1. Homework 4
due 11/25

2. 1. Homomorphism
1. Let A = L(M) where M is the DFA given in the figure, and h be a homomorphism from {a,b}*  {a,b}* with h(a) = ab and h(b) = ba. 1.1 Find a DFA N1 such that L(N) = h(A), 1.2 Find a DFA N2 such that L(N) = h-1(A) Give your answer Using state-transition diagram 2 a b 1 3

3. 2. Pumping lemma and closure properties of regular languages
2. Using (game-theoretical version of ) pumping lemma to show the following two sets are not regular:
A1 = { x x | x  {0,1}* }
A2 = {an bm | 0  n  m  2n+10 }
Using closure properties of regular languages to show that :
2.3 If A1(the language given at 2.1) is not regular, then neither is the language A3={ xcx | x  {0,1}* }
2.4 If the language Prime={ ap | p is a prime number } is not regular, then neither is the language A4 = { ambnct | (m+n+t) is not a prime number }.

4. 3. Which of the following sets are regular ? Give your reason.
3. Testing Regular sets
3. Which of the following sets are regular ? Give your reason.
{an b m | n  0 and m  0 }
{an b m | n  m, n  0 and m  0 }
{ak y | k  0, y  {a,b}* and y contains at least k a’s }
{xcy | x, y  {a,b}* }
{an b m | n +10  m }
{an bm | n > m and m < 20 }
{an b m | n > m > 20 }

5. 4. Elimination of inaccessible states
4. For the given FA, find all states which are inaccessible.
4.1 Draw a state transition diagram for the DFA.
4.2 Draw the state transition diagram for the
resulting DFA after removing of inaccessible states. a b 1 1 3 2F 6 5 7 4F 6F 2 8 4

6. 5. Minimization of DFA a b 1 F 6 4 2F 7 5 3 2 8 1
5. Apply the minimization algorithm to find a minimal DFA equivalent to the right one.
5.1 Draw the chart table
5.2 Mark each cell corresponding to a pair {p,q} of states p,q that are not equivalent with a number indicating the length of a minimal string which witnesses their inequivalence.
5.3 Draw a state-transition table (or diagram) for the resulting DFA and indicate clearly which equivalence class of the old DFA corresponds to each state of the new one. a b
1 F 6 4 2F 7 5 3 2 8 1 2 3 4 5 6 7 8 1

7. 6. Consider the following context free grammar G:
6. CFG and CFL
6. Consider the following context free grammar G:
C  ABC | AB A  Aa | a B  bA | b
6.1 If G=(N, S, S, P), then
N = __________; S = _______________ ;
S = ___________ and P = {________________}.
6.2. Define L(X) = {xS* | X*x}. Find 3 regular expressions
representing L(A), L(B) and L(C), respectively.
6.3 Which of the following strings are in L(G) ?
(a) aabaab (b) aaaaba (c) aabbaa (d) abaaba
6.4 Give a derivation for each of the above strings which are in
L(G).

8. 7.1 A1 = {w | w contains at least three 1s.}
CFG design
7. Give context-free grammars that generate the following languages. Suppose the alphabet of terminals is {0,1}.
7.1 A1 = {w | w contains at least three 1s.}
7.2 A2 = {w | w starts and ends with the same symbol.}
7.3 A3 = { w | the length of w is odd and its middle symbol is 0.}
7.4 A4 = {w | w contains as many 0s as 1s.}
7.5 A5 = {w  {a,b,c}* | #a(w) + #b(w)  #c(w) }.
// #a(w) is the number of a’s occurring in w.
7.6 A6 = {anbnck | 3  k}

9. 8. Let G be the context-free grammar (CFG) given below: S  (S) S | e
8. Verification of CFG
8. Let G be the context-free grammar (CFG) given below:
S  (S) S | e
Assume the alphabet S = {(, )}. Show that
If x  L(G), then x is a balanced parenthesis.
If x is a balanced parenthesis, then xL(G).
Note that x is a balanced parenthesis iff
1. it has the same number of occurrences of ‘(‘ and ‘)’ and
2. for every proper prefix y of x, occurrences of ‘(‘ in y is no less
than occurrences of ‘)’, i.e., L(y)  R(y)).