SUBJECT: DISCRETE STRUCTURE & APPLICATIONS
CODE: BCT2083
FACULTY OF INDUSTRIAL
SCIENCES & TECHNOLOGY
TOPIC: Chapter 5 Modeling Computation
TUTORIAL 5
DURATION: 2 weeks (week 14-15)
Week 14
1.
Suppose A  01. Describe all strings belonging to A.
2.
Suppose a phrase-structure grammar has productions S  S0, S  A1, A  0. Find a
derivation of 01, 0100, and 010.
3.
Suppose a phrase-structure grammar has productions S  1S0, S  0A, A  0. Find a
derivation of 00, 1000, and 110000.
4.
Suppose a phrase-structure grammar has productions S  S11, S  0A, S  A1, A  0. Find a
derivation of 01, 0011 and 011111.
5.
Let G be the phrase-structure grammar with vocabulary V  ABabS, terminal element set T
 ab, start symbol S, and production set P  S  ABaS  BaA  aBAB  bB  ab.
Which of these are derivable from ABa?
a) ba
6.
babaa
d) b
e) aababa
f
b) aab
c) bba
Let G be the phrase-structure grammar with vocabulary V  ABabS, terminal element set T
 ab, start symbol S, and production set P  S  ABaS  BaA  aBAB  bB  ab.
Which of these are derivable from S?
a) ba
8.
c) aba
Let G be the phrase-structure grammar with vocabulary V  ABabS, terminal element set T
 ab, start symbol S, and production set P  S  ABaS  BaA  aBAB  bB  ab.
Which of these are derivable from A?
a)
7.
b) abb
b) ab
c) baab
d) aababa
e) aba
Let G be the phrase-structure grammar with vocabulary V  AB01S, terminal elements T 
01, start symbol S, productions P  S  AB0AB  1A  0B  AB. Which of these are
derivable from S?
a) 000
b) 11 c) 0000
d) 110
e) Both 010 and 0111
9.
The productions of a phrase-structure grammar are S  S1, S  0A, A  1. Find a derivation
of 0111.
10.
What language is generated by the phrase-structure grammar if the productions are S  S11,
S  , where S is the start symbol?
11.
What is the language generated by the grammar with productions S  SA, S  0, A  1A, and
A  1, where S is the start symbol?
BCT2083
12.
Tutorial 5
In the questions below let V  SAB01 and T  01. For each set of productions determine
whether the resulting grammar G is
(i)
(ii)
(iii)
(iv)
a)
b)
c)
d)
e)
f)
type 0 grammar, but not type 1,
type 1 grammar, but not type 2,
type 2 grammar, but not type 3,
type 3 grammar.
S  A10, AB  0.
S  B, A  B, B  A.
S  AB, A  0B1, 0B1  0.
S  1A, A  1, S  .
S  1AB, AB  0B, B  0, A  1B.
S  0B, B  1A, B  0, A  0B.
13.
Let G = (V, T, S, P), where V = {a, c, A, B, S}, T = {a, c}, S is a start symbol and productions P
= {S → ABc, A → BB, B → ac, A → Bc}. If word w = acacacc belongs to the language
generated by this productions, construct the derivation tree for w. Then, write the language of
this grammar.
14.
The production rules of a grammar for simple arithmetic expression are:
‹ expression › ::= ‹ digit ›| (‹ expression ›) | + (‹ expression ›) |
− (‹ expression ›) | ‹ expression › ‹ operator › ‹ expression ›
‹ digit › ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
‹ operator › ::= + | − | * | / | ↑


Construct a derivation tree for arithmetic expression 5  2  3 and 8   2 / 3 .
15.
Draw the state diagram for the Finite State Machine with the following state table.
f
input
State
s0
s1
s2
s3
16.
0
s0
s2
s2
s1
g
input
1
s1
s2
s0
s2
0
1
1
0
1
1
0
0
1
1
Construct a state table for the Finite State Machine with the following state diagram
x,0
x,1
x,1
y,0
s0
s1
s2
y,0
y,1
2
BCT2083
17.
Tutorial 5
Given the finite state machines with the following state table. Determine the output string if the
input strings is 01101.
f
input
State
0
s0
s2
s2
s0
s1
s2
g
input
1
s1
s2
s0
0
a
a
b
1
b
a
a
18.
What is the output produced by the following finite-state machine when the input string is
11101?
19.
In the questions below determine the output for each input string, using this state table.
s0
s1
s2
s3
a) 1111. b) 10111.
f
input:
0
s1
s2
s1
s3
c) 000.
g
input:
1
0
s2
1
s0
0
s3
1
s1
0
1
1
0
1
1
d) 11000.
Week 15
20.
Suppose that A  11101 and B  010. Find AB.
21.
Suppose that A  11101 and B  010. Find BA.
22.
Let A  110. Which strings belong to A?
23.
Let A  011. Find A2 and A3.
24.
Find the Kleene closure of A  1.
25.
Find the Kleene closure of A  00.
26.
Find the Kleene closure of A  012.
3
BCT2083
Tutorial 5
27.
Find the set recognized by the following deterministic finite-state machine.
28.
Find all strings recognized by the following deterministic finite-state automaton.
29.
Find all strings recognized by the following deterministic finite-state automaton.
30.
Find the language recognized by the following nodeterministic finite-state automaton.
31.
Find the language recognized by the following nodeterministic finite-state automaton.
4
BCT2083
Tutorial 5
ANSWERS CHAPTER 5: Modeling Computation
1.
A consists of all strings of 0s and 1s, including the empty string.
5.
6.
7.
8.
a and e
b
a, d, e
e
10.
The language generated is the set of all strings consisting of an even number of 1s and
no other symbols.
11.
The set of all bit strings that consist of a 0 followed by an arbitrary number of 1s.
12.
a) i.
13.
L(G) = { amc4, m positive integers}
17.
ababa
18.
10000.
19.
The strings in A are those in which each 0 is preceded by at least one 1.
20.
AB  1011011100100110.
21.
BA  0101100110110111001.
22.
a) 1110
b) iii. c) i.
d) iv. e) ii. f) iv.
b) 11011
c) 101
d) 11000
 000111101111and 000001101111011011001101111110111111.
 1n  n  012….
 02n  n  012….
26.
All strings of 0s, 1s, and 2s.
27.
The set represented by (01).
28.
All bit strings with no 1s.
29.
All bit strings with an even number of 1s.
 00010.
 101n01n0  n  0.
5