Chapter 3 Regular Languages and Regular Grammars

1. Chapter 3Regular Languages andRegular Grammars

2. 3.1: Regular Expressions (1) • Regular Expression (RE): • E is a regular expression over  if E is one of: • e • a, where a   • If r and s are regular expressions (REs), then the following expressions also regular: • r | s (r or s) • rs  (r followed by s) • r* (r repeated zero or more times) • Each RE has an equivalent regular language (RL)

3. 3.1: Regular Expressions (2) • Regular Language (RL): • L is a regular language over  if L is one of: •  empty set • {e} a set that contains empty string • {a} where a   • If R and S are regular languages (RL), then the following languages also regular: • R  S = {w | w  R or w  S} • RS = {rs | r  R and s  S} • R* = R0  R1  R2  R3  …

4. 3.1: Regular Expressions (3) precedence • Rules for Specifying Regular Expressions: • e is a regular expression  L = {e} • If a is in , a is a regular expression  L = {a}, the set containing the string a. • Let r and s be regular expressions with languages L(r) and L(s). Then • a.r | s is a RE  L(r)  L(s) • b.rs is a RE  L(r) L(s) • c.r* is a RE  (L(r))* • d. (r) is a RE  L(r), extra parenthesis

5. 3.1: Regular Expressions (4) • Examples: • {0, 1}{00, 11} = {000, 011, 100, 111} • {0}* = {e, 0, 00, 000, 0000, …} • {e}* = {e} • {10, 01}* = {e, 10, 1010, 101010, …, 01, 0101, 010101, …, 1001, 100101, 10010101, …, 0110, 011010, 01101010, …} • Notational shorthand: • L0 = {e} • Li = LLi-1 • L+ = LL*

6. 3.1: Regular Expressions (5) • Let L be a language over {a, b}, each string in L contains the substring bb • L = {a, b}*{bb}{a, b}* • L is regular language (RL). Why? • {a} and {b} are RLs • {a, b} is RL • {a, b}* is RL • {b}{b} = {bb} is also RL • Then L = {a, b}*{bb}{a, b}* is RL

7. 3.1: Regular Expressions (6) • Let L be a language over {a, b}, each string in L • begins and ends with an a • contains at least one b • L = {a}{a, b}*{b}{a, b}*{a} • L = a*b*a • L is regular language (RL). Why? • {a} and {b} are RLs • {a, b}* is RL • Then L = {a}{a, b}*{b}{a, b}*{a} is RL

8. 3.1: Regular Expressions (7) • L = {a, b}*{bb}{a, b}* • RE = (a|b)*bb(a|b)* • L = {a}{a, b}*{b}{a, b}*{a} • RE = a(a|b)*b(a|b)*a • This RE = (a)|((b)*(c)) is equivalent to a|b*c • We say REs r and s are equivalent (r=s), iff r and s represent the same language • Example: r = a|b, s = b|a r = s Why? • Since L(r) = L(s) = {a, b}

9. 3.1: Regular Expressions (8) • Let = {a, b} • RE a|b L = {a, b} • RE (a|b)(a|b)  L = {aa, ab, ba, bb} • RE aa|ab|ba|bb same as above • RE a*  L = {, a , aa, aaa, …} • RE (a|b)*  L = set of all strings of a’s and b’s including  • RE (a*b*)* same as above • RE a|a*b L = {a,b,ab,aab,aaab, …}

10. 3.1: Regular Expressions (9) • Algebraic Properties of regular Expressions AXIOM r | s = s | r r | (s | t) = (r | s) | t (r s) t = r (s t) r ( s | t ) = r s | r t ( s | t ) r = s r | t r r = r  = r r* = (r*)* = ( r |  )+ = r+ |  r** = r* r+ = r r*

11. 3.1: Regular Expressions (10) abc • concatenation (“followed by”) a | b | c • alternation (“or”) * • zero or more occurrences + • one or more occurrences

