Prove that fork∈N, (Ak,≤k) is a wqo

SS 2014 21.05.2014 Exercises to the lecture

Concurrency Theory Sheet 5

Roland Meyer, Viktor Vafeiadis Delivery until 27.05.2014 at 12h Exercise 5.1 (Well-quasi orderings)

a) Prove or disprove that (N,|) is a well-quasi ordering where a|b means ”adividesb”.

b) Let (A,≤) be a wqo. Prove that fork∈N, (A^k,≤^k) is a wqo.

The ordering ≤^k is obtained by component-wise application of ≤on vectors in A^k. Thus, (a₁, . . . , a_k)≤^k(a⁰₁, . . . , a⁰_k) if and only ifa_i ≤a⁰_i fori∈ {1, . . . , k}.

c) Given a set A and a set W ⊆ P(A×A), so that (A, w) is a wqo for all w ∈ W. Show that (A,(S

w∈W w)⁺) is a wqo.

Exercise 5.2 (Upward-closed sets)

a) For a finite alphabet Σ and w1, w2 ∈Σ^∗, let w1≤w2 if and only ifw1 is a subword ofw2 as defined in the lecture.

Show that for any language L ⊆Σ^∗, the languagesL ↑ and L ↓ are regular.

b) Let (A,≤) be a wqo andM1, M2 ⊆Afinite. Show that it is decidable ifM1↑=M2↑.

Exercise 5.3 (Well-Structured transition systems)

a) Consider a