Cursus: M1, computer-science Code UE: JEIN8602

Cursus: M1, computer-science Code UE: JEIN8602

Subject: Formal languages theory Date: 20 March 2016

Duration: 3H

Documents: authorized

Lectures by: Mr G´eraud S´enizergues

The exercises are independant one from each other.

It is not required to solve all the exercises. Every correct solution to an exercise will give (around) 4 points.

Exercice 1 [/4] We consider the finite automaton A described on figure 1. Note that 0 is

0

g c

e

f

1 b 2 3

a d

Figure 1: finite automaton A the only initial state and 3 is the only final state.

0- Describe accepting computations of A over the words:

abd, agabd, af bd, abcbd, e.

1- Construct a regular expression for the language L

recognized by the automaton A.

Explain the successive steps of your construction.

We consider the monoid homomorphism h : {a, b, c, d, e, f, g}

→ {a, b}

defined by:

h(a) = a, h(b) = ba, h(c) = abb, h(d) = b, h(e) = bb, h(f) = aa, h(g) = b.

2- Give a regular expression for the language h(L

).

3- Construct a finite automaton recognizing the language h(L

).

Exercice 2 [/4] Let us consider the regular expression:

e := (((ab) ∪ (bc))

a)

c

Construct, by Glushkov’s method, a finite automaton recognizing L

.

Exercice 3 [/4] We consider the finite automaton B described on figure 2. Note that 0 is

0

1

2

3 4

5 6

a

a a c

a

b c

a b b a, b

Figure 2: finite automaton B the initial state and 5 is the only final state.

0- Check that the following words belong to L

:

acaaa, acbaa, baa, bcbaa

and that the following words do not belong to L

:

acc, bcc, bcab, acbaaa.

1- Is B deterministic ?

2- Is every state of B accessible ?

3- Is B complete ? Is B minimal ? If not, transform B into a minimal deterministic complete automaton C that recognizes the same language.

Exercice 4 _[/5]

1- For each of the following languages over the terminal alphabet {a, b, c}, construct a context- free grammar that generates the language L

:

L

:= {a

bc

| n ≥ 1}

L

:= {(ab)

c

| p ≥ q ≥ 1}

L

:= {uc u ˜ | u ∈ {a, b}

}

we recall that ˜ u designates the “ mirror-image” (also called “ reversal”) of the word u: for example, if u = abbabbb then ˜ u = bbbabba.

Let us consider the language

L

:= {ucv | u, v ∈ {a, b}

, v 6= ˜ u}.

2

2- Check that the following words belong to L

:

abacaaa, babacab, bacabab.

3- Let w be some word in {a, b}

c{a, b}

. 3.1- Show that if :

w = v

xv

cv

yv

for some v

, v

, v

, v

∈ {a, b}

, x, y ∈ {a, b} such that v

= ˜ v

, x 6= y, then w ∈ L

.

3.2- Show that if :

w = v

xv

cv

for some v

, v

, v

∈ {a, b}

, x ∈ {a, b} such that v

= ˜ v

, then w ∈ L

.

4- Let w ∈ L

. Does w necessarily have one of the two forms described in 3.1 or 3.2 ? If not, describe the missing possible forms of w.

5- Construct a non-ambiguous context-free grammar generating the language L

.

Exercice 5 [/5] We consider the context-free grammar G := (A, N, R) where A = {a, b, c}, N = {S

, S

, S

, S

, S

, S

} and R consists of the following 15 rules:

S

→ aS

S

S

→ bS

S

S

→ aS

S

→ aS

S

S

→ bS

c

S

→ bS

S

→ aS

S

S

→ cS

S

→ bS

S

S

→ aS

S

S

→ b S

→ aS

S

S

→ bS

S

→ aS

.

S

→ aS

.

The start symbol of G is S

.

1- What are the productive non-terminals of G ? 2- What are the useful non-terminals of G ?

3- Transform the grammar G into an equivalent grammar G

where every non-terminal is productive and useful.

4- Is the language L(G, S

) empty ? 5- Is the language L(G, S

) infinite ? 6- Is the language L(G, S

) rational ?

3

Exercice 6 _[/4] We consider the context-free grammar G := (A, N, R) where A = {a, a, b, ¯ ¯ b}, N = {S} and R consists of the following rules:

r1: S → aS¯ a r2: S → ¯ aSa r3: S → bS ¯ b r4: S → ¯ bSb r5: S → SS r6: S → ε

The start symbol of G is S.

1- Show that abaa¯ a¯ a ¯ b¯ a ∈ L(G, S). Give a leftmost derivation from S to abaa¯ a¯ a ¯ b¯ a.

Give a rightmost derivation from S to abaa¯ a¯ a ¯ b¯ a.

2- Show that a¯ ab ¯ b ∈ L(G, S). Give a leftmost derivation from S to a¯ ab ¯ b.

Give a rightmost derivation from S to a¯ ab ¯ b.

Give a derivation-tree for a¯ ab ¯ b.

(The student can choose, among the various notions of syntax-tree, abstract syntax-tree, etc...

his/her favorite notion and compute the corresponding tree).

3- Show that a¯ aa¯ a ∈ L(G, S).

Give two different derivation-trees for a¯ aa¯ a.

4- Is the grammar G ambiguous ?

5- Let G

be the context-free grammar obtained from G by removing rule r5.

Is G

ambiguous ? Does G

generate the same language as G ?

4