12. 3.1: Regular Expressions (11) • All strings of 1s and 0s (0 | 1)* • All strings of 1s and 0s beginning with a 1 1 (0 | 1)*

13. 3.1: Regular Expressions (12) • All strings containing two or more 0s (1|0)*0(1|0)*0(1|0)* • All strings containing an even number of 0s (1*01*01*)* | 1*

14. 3.1: Regular Expressions (13) • All strings containing an even number of 0s and even number of 1s ( 0 0 | 1 1 )*(( 0 1 | 1 0 )( 0 0 | 1 1 )*( 0 1 | 1 0 )( 0 0 | 1 1 )*)* • All strings of alternating 0s and 1s (  | 1 ) ( 0 1 )* (  | 0 )

15. 3.1: Regular Expressions (14) • Strings over the alphabet {0, 1} with noconsecutive 0's • (1 | 01 )* (0 | ) • 1*(01+)* (0 | ) • 1*(011*)* (0 | ) • Strings over the alphabet {a, b} with exactly three b's • a*ba*ba*ba* • Strings over the alphabet {a, b, c} containing (at least once) bc • (a|b|c)*bc(a|b|c)*

16. 3.1: Regular Expressions (15) • Strings over the alphabet {a, b} in which substrings ab and ba occur an unequal number of times • (a+b+)+ | (b+a+)+

17. 3.1: Regular Expressions (16) • Describe the following in English: • (0|1)* • all strings over {0, 1} • b*ab*ab*ab* • all strings over {a, b} with exactly 3 a’s

18. 3.1: Regular Expressions (17) • Examples of RE: • 01* • {0, 01, 011, 0111, …..} • (01*)(01) • {001, 0101, 01101, 011101, …..} • (0 | 1)* = {0, 1, 00, 01, 10, 11, …..} • i.e., all strings of 0 and 1 • (0 | 1)* 00 (0 | 1)* = {00, 1001, …..} • i.e., all 0 and 1 strings containing a “00”

19. 3.1: Regular Expressions (18) • More Examples of RE: • (1 | 10)* • all strings starting with “1” and containing no “00” • (0 | 1)*011 • all strings ending with “011” • 0*1* • all strings with no “0” after “1” • 00*11* • all strings with at least one “0” and one “1”, and no “0” after “1”

20. 3.1: Regular Expressions (19) • What languages do the following RE represent? • (1 | 01 | 001)*( | 0 | 00) • ((0 | 1)(0 | 1))* | ((0 | 1)(0 | 1)(0 | 1))*

21. 3.1: Regular Expressions (20) • Home Study: • Construct a RE over ={0,1} such that • It does not contain any string with two consecutive “0”s • It has no prefix with two or more “0”s than “1” nor two or more “1”s than “0” • The set of all strings ending with “00” • The set of all strings with 3 consecutive 0’s • The set of all strings beginning with “1”, which when interpreted as a binary no., is divisible by 5 • The set of all strings with a “1” at the 5th position from the right • The set of all strings not containing 101 as a sub-string

22. 3.1: Regular Expressions (21) • Strings over the alphabet {(, )} where the parentheses are balanced • Not regular   • L = {anbn | n 0} • L = {e, ab, aabb, aaabbb, aaaabbbb, …} • Not regular  

23. 3.2: Connection Between RE & RL (1) • A language L is called regular if and only if there exists some DFA M such that L = L(M). • Since a DFA has an equivalent NFA, then • A language L is called regular if and only if there exists some NFA N such that L = L(N). • If we have a RE r, we can construct an NFA that accept L(r).

24. 3.2: Connection Between RE & RL (2) 0. For  in the regular expression, construct NFA start L = { } =  1. For  in the regular expression, construct NFA  start L = {e} 2. For a   in the regular expression, construct NFA a start L = {a}

25. 3.2: Connection Between RE & RL (3) 3.(a) If s and t are regular expressions, Ms and Mt are their NFAs. s|t has NFA: L = {L(Ms)  L(Mt)} Ms e e start i f e e Mt where i and f are new start / final states, and e-moves are introduced from i to the old start states of Ms and Mt as well as from all of their final states to f.

