Exercise 10.1 PDS Modeling

(1)

Applied Automata Theory (WS 2014/2015) Technische Universit¨ at Kaiserslautern

Exercise Sheet 10

Prof. Dr. Roland Meyer, Reiner H¨ uchting Due: Tue, Jan 20

Exercise 10.1 PDS Modeling

Give a pushdown system that models the following program where r() is a function that randomly returns 0 or 1. Describe how you represent the program counter and the value of the variable x. Hint: In a first step, assign each line in the code a unique label.

void m() x=1;

if(r()==1) s();

void s() x=0;

if(r()==1) m();

int x=0;

void main() m();

Exercise 10.2 pre

^∗

Computation

Use the algorithm given in the lecture notes to determine pre

^∗

(C) in the pushdown system below, where C = {(q, w) | w ∈ {γ, δ}

^∗

and |w| is even}.

p q

γ/ε

γ/ε δ/ε

Exercise 10.3 pre

^∗

Computation

Compute A

pre^∗

starting from A (left) and P (right) given below.

p

q r

s

1

s

₂

γ γ

p

q

r γ/δ

1

γ δ

1

/δ

2

γ δ

₁

/ε

δ

₂

/δ

₁

(2)

Exercise 10.4 B¨ uchi Pushdown Systems

Solve the accepting run problem for the B¨ uchi-pushdown system over Γ = {γ, δ} below:

p q r

δ/γδ γ/δγ

δ/ε γ/δ

δ/γδ

δ/ε

(a) Find all (s, γ) ∈ Q × Γ such that (s, γ) →

⁺

(r, u) →

^∗

(s, γ · v) for some u, v ∈ Γ

^∗

. (b) Compute A

_pre^∗_(C)

for C = {(s, γ · Γ

^∗

Exercise 10.1 PDS Modeling

Applied Automata Theory (WS 2014/2015) Technische Universit¨ at Kaiserslautern

Exercise Sheet 10

Prof. Dr. Roland Meyer, Reiner H¨ uchting Due: Tue, Jan 20

Exercise 10.1 PDS Modeling

Give a pushdown system that models the following program where r() is a function that randomly returns 0 or 1. Describe how you represent the program counter and the value of the variable x. Hint: In a first step, assign each line in the code a unique label.

void m() x=1;

if(r()==1) s();

void s() x=0;

if(r()==1) m();

int x=0;

void main() m();

Exercise 10.2 pre

Computation

Use the algorithm given in the lecture notes to determine pre

(C) in the pushdown system below, where C = {(q, w) | w ∈ {γ, δ}

and |w| is even}.

p q

γ/ε

γ/ε δ/ε

Exercise 10.3 pre

Computation

Compute A

starting from A (left) and P (right) given below.

p

q r

s

s

γ γ

p

q

r γ/δ

γ δ

/δ

γ δ

/ε

δ

/δ

Exercise 10.4 B¨ uchi Pushdown Systems

Solve the accepting run problem for the B¨ uchi-pushdown system over Γ = {γ, δ} below:

p q r

δ/γδ γ/δγ

δ/ε γ/δ

δ/γδ

δ/ε

(a) Find all (s, γ) ∈ Q × Γ such that (s, γ) →

(r, u) →

(s, γ · v) for some u, v ∈ Γ

. (b) Compute A

for C = {(s, γ · Γ

) | (s, γ) is a configuration found in (a)}.

Note: the lowercase γ ∈ Γ in (a) and (b) is an arbitrary symbol.