages and FSA
Kleene closure:
repetition operator: zero or more times
use empty transitions for loopback and bypass

11. Regular Languages and FSA
Union: aka disjunction
Non-deterministically run both FSAs at the same time, accept if either one accepts

12. Regular Languages and FSA
Other closure properties:
Let’s consider building the FSA machinery for each of these guys in turn…

13. Regular Languages and FSA
Other closure properties:
Final state?

14. Regular Languages and FSA
Other closure properties:
Final state?

15. Regular Languages and FSA
Other closure properties:
Example: Σ* = {a,b}
need arcs for each character in Σ

16. Regular Languages and FSA
Other closure properties:
reverse arrows and swap initial/final

17. Regular Expressions from FSA
Recall textbook Exercise: find a RE for
Examples (* denotes string not in the language):
*ab *ba
bab
λ (empty string) bb
*baba
babab

18. Regular Expressions from FSA
Draw a FSA and convert it to a RE: b b
[Powerpoint
Animation]
> 1 2 3 4 b a b ε b* b
( )+
ab+
= b+(ab+)*
| ε

19. Regular Expressions from FSA
Example Perl implementation:
$s = "ab ba bab bb baba babab";
while ($s =~ /\b(b+(ab+)*)\b/g) {
print "<$1> match!\n"; }
Output:
perl test.perl
<bab> match!
<bb> match!
<babab> match!
Note: this doesn’t include
the empty string case
Note: recall /../g global flag for multiple matches

20. Converting FSA to REs Example: State by-pass method:
Give a RE for the FSA:
State by-pass method:
Delete one state at a time
Calculate the possible paths passing through the deleted state
Add the regex calculated at each stage as an arc
e.g.
eliminate state 3
then 2… 1 2 3
>

21. Converting FSA to REs Answer: (0(1+0|1)*1+1 | 1)* eliminate state 32 3
>
eliminate state 3
eliminate state 2 1 1
> 1
> 1
0(1+0|1)*1+1 2
1+1
1+0 1
>
0(1+0|1)*1+1 | 1
Answer: (0(1+0|1)*1+1 | 1)*
[Powerpoint animation]