26. 3.2: Connection Between RE & RL (4) start Ms Mt Mt Ms i i f f start e e e 3.(b) If s and t are regular expressions, Ms, Mt their NFAs. st (concatenation) has NFA: L = {L(Ms)L(Mt)} overlap Alternative: where i is the start state of Ms (or new under the alternative) and f is the final state of Mt (or new). Overlap maps final states of Ms to start state of Mt

27. 3.2: Connection Between RE & RL (5) start e e Ms i f 3.(c) If s is a regular expressions and Ms its NFA, s* (Kleene star) has NFA: L = {L(Ms)*} e e where : i is new start state and f is new final state e-move i to f (to accept null string) e-moves i to old start, old final(s) to f e-move old final to old start (WHY?)

28. 3.2: Connection Between RE & RL (6) a b start start • Build an NFA-e that accepts (a|b)*ba a b ba a|b b a start q1 e a e e start e b e

29. 3.2: Connection Between RE & RL (7) • Build an NFA-e that accepts (a|b)*ba (a|b)* e a e e e e e b e e

30. 3.2: Connection Between RE & RL (8) • Build an NFA-e that accepts (a|b)*ba e a e e e e e b e e e b a e

31. 3.2: Connection Between RE & RL (9) r13 r5 | r12 r3 r4 r11 r10 ) ( a a r9 r1 r2 r7 r8 | r0 c * r6 * b b c (ab*c) | (a(b|c*)) Decomposition for this regular expression: What is the NFA? Let’s construct it !

32. 3.2: Connection Between RE & RL (10) r3: r0: r2: a b c e e b e r1: e r4 : r1 r2 e e e b e c a e b e c r5 : r3 r4 e e

33. 3.2: Connection Between RE & RL (11) r7: b e e c e r11: a r8: e r6: c b e e e e c e r9 : r7 | r8 e r10 : r9 b e e e a e c e r12 : r11 r10 e

34. 3.2: Connection Between RE & RL (12) r13 : r5 | r12 e a e b e c 2 3 4 5 6 7 e e e 17 1 b 10 11 e e e e e a e c e 8 9 12 13 14 15 16 e

35. 3.2: Connection Between RE & RL (13) a b c S1 S0 S1 S0 S1 S0 b S2 S1 S0 S5 c S4 S3  b S3 S2   S0 S1 S7 S6 c   S5 S4  Let’s try a ( b | c )* 1. a, b, & c 2. b | c 3. ( b | c )*      

36. 3.2: Connection Between RE & RL (14)  b S5 S4   a  S1 S0 S2 S3 S9 S8 c   S7 S6  b | c a S1 S0 4. a ( b | c )*  

37. 3.2: Connection Between RE & RL (15) NFA’s : start a 1 2 start a b b 3 4 5 6 a b start 7 8 b Let : a abb a*b+ 3 patterns

38. 3.2: Connection Between RE & RL (16) 1 0 NFA for : a | abb | a*b+  a 2  a b b start 3 4 5 6 a b  7 8 b

39. Regular Expression to NFA-e (a | ba)*a

40. First Parsing Step concatenate (a|ba)* a

41. Second Parsing Step concatenate * a a|ba

42. Third Parsing Step concatenate * a | a ba

43. Fourth Parsing Step concatenate * a | a concatenate b a

44. Identify Leaf Nodes concatenate + a | a concatenate b a

45. Convert Leaf Nodes a a a b concatenate * | concatenate

46. Identify Convertible Node(s) a a a b concatenate * | concatenate

47. Convert Node a a b a concatenate * | e

48. Identify Convertible Node a a b a concatenate * | e

49. Convert Node a a a concatenate * e e b e

50. Identify Convertible Node a a a concatenate * e e b e