1. Lecture 10 Closure Properties of Regular Languages
CSCE 355 Foundations of Computation
Lecture 10 Closure Properties of Regular Languages
Topics:
Extended RegExpr Thompson Construction
June 13, 2015

2. Last Time: Readings 2.3 New: Mutual Induction Proof revisited
Languages denoted by regular expressions
Examples
Ruby Regular Expressions
TEST 2 – Monday
New:
Readings section 3.1,3.2(skip DFA RE), 3.3

3. Homework English descriptions  Regular expressions
3.1.1b – The set of strings of 0’s and 1’s such that the tenth symbol from the right is a 1.
3.1.1c - The set of strings of 0’s and 1’s with at most one pair of consecutive 1’s.
3.1.2 b The set of strings of 0’s and 1’s such that the number of zeroes is divisible by 5.
Regular expressions  English descriptions
3.14b (0*1*)*000(0+1)*
3.14c (0+10)*1*
Regular expressions  εNFA (construction)
3.2.4b (0+1)01
3.2.4c 00(0+1)*
3.14b, 3.14c convert to εNFA
Text matching re for matching phone numbers (international etc)

4. Regular expressions for Addresses
Why?
So we can automatically read messages and extract addresses and insert into DB
Joe Shmo
532 Spring Lake Road
Columbia, SC 29209
Let’s start with Street Names
Spring Lake Road
Road | Rd. | Rd | avenue | Street | …

5. McNaughton-Yamada Construction Examples

6. McNaughton-Yamada Construction Examples
Extended regular expressions

7. Lex and Flex Regular definitions Pattern match “.” Last pattern
Examples
digit = [0-9]
alpha= [a-zA-Z_]
id = {alpha} ({alpha} | {letter} | ‘_’)* %%
id {install_InSymbolTable(yytext); return(ID);}
while { return (KEYWHILE);}

8. Compilation

9. DFA RegExpr  NFA DFA
Regular Languages

10. Not every language can be recognized by a DFA!

11. Algebraic Laws for Arithmetic Expressions
Properties
Equivalence of two expressions

12. Algebraic Laws for Regular Expressions
What does it mean re1=re2 ?

13. Languages are Sets So Properties are Inherited
Commutativity Law for alternation
AUB = BUA so
r l s = s l r
Associativity Law for alternation
(r + s) + t = r + (s + t)
Associativity Law for concatenation
(rs) t = r (st)

14. Commutivity

15. Identity/Annihilator

16. Distributive Laws In algebra
Theorem 3.11 If L, M, and N are any languages, then
L(M U N) = LM U LN
Proof: w is in L(M U N) if and only if w is in LM U LN
w is in L(M U N) 
w is in LM U LN 

17. Factoring and Simplifying
0 + 01* =
0(ϵ + 1*) =
01*

18. Checking for Equality of Two Regular Expressions
s = r ?
(r+s)* = r* s*

19. Idempotent Law
x + x ?
xx ?

20. Laws involving Closures
Ф* = ϵ
ϵ* = ϵ
L+ = L L*
L* = L+ + ϵ L?

21. Is L intersection R regular?

22. Is the Complement of a Regular Lang. regular?

23. Not all Languages are regular
L = {0n1n | n >= 1}

24. Every finite language is Regular

25. Discovering Laws for Regular Expressions
(L +M)* = (L*M*)*
Concretize expressions (only work for pure regular expressions, not intersections.)

26. Theorem 3.14
Theorem 3.14 The following test identifies true laws for regular expressions:
Test for a Regular-Expression Algebraic Law
to Test if regular expressions E = F is true (L(E) = L(F))
Convert E and F to concrete regular expressions C and D by replacing Variables by a concrete symbol
Test whether L(C) = L(D) if so the E = F is true!

28. Pumping Lemma