Examination
Formal Languages and Automata Theory
TDDD14 (Formella Språk och Automatateori)
2012–01–14, 08.00 – 12.00
1. Allowed help materials
• A sheet of notes - 2 sided A5 or 1 sided A4.
The contents is up to you.
The notes should be signed in the same way as the exam sheets
and returned together with the exam.
• English dictionary
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Tillåtna hjälpmedel:
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• Ett papper med valfria anteckningar - 2 sidor A5 eller 1 sida A4.
Anteckningarna ska signeras på samma sätt som tentamensarken och bifogas tentamen vid inlämnandet.
• Engelsk ordbok
2. You may answer in Swedish or English.
3. Total number of credits is 32:
3: 16 p, 4: 21 p, 5: 26 p.
4. Jour (person on duty): Jonas Wallgren.
GOOD LUCK !
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Make sure that you justify your answers! Unexplained answers will be
granted 0 points. (For example, assume that you are writing a grammar for
a given language. Then you should also explain that the grammar indeed
generates the language).
1. (2p) Using a standard method
construct a DFA equivalent to the
NFA given by the table. (Its set
of states is Q = {1, 2, 3}, the input
alphabet Σ = {a, b, c}, the start
state is 1, and 3 is the only final
state.)
ε
a
b
c
→1
{2, 3} ∅ {2} {3}
∅
{1} {3} {1, 2}
2
3F
∅
∅
∅
∅
2. (2p) (a) Consider regular expressions
α = (a + b)(ab + ba)∗
∗
and β = (ab)∗ (ba)∗ (a + b).
Give an example of a string in L(α)\L(β), and of a string in L(β)\L(α),
if such string exists. (If a required string does not exist, you should
explain that this is indeed the case.)
(b) The same for
∗
γ = b(ab + aab)∗ b and δ = ba + baa bb.
Remember that \ denotes the set difference.
3. (2p) Consider the NFA given by the table. Using a standard method,
construct a regular expression defining the same language. (The set
of states of the NFA is Q = {A, B, C, D}, the input alphabet Σ =
{0, 1, 2, 3}, the start state is A, and C is the only final state.)
0
1
2
3
→A
{B, C} ∅
∅
∅
B
∅
{A} {D} ∅
CF
∅
∅
∅
{A}
D
{D}
∅ {C} ∅
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4. (2p) Using a standard method,
construct the minimal DFA equivalent to the DFA given by the table. (Its set of states is Q =
{A, B, C, D, E, F, G, H}, the input alphabet Σ = {0, 1}, the start
state is A, and {C, F } is the set of
final states.
→A
B
CF
D
E
FF
G
H
0
1
B
C
D
E
F
A
H
F
E
F
H
H
C
B
B
C
5. (2p) Construct a context-free grammar for the language
L = { am+n bm+k ck+j dl+n | j > l, m, n, k, l ≥ 0 }
Make it clear what are the sets generated by the nonterminal symbols
of your grammars, and what is the role of each production.
6. (6p) For each of the following languages answer whether it is regular,
context-free but not regular, or not context-free. A brief, informal explanation is sufficient.
(a,b) L1 is the language generated by the context-free grammar
S → ASB | A → a | aa
B→b|c|
and L2 is generated by
S → ASB | d
A→a|b
B → bB | cB | .
In the grammars, S, A, B are nonterminal symbols, a, b, c, d are
terminal symbols, and S is the start symbol.
(c) L3 = { xei y | x ∈ {a, b}∗, y ∈ {c, d}∗, |x| > |y|, |x| < 2|y|, i ≥ 3 },
(d) L4 is the image of L3 under the homomorphism h : {a, b, c, d, e}∗ →
{0, 1, 2}∗ such that
h(a) = 01,
h(b) = h(c) = ,
h(d) = 11,
h(e) = 2.
(e) L5 = { xei y | x ∈ {a, b}∗ , y ∈ {c, d}∗ , |x| > i, i < 2|y| },
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7. (1p) Choose a grammar from problem 6 (a,b) and explain whether it
is ambiguous.
Comment: If it turns out to be unambiguous then an informal explanation is sufficient.
8. (3p) Prove that a language
L6 = { a(bj a)j | j > 0 }
is not regular or that it is not context-free. Use the appropriate pumping lemma or employ reasoning similar to the proof of the lemma.
9. (2p) Explain briefly the notion of a universal Turing machine (UTM).
What is the language accepted by it? To which class of languages does
it belong?
10. (3p) We say that a state of a Turing machine M is useless if it is never
reached in any computation of M .
We say that a Turing machine M writes A on some input (where A is
a symbol from the tape alphabet of M ) when M writes A on the tape
at some step of some computation of M . (Assume that this includes
writing A into a cell already containing A.)
Show that the problem “a state q of a Turing machine M is useless” is
undecidable. Use the fact that it is undecidable whether a given Turing
machine M writes a given symbol A on some input.
In other words, show that the language of the considered problem, this
means
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Turing machine M
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,
US = hM, qi does not reach state q
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on any input
is not recursive, using the fact that the language
Turing machine M
WP = hM, Ai on some input writes A
is not recursive.
11. (3p) Which of the following statements are true, which are false?
Justify your answers.
(a) There exists a homomorphism h which makes all the strings longer
(this means that |x| < |h(x)| for each string x over the appropriate
alphabet, where |w| is the length of string w).
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(b) If L1 and L2 are a regular languages and L1 = L2 L3 then L3 is a
regular language.
(c) There exists an ambiguous context-free grammar which is LR(1).
12. (1p) Consider two context-free grammars
S → Ac
A → | aAb | b
S → Ab
A → | aAb | b
(S is the start symbol of the grammar, a, b, c are the terminal symbols,
and A is a nonterminal symbol.) One of them is LL(1) and the other
is not. Explain which.
13. (3p) In an attempt to construct LR parsers for certain grammars,
we applied the standard method of constructing a DFA for the viable
prefixes of a grammar. Some fragments of the obtained DFA’s are given
below.
Complete the missing items in the given states, the missing lookahead
sets and the missing symbols labelling the arrows. In each case answer
the following questions. Justify your answers.
• Does the fragment of a DFA satisfy the conditions for the grammar
to be LR(0)?
• The same question about the conditions for LR(1).
You may skip adding missing items or lookahead sets if they are not needed to
answer the questions. For instance if you find the items in some state to violate the
LR(1) conditions then you do not need to complete the other states.
a, b, (, ), + are terminal symbols and S, A are nonterminal symbols of the grammars;
S is the start symbol.
(a)
S → • Ab { $ }
S → •A {$}
A
a
−→
←−
The productions of the grammar are S → Ab | A, A → aAb | a.
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(b)
|
↓
A → ( • A)
{ +, ) }
A → ( • A+A) { +, ) }
←−
−→
y
The productions of the grammar are A → b | (A) | (A + A) and
S → A.
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