Examination
Formal Languages and Automata Theory
TDDD14
(Formella Språk och Automatateori)
2012–10–24, 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 31:
3: 16 p, 4: 21 p, 5: 26 p.
4. Jour (person on duty): Wlodek Drabent, tel. (013 28) 89 29.
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) Consider the NFA whose
transition function is given by
the table. (Its set of states
is Q = {A, B, C, D, E, F }, the
input alphabet Σ = {0, 1}, the
start state is A and the final state are D, F .) Using a
standard method construct an
equivalent DFA.
1
2
1
→A
{B, E}
∅
∅
∅
{B, C} {B}
B
C
∅
{D}
{D}
∅
∅
∅
DF
E
∅
{E} {E, F }
∅
∅
∅
FF
2. (2p)
0
0
3
→ A F {B} {C}
{D} {A}
B
C
{B} {A}
D
{C}
3. (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}, the input
alphabet Σ = {0, 1}, the start
state is A, and {F, G} is the set
of final states.)
Using a standard method, construct a regular expression defining the same language as the
given NFA. (Its set of states is
Q = {A, B, C, D}, the input alphabet Σ = {0, 1, 2, 3}, the start
state and the final state is A;
the missing positions of the table
stand for ∅.)
→A
B
C
D
E
FF
GF
0
1
B
C
D
E
C
B
F
F
G
G
G
G
G
G
4. (2p) Let L be the language accepted by the DFA from the previous
problem. Consider the relation ≡L on strings over Σ = {0, 1}, defined
by
x ≡L y ⇔ ∀z ∈ Σ∗ (xz ∈ L ⇔ yz ∈ L).
How many equivalence classes does ≡L have? Why? Choose one of the
equivalence classes, and give a DFA defining it.
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5. (1p) Construct an ambiguous context-free grammar. The grammar
should have at least two nonterminal symbols, and neither -productions
nor unit productions.
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.
Remember that wR denotes the string w reversed, and #a(w) denotes the number
of occurrences of symbol a in w.
(a) L1 =
∗ #a(w) is even, #b(w) ≡ 1 mod 3
w ∈ {a, b, c, d} .
w does not have a substring abc
(b) L2 = { w ∈ {a, b, c, d}∗ | #a(w) is even, #b(w) < #c(w)} .
(c) L3 = { w ∈ {a, b, c, d}∗ | #a(w) < #b(w) < #c(w)} .
(d) L4 = { w ∈ {a, b, c, d}∗ | #a(w) < #b(w) < #c(w) < 6 } .
(e) L5 = { wR x ∈ {a, b, c, d}∗ | w = 2(x) }, where 2(a1 a2 · · · an ) =
a1 a1 a2 a2 · · · an an , for any n ≥ 0 and a1 , a2 , . . . , an ∈ {a, b, c, d}∗
(i.e. each symbol ai is replaced by ai ai ).
7. (5p)
One of the languages
L6 = { a p b q+r c s d q+t e p+r | p, q, r, t ≥ 0, s > t },
L7 = { a p+q b r c s d q+r e s | p, q, r, s ≥ 0 }
is context-free and the other is not. Build a context-free grammar for
the one that is. For the other one provide a proof that it is not regular,
or that it is not context-free.
For the grammar, explain the set generated from each its nonterminal
symbol, and the role of each production. For the proof, use the appropriate pumping lemma or employ reasoning similar to the proof of the
lemma.
Hints: Begin from finding the context-free language. Confirm that it
is context-free by constructing a grammar. In the proof it is useful to
find a simple string to pump.
8. (3p) Which of the following statements are true, which are false?
Justify your answers.
(a) Let NFA M 0 be obtained from an NFA M by swapping its final and
non-final states: M = (Q, Σ, ∆, S, F ) and M 0 = (Q, Σ, ∆, S, Q\F ).
Then L(M 0 ) = Σ∗ \ L(M ).
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(b) There exists a language L such that for every homomorphism h
the language h(L) is not regular.
(c) If languages L1 , L2 are recursively enumerable then LR
1 ∩ L2 is
recursively enumerable.
9. (1p) Explain briefly the notion of the language of a decision problem.
Hint: The notion is used below.
10. (3p) Show that the problem “a Turing machine M rejects string x” is
undecidable. Use the fact that the halting problem is undecidable (i.e.
it is undecidable whether M halts on input x).
In other words, show that the language of the problem
RP = { hM, xi | Turing machine M rejects input x }
is not recursive, using the fact that the language
HP = { hM, xi | Turing machine M halts on input x }.
is not recursive.
11. (1p) Consider a grammar with productions
S → aB | bA | A → aS | bAA
B→b
(where a, b are terminal symbols, and S is the start symbol). Explain
why it is not LL(1).
12. (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).
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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 the terminal symbols and S, A, B, C are the nonterminal symbols of the
grammars; S is the start symbol.
(a)
|
↓
C → b•A {$}
A
b
←−
−→
The productions of the grammar, starting with C and A are
C → aB | bA, A → aC | bAA
(b)
A → b • AA { a, b }
−→
a
a
−→
−→
The productions of the grammar, starting with C and A are
C → | aB | bA, A → aC | bAA.
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