Problem 3: And-Or Graphs and Execution Trees

Concurrency Theory(WS 2011/12) Out: Tue, Jan 10 Due: Mon, Jan 16

Exercise Sheet 11

Jun.-Prof. Roland Meyer, Georgel C˘alin Technische Universit¨at Kaiserslautern

Problem 1: Reachability of Upward-Closed Sets

Let(Γ, γ₀,→,≤)be a well-structured transition system andI ⊆Γan upward-closed set.

(a) Prove thatR(γ₀)∩I =∅if and only ifR(γ₀)↓ ∩I =∅.

(b) LetR(γ₀)∩I 6=∅in(Γ, γ₀,→). Prove that there existsΓ⁰ ⊆Γfinite withγ₀ ∈Γ⁰such thatR(γ₀)∩I 6=∅in(Γ⁰, γ₀,→ ∩(Γ⁰×Γ⁰)).

Problem 2: Adequate Domain of Limits for LCSs

(a) Show that symbolic configurations(q, R)withR:C →SRE are an adl.

(b) Argue why the above adequate domain of limits (adl) is effective for LCSs.

Problem 3: And-Or Graphs and Execution Trees

Give four And-Or graphs with the following properties: the first one has infinitely many exe- cution trees, the second one has more than one but finitely many execution trees, the third has a unique execution tree with infinitely many branches, and the last has a unique execution tree with finitely many branches.

Problem 4: Expand, Enlarge and Check

Consider the lossy channel systemLCS:

q0 c!0 q1 q2

d!1

c!1

c?0

d!1 d!1

together withΓ ={(q₀, ε),(q₁, ε),(q₂, ε)}and limit domains L₀ =

>, q₀, 1^∗

ε

, q₀, ε

1^∗

, q₁,

(0 + 1)^∗ 0^∗.1^∗

, q₁,

(0 + 1)^∗ 1^∗.0^∗

L1 =L0∪ q0,

1^∗ 1^∗

, q1,

1^∗.(0 +ε) 1^∗

, q2,

ε 1^∗

.

(a) ComputeOver(LCS,Γ, L₀). Provide an execution tree.

(b) ComputeOver(LCS,Γ, L₁). Argue why configuration(q₂, 1

ε

)is not coverable.