Transparency No. 2-1

Formal Languageand Automata Theory

Homework 2

Homework

Transparency No. 2-2

DFA definition

1. Given the DFA :

(a) Draw a state transition diagram for the DFA M. (b) Let M = (Q, , , s, F). Then what are the contents of Q, , , S and F, respectively.

Q = _______________________ = _____________________ s =_______________________ F =______________________ = _____________

(c) List all strings of length 3 which are accepted by the machine.

M c d 0 3 11 2 32F 1 23F 3 0

Homework

Transparency No. 2-3

DFA design

Answer problem 2 & 3 using state transition diagram. Note #a(x), where a is a symbol, denotes the number of a's appearing in x.

2. Design deterministic finite automata (FDA) for each of the following sets:(a) the set of strings in {a,b,c}* containing the substring abc;(b) the set of strings in {a}* whose length is divisible by either 3 or 5;(c) the set of strings x in {0,1}* such that #0(x) is odd and #1(x) is a multiple of three;(d) the set of strings over the alphabet {a,b} containing at least three occurrences of three consecutive b's, overlapping permitted (e.g., the string bbbbb should be accepted);(e) the set of strings overt he alphabet {0; 1; 2 } that are ternary (base 3) representations, leading zeros permitted, of numbers that are not multiples of four. (Consider the empty string a representation of zero.)

3. Let = {a,b}. Construct DFA's for the following languages :(a) A1 = { a, ab, ba, bbaa}. (b) A2 = { x | x contains at most two a's, i.e., #a(x) 2}(c) A3 = { xaba | x {a,b}* }(d) A4 = { x b y | |x| 2, |y| > 0 and x,y {a,b}* }

Homework

Transparency No. 2-4

Regular languages are closed under basic set operations4. Use state transition table for your answer.(a) Given the following two DFAs M1 and M2 :

Construct two DFAs M3 and M4 such that 1. L(M3) = {a,b}* - L(M1) and 2. L(M4) = L(M1) U LM(2) -- use the product construction.(b) : Find a DFA M5 for the language B = { x {a,b}* | x contains substring ‘bab' but not

‘bb' } by firstly1. construct two DFA M51 and M52 such that L(M51) = { x {a,b}* | x contains substring

‘bab' } and L(M52) = { x {a,b}* | x contains substring ‘bb' } and then2. use the product construction to construct M5 such that L(M5) = L(M51) - L(M52) = B.

M1 a b

>pF p r

q p q

rF q p

M2 a b

>1 1 2

2F 2 1

Homework

Transparency No. 2-5

NFA to DFA5. Let N1={Q,,,S,F) be an NFA with the following transition table:

(a) : What is the value of Q,,,S and F, respectively?(b) : Draw a state transition diagram for N1.(c) : Final all strings of length 3 accepted by N1.(d) : Find a DFA M1 equivalent to N1. Give your answer using transition table. Remember

to name each state of M1 with the corresponding state subset of N1.

6. Let M = (Q; ; ; s; F) be an arbitrary DFA. Prove by induction on |y| that for all strings x,y * and q Q,(q, xy) = ((q, x), y) where is the extended version of defined on all strings described in ch2.

Q\ a b>0 {0,2} {0}1 {1} {2}

2F {1,3} {3}3F {0} {1}