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Give a regular expression for the language on ∑={a,b,c} containing no run of a’s of length greater than two.17.. Give a regular expression for the language on ∑={a,b} containing all str

1 Let L = {ab,aa,baa} Which of the following strings are in L*:

4 Give dfa’s for the sets consisting of

a all strings with exactly one a.

b all strings with no more than three a’s

5 Give a dfa for the language L = {ab5wb4: w ∈ {a,b}*}

6 Find dfa’s for the following languages on ∑ = {a,b}

8 Show that the language L = {a n : n ≥ 0, n ≠ 4} is regular

9 Find δ*(q0, a) and δ*(q1,λ) for the following nfa

10 For the following nfa, find δ*(q0,1010) and δ*(q1,00)

11 Find an nfa with three states that accepts the language {ab,abc}*

12 Convert the following nfa into dfa

13 Convert the following nfa into dfa

14 Find all strings in L((a+b)*b(a+ab)*) of length less than four.

15 Find a regular expression for the set {a n b m : (n+m) is even}

16 Give a regular expression for the language on ∑={a,b,c} containing no run of a’s of length greater than two.

17 Give a regular expression for the language on ∑={a,b} containing all

strings not ending in 01

18 Give a dfa that accepts L((a+b)*b(a+ab)*)

19 Find regular expressions for the language accepted by the following automaton:

20 Construct a dfa that accepts that language generated by the following grammar

S  abA

A  baB

B  aA | Bilbo

21 Construct left-linear grammar for the language in Problem 8

22 Construct right-linear and left-linear grammars for the language L = {a n b m ,n≥2,m≥3}

23 Construct a dfa that accepts that language generated by the following grammar

S  abA

A  baB

B  aA | bb

24 Construct left-linear grammar for the language in Problem 2

25 Construct right-linear and left-linear grammars for the language L = {a n b m ,n≥2,m≥3}

26 Find intersection of the two following languages: L1 = {a n b 2m c 2k } and L2

= {a 2n b 4m c k}

27 The nor of two languages is defined as follows:

nor(L1,L2) = {w: w ∉L1 and w ∉L2 }

Show that nor is closed for regular lanaguages

28 Let L1 = L(a*baa*) and L2 = (aba*) Find L1/L2

29 Suppose L1∪L2 is regular and L1 is finite Can we conclude that L2 is regular?

30 Prove that the following languages are not regular

32 Show a derivation tree, together with the corresponding leftmost and

rightmost derivations of the string aabbbb with the grammar

S  AB | λ

A  aB

B  Sb

33 Find a context-free grammar for the set of all regular expressions on the

alphabet {a,b}* Give a derivation tree for (a+b)*+a+b

34 Find an s-grammar for L(aaa*b+b)

35 Show that every s-grammar is unambigous

36 Show that the following grammars is ambigous

S  AB | aaB

A  a | aA

B  b

S  aSbS|bSaS|λ

37 Is it possible for a regular grammar to be ambigous?

38 Find an s-grammar for L(aaa*b+b)

39 Show that every s-grammar is unambigous

40 Consider the following grammar:

S  aSb | SS | λ

Eliminate left-recursive for the grammar

Show a derivation for the string w = aabbab using both orignal and rewriten

43 Transform the following grammars to Chomsky normal form:

51 Show that L = {a n b 2n} is a deterministic context-free language

52 Let L = {ab,aa,baa} Which of the following strings are in L*:

abaabaaabaa, aaaabaaaa, baaaaabaaaab, baaaaabaa?

54 Which of the strings 0001, 01001, 0000110 are accepted by thefollowing nfa:

0001

01001

55 Give dfa’s for the sets consisting of

a all strings with exactly one a.

b all strings with no more than three a’s

56 Give a nfa for the language L = {ab5wb4: w ∈ {a,b}*}

57 Find dfa’s for the following languages on ∑ = {a,b}

a L = {w: |w| mod 3 = 0}

b L = {w: |w| mod 5 ≠ 0}

c L = {w: n a (w) mod 3 > 1}

58 Consider the set of strings on {0,1} in which every 00 is followedimmediately by 1 For example 101, 0010, 0010011001 are in the language,but 0001 and 00100 are not Construct an accepting dfa

59 Show that the language L = {a n : n ≥ 0, n ≠ 4} is regular

60 Find δ*(q0, a) and δ*(q1,λ) for the following nfa

62 Find an nfa with three states that accepts the language {ab,abc}*

63 Convert the following nfa into dfa

a

b

a

b

Optimize the DFA if you want (read Section 1.3 for DFA optimization):

Thus, the language is, surprisingly, (0+1)*

64 Find all strings in L((a+b)*b(a+ab)*) of length less than four.

Method 1:

A string of the given language can be considered as a sequence of patterns

each of which is either ax, x or aax (when a run of 2 a’s appears, it must be followed immediately by b to make sure that the run stops with no more a) where x is either b or c The sequence can stop exceptionally by either aa or

a.

