We extend lossy channel systems with a transition that, given a channel c and a word w ∈ Σ ∗ checks whether w is contained in c as a subword:

WS 2019 / 2020 28.01.2020 Exercises to the lecture

Concurrency Theory Sheet 9 Dr. Prakash Saivasan

Peter Chini Delivery until 04.02.2020 at 12:00

Exercise 9.1 (Lossy Channel Systems with conditionals)

We extend lossy channel systems with a transition that, given a channel c and a word w ∈ Σ ^∗ checks whether w is contained in c as a subword:

q check w in c p

Formally, we extend the transition relation on configurations of a lossy channel systems by the rule:

(q, W ) → (p, W ) if q −−−−−−−−→ ^{check w in c} p and w ≤ W (c).

Given an extended lossy channel system L = (Q, q 0 , C, M, → _L ), construct a lossy channel system L ⁰ = (Q ⁰ , q ₀ , C, M ⁰ , → _L

) with Q ⊆ Q ⁰ and M ⊆ M ⁰ such that:

(q 1 , W ) → ^∗ _L (q 2 , W ⁰ ) if and only if (q 1 , W ) → ^∗ _L

(q 2 , W ⁰ ), for each q 1 , q 2 ∈ Q and W, W ⁰ channel contents.

Exercise 9.2 (Ideals)

Let (C, ≤) be a well quasi ordering. An ideal is a subset I ⊆ C which is non-empty, downward closed, and directed. Directed means that for each x, y ∈ I there exists a z ∈ I such that x ≤ z and y ≤ z.

a) Let (A, ≤ _A ) and (B, ≤ _B ) be two wqos and (A × B, ≤) their product. Show that a subset J ⊆ A × B is an ideal if and only if J = I A × I B where I A , I B are ideals in A and B , respectively.

Hint: For one direction you need to show that J = proj _A (J ) × proj _B (J ), where proj _A denotes the projection to A: proj _A (a, b) = a.

b) Show that the ideals of ( N , ≤) are N itself and all sets of the form n ↓ for n ∈ N . c) Derive that the ideals of N ^d are of the form M ↓, where M is a generalized marking,

a vector M ∈ (N ∪ {ω}) ^d .

Delivery until 04.02.2020 at 12:00