Automaten und formale Sprachen

Universit¨ at Duisburg-Essen SS 2012 Ingenieurwissenschaften / Informatik April 16, 2012

Professor: Dr. Sander Bruggink Exercise sheet 1

Teaching assistent: Jan St¨ uckrath Deadline: 23rd April 2012

Automaten und formale Sprachen

Exercise 1 Set notation (7 points)

(a) Determine the power set P (M

) of the following sets:

1) M

= {a, (a, b), b} (1 p)

2) M

= {1} × {1, 2} (1 p)

3) M

= M

∩ M

(1 p)

(b) Give the following sets by means of the set notation presented in the lecture:

1) The set of all even, natural numbers. (1 p)

2) The set of all subsets of integers, which have an odd number of elements. (1 p)

3) The set of all 3-tuples of integers, where the sum of all three integers is less than

zero. (1 p)

4) The set of all pairs (2-tuples), where both elements are subsets of the natural numbers and the first element is a subset of the second element. (1 p)

Exercise 2 The art of proving (7 points)

In the following multiple mathematical statements are listed. Prove or contradict each state- ment, as shown in the lecture.

(a) Prove the follwoing statements:

1) For all sets A, B it holds that: P (A) ∩ P (B) = P (A ∩ B ). (3 p)

2) The relation < (really less) is no ordering relation. (1 p)

(b) Contradict the following statements:

1) For all sets A, B it holds that: P (A) ∪ P (B) = P (A ∪ B ). (1.5 p)

2) There are two natural numbers n 6= m, such that n ≤ k and m ≤ k hold for all natural numbers k, i.e. there are two smallest natural numbers. (1.5 p)

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Exercise 3 Adventure (Level 1) (6 points) You are an adventurer on a treasure hunt and you are given the map below, over the alphabet Σ:

Dragon (d), Arch (b), Door (t), Sword (w), Treasure (a), River (f ) and Key (l). Your mission is as follows: You start at the start state and you have to reach the end state, where the following conditions must be fulfilled.

(S) You must find two treasures. If you reach the same treasure field several times, you gain a new treasure every time.

(T) You can only go through doors, if you have found a key before. Every key applies to every door.

(D) If you meet a dragon, you must jump into a river directly afterwars, otherwise the dragon will set you on fire. This is not needed anymore, once you have found a sword, because you can kill the dragon early enough in this case. If you reach the same dragon field several times, there is a new dragon every time.

1 2 3 4 5 6

10 14

7 8 15

9 13 11 12

(a) Explain, why the following words cannot be a solution for the given adventure (i.e. why does these words not fulfill the conditions (S), (T), (D) respectively). (1.5 p)

• (tdf aad)

• (bdf ltwad)

•

(bdf ltdf aadf )

(b) Find all shortest solutions for the given adventure. A solution is a shortest one, if there is no other solution with a fewer number of symbols. (1 p)

(c) Does the given adventure have finitely or infinitely many solutions? Justify your answer!

(2 p)

(d) Give an own adventure that has at least one solution and where the treasure rule (S) is satisfied on all paths from an initial state to a final state. (1.5 p)

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The solutions to this exercise sheet must be submitted before Tuesday, 23rd April 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. 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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