The corresponding regular expression:

(a(b+c)+(b+c)+aa(b+c))(aa+a+λ)

Method 2:

One can easily construct a DFA for the language as follows:

The required regular expression can then be derived from the DFA, which should be the same as that given from Method 1

67 Give a regular expression for the language on ∑={a,b} containing all

strings not ending in ab

(a+b)*(aa+ab+ba) + b + λ

68 Give a dfa that accepts L((a+b)*b(a+ab)*)

69 Find regular expressions for the language accepted by the following automaton:

74 The nor of two languages is defined as follows:

Since L1 is finite, so is L2 ∩ L1 Hence L2 ∩ L1 is also regular (3).

From (1),(2) and (3), L2 is regular.

Note: If L1 is regular but not finite, we could not conclude anything about L2since the fact (3) above can not be claimed to complete the proof One can

easily find a counter-example showing that whereas L1∪L2 and L1 are

regular, L2 can still be not regular.

77 Prove that the following languages are not regular

a L = {a n b m , n ≤ m}

b L = {w: n a (w) ≠ n b (w)}

Read Linz’s book

78 Find context-free grammars for the following languages

79 Show a derivation tree, together with the corresponding leftmost and rightmost

derivations of the string aabbbb with the grammar

S  AB | λ

A  aB

B  Sb

S => AB => aBB => aSbB => aABbB => aaBBbB => aaSbBbB => aabBbS => aabSbbB

=> aabbbB => aabbbSb => aabbbb

80 Find a context-free grammar for the set of all regular expressions on the alphabet

{a,b}* Give a derivation tree for (a+b)*+a+b

E  E + T | T

T  T.F | F

F  (E) | E*| a | b

The tree is left for students to construct

81 Show that the following grammars is ambigous

S => aSbS => abSaSbS =>* abab

S => aSbS => aSbaSbS =>* abab

Draw the corresponding trees for the above derivations to observe the ambiguities.

82 Is it possible for a regular grammar to be ambigous?

S  aAB | aaB

A  a

B  b

83 Find an s-grammar for L(aaa*b+b)

S aS1 | b

S1  aA

A  aA | b

84 Show that every s-grammar is unambigous

Consider an s-grammar Gs If Gs is ambigous, there is at least a string s = a1 a2…a n ∈

L(G s) of which two distinct derivation tree exits, namely T1 and T2 Let D1 and D2 be the two leftmost derivations of T1 and T2 respectively Since T1 and T2 are distinct, D1 and D2

must be different However, since Gs is an s-grammar, w should have only one unique

leftmost derivation as S=> a1 x1 => a1a 2 x2 =>…=> a1a2…an, hence the conflict.

85 Consider the following grammar:

S  aSb | SS | λ

Eliminate left-recursive for the grammar.

S aSbT | T

T  ST | λ

Show a derivation for the string w = aabbab using both orignal and rewriten grammars

S => SS => aSbS =>aaSbbS => aabbS => aabbaSb => aabbab

S => aSbT => aaSbTbT => aaTbTbT => aabTbT => aabbT => aabbST => aabbaSbTT => aabbabTT => aabbabTT => aabbabT => aabbab

86 Eliminate useless and unit productions for the following grammar:

87 Eliminate all λ -production from

Convert to Chomsky form:

S  A’S 1 | A’S3 | A’S4 | A’B’

(One can realize that A2  a now becomes useless and removable from the grammar)

90 Prove that the following ndpa does not accept any string not in {ww R}

δ(q1,a,a) = {(q1,λ )}

δ(q1,b,b) = {(q1,λ )}

δ(q1,λ,z) = {(q2,λ )}

One can observe that the following move sequences cannot happen on the ndpa:

(q1,x,y) |* (q0,x’,y’) ∀x,x’∈ ∑*, y,y’∈ ∑ *; and

(q2,x,y) |* (q1,x’,y’) ∀x,x’∈ ∑*, y,y’∈ ∑ *.

Thus any string w accepted by the ndpa must be of the sequence form as follows: