Solution Exercise 1'43

1
Solution Exercise 1.43

• Input Automaton A accepting L
• Output Automaton AD accepting DROP-OUT(L)
• Process
• 1) Let A be a duplicate of A
• 2) Let q1, , qn be the states in A, then rename
  the states in A as q1, , qn
• 3) For every transition
• ((qk,?),qc) in A
• Construct a new transition
• ((qk,e),qc )
• 4) The automation AD is formed by A and A, plus
  the transitions in (3), with the starting state
  of AD being the starting state of A

a
A
r
r
s
q
q
gt
b
b
b
and removing from the list of favorables all
states in A
2
Solution Exercise 1.60
a
qk1

q
q3
q1
q
q2
gt
a,b
a,b
a
a,b
b
k-1 states
3
Nonregular Languages
L akbk k ?N
Lets show that is regular

• find a regular expression for L, or
• automaton that accepts L

How can we proof that L is nonregular?
4
Pumping Lemma (Theorem 1.70)

• Lemma. Let L be a regular language. There exists
  an integer p gt 0 such that for any string w ? L
  with w ? p, then there exist strings x, y, and
  z such that
• w xyz
• y ? e
• xy ? p
• xyiz ?L for i 0, 1, 2,

5
Sketch of Proof (1)
Lets analyze how words are accepted by finite
automata. Lets take a generic word
w ?1 ?2 ?m
If w is accepted by a DFA D. What does it mean
6
Sketch of Proof (2)
gt
Let p of states in D. If m gt p, what does
this means for these s1, s2, , sm?
z
Thus, w can be split into 3 parts x, y and z
7
Exercises

• Lets proof that
• L akbk k ?N
• is nonregular using the pumping lemma

• Given a word w, wR denotes the reverse of the
  word. Lets proof that
• Palindromes w ? ? w wR
• is nonregular using the pumping lemma (?
  a,b)
• Note In Example 1.73 of the book (read it!)
  only case Number 1is necessary to analyze. The
  other two cases cannot happen because xy ? p
  (the text below the cases states it so too but at
  least one of the selected answers to exercises
  again analyze more cases than it is necessarily
  -e.g., Exercise 1.29 (a) -)

8
Homework (Optional Wednesday!)

• Exercise 1.29 (a), (b)
• Problem 1.46 (b)
• Problem 1.53
• (Hint you can pick any numbers x, y and z as
  long as
• x yz, and
• Viewed as an string, x yz has at least p
  characters
• So what you are trying to do is to take some
  numbers x, y and z so when you pump the
  summation longer adds up)