Universit¨at Duisburg-Essen SS 2012
Ingenieurwissenschaften / Informatik June 5, 2012
Professor: Dr. Sander Bruggink Exercise sheet 6
Teaching assistent: Jan St¨uckrath Deadline: 11 June 2012
Automaten und formale Sprachen
Exercise 16 Pumping Lemma for advanced learners (6 points) Show by means of the Pumping Lemma for regular languages, that the following languages are not regular:
(a) L1 ={anbm |n, m∈N0∧n ≤m} (3 p)
(b) L2 ={anbman|n, m∈N0} (3 p)
Exercise 17 Regular languages and Myhill-Nerode equivalence (8 points) Show by means of the Myhill-Nerode Theorem, whether the following languages over the al- phabet Σ ={a, b} are regular or not:
(a) L1 ={anbm |n, m∈N0∧n ≤1≤m} (3 p)
(b) L2 ={ambn |n, m∈N0∧0< m < n} (2 p)
(c) L3 ={a`bman |n, m, `∈N0∧(` ≤m∨`≤n∨m≤n)} (3 p)
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Exercise 18 Equivalence of regular languages (6 points) Let the following deterministic finite automata M1 and M2 be given:
M1 : s2
s1
s3
s4
s5
a, b
a b
a b
a, b
a
b
M2: t1
t2
t4 t3
t6
t7
t5 t8
a
a, b b
a b
a
b
a b
a
b a
b a
b
Check if both deterministic finite automata are equivalent. Two finite automata are equivalent, if the following holds:
T(M1) = T(M2).
First of all construct the minimal automata ofM1 andM2 by means of the algorithm presented in the lecture and motivate with the aid of the minimal automata, whyM1 and M2 are (not) equivalent.
(Note: Indicate all intermediate steps of the algorithm. Submissions without intermediate steps donot achieve points!)
The solutions to this exercise sheet must be submitted before Monday, 11 June 2012 at 16:00. Put your solutions in the letterbox labeled Automaten und formale Sprachen adjacent to room lf, or hand them in through the online moodle-platform. If you hand in online, please upload your solutions as a single pdf-file. Your name, student number, group number and the lecture name (“Automaten und formale Sprachen”) must be clearly written on your solutions.
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