Concurrency Theory(WS 2011/12) Out: Tue, Dec 6 Due: Mon, Dec 12
Exercise Sheet 8
Jun.-Prof. Roland Meyer, Georgel C˘alin Technische Universit¨at Kaiserslautern
Problem 1: Petri Nets and LCSs as WSTS
(a) The transition system of Petri net N = (S, T, W, M0) is T S(N) := (N|S|, M0,→), where transitionM1 →M2 exists ifM1[tiM2 for somet ∈T. Prove that(Nk,≤)is a wqo for anyk ∈Nand thatT S(N)is well-structured for any netN.
(b) Consider some lcsL=hQ, q0, C, M,→i. Prove that(Q×M∗C,≤), with≤as defined in the lecture, is a wqo and(T S(L),≤), withT S(L) := (Q×M∗C, γ0,→), is well structured.
Problem 2: Upward-Closed Sets by Minimal Elements
Let(A,≤)be a wqo and letI ⊆ Abe an upward closed set. Prove Lemma 6.2 given in class:
ifM in(I)is a finite set of minimal elements ofI, thenI =M in(I)↑.
Problem 3: Parallel Composition of WSTS
Consider two wstsT S1 = (Γ1, γ0,→1,≤1)andT S2 = (Γ2, γ0,→2,≤2). Define their parallel composition to beT S1 kT S2 := (Γ1 ×Γ2,(γ0, γ0),→)where
(γ1, γ1)→(γ2, γ2)ifγ1 →1 γ2 andγ1 →2 γ2. Prove that(T S1 kT S2,≤1×2)is a wsts.
Problem 4: LCS Variation remains Well Structured
Consider another type of lcsL= (Q, q0,{c}, M,→)withca channel carrying natural numbers as content, i.e.,M =N. Take the ordering≤∗⊆M∗×M∗given in Higman’s lemma.
(a) Prove that(Q×M∗,C), withCdefined by(q, w)C(q, w0)iffw≤∗ w0, is a wqo.
(b) The transitions inLare given byq →!n q0 andq →?n q0 withn ∈ N. The first appendsn to the channel, the second receives a numbern0 ≥nwithn0 ∈Nfrom the head of the channel.
The channel is supposed to be lossy. Formalise the transition relation between configurations.
(c) Prove that((Q×M∗,(q0, ),→),C)is a wsts.
