Concurrency Theory (WS 2011/12) Out: Tue, Dec 13 Due: Mon, Dec 19
Exercise Sheet 9
Jun.-Prof. Roland Meyer, Georgel C˘alin Technische Universit¨at Kaiserslautern
Problem 1: Reachability of Upward-Closed Sets
Consider wsts (Γ, →, γ 0 , ≤). Let pre j (I) :=
j times
z }| {
pre(. . . pre(I ) . . .) for upward closed set I ⊆ Γ.
(a) Show that I j = S j
l=0 pre l (I) with I j as it has been defined in the lecture.
(b) Prove that I is reachable from γ in ≤n steps if and only if γ ∈ I n .
Problem 2: Downward Closure of Automata Languages
Compute L(A)↓ for the following automata A:
q 0 b q 1 q 2 q 3
b
a b
a b
a
Give a general procedure which given an arbitrary automaton A computes L(A)↓.
Problem 3: SRE Inclusion
Use the algorithm given in the lecture to check whether the following SRE inclusions hold:
(a) (a + n + s) ∗ (t + a + n) ∗ ⊆ (s + a + n + t + a) ∗
(b) (r + )(p + )(n + t) ∗ ⊆ p ∗ (r + )(s + )(n + t) ∗ + (p + )r ∗ (n + e + t) ∗ (b) (r + )(p + )(n + t) ∗ ⊆ (p + r + e) ∗ (s + )(n + t) ∗
Problem 4: Coverability for Lossy Channel Systems
Consider the lcs depicted in the figure below.
q 0 N !1 q 1 q 2 q 3 q 4
A!1 N !0
A?0
N ?1 N !0
N ?0
Determine if configurations (q 4 ,
N entry
↓
0 ε
↑
