1: [1,5

(1)

LIACS

Leiden University

Examination Fundamentele Informatica

2

13 January 2016,10:00

-

13:00

Question

1:

_[1,5

pointsl

a) Give a

deterministic

finite

automaton recognizing

the

language

Ll : _{ * e _{a,b}*

_|

bb is

a subshing of

x _).

b)

Give a deterministic

finite

automaton recognizing the language

I-z: _{ x e _{a,b}*

_Ibu is not a substring of

x).

c)

Use ^the

product

construction

to

give a deterministic finite automaton recognizing the language Lr \ Lz.

Question 2:

Construct aminimal deterministic finite automaton equivalent to the following one:

[1,5

pointsl

b a

Question

3:

_[2

pointsl

Find ^aregular expression corresponding to each of the

following

subsets

of

_{a,b}

*

_:

a)

The language of all strings containing exactly two b's

b)

The language of all strings containing an even number of

b's

c)

The language of all strings not containing the substring bb

d)

The language of all strings in which every b is followed immediately by aa.

Question

4:

_[2points]

a)

Usethepumping lemma to show thatthe language

L: _{

^w1cw2

_lw e _{a,b}*

and

lwll :

_lw2l_}

over the alphabet {

a,b,

c

}

is not regular.

b)

Give apushdown automaton accepting by empty stackrccognizingthe above language L. Use only one single stack symbol X, which must necessarily be the

initial

stack symbol.

Question

5:

_[1,5

pointsl

a)

Give a regular grammar generatingthe

languageL:

_{

x e _{{a,b}* |}

^x^ends

_{withbb }.}

b)

Construct a non-deterministic finite automata corresponding to your regular grammar above.

c)

Give an example of an ambiguoøs context free grammar and show why

it

is so.

Question

6:

_[1,5

pointsl

Give context free grammars generating the

following

languages over the alphabet { ^a,^{b, c}

}:

a)

L1

:

_{aob-_ckl_{0 < n}

(

_{m andk}_>_0}

b) Lz:

_lanb-ck| 0 < n+2m <

k)

c) ^L3:L1uL2

b

The final score is given by the sum of the points obtained.

1: [1,5

LIACS

Examination Fundamentele Informatica

-

1:

pointsl

a) Give a

finite

the

Ll : { * e {a,b}*

bb is

x ).

b)

finite

I-z: { x e {a,b}*

x).

c)

product

to

pointsl

3:

pointsl

following

of

*

a)

b)

b's

c)

d)

4:

a)

L: {

lw e {a,b}*

lwll :

a,b,

}

b)

initial

5:

pointsl

a)

languageL:

x e {a,b}* |

withbb }.

b)

c)

it

6:

pointsl

following

}:

a)

:

(

b) Lz:

k)

c) L3:L1uL2

Ll : _{ * e _{a,b}*

x _).

I-z: _{ x e _{a,b}*

L: _{

_lw e _{a,b}*

x e _{{a,b}* |}

_{withbb }.}

c) ^L3:L1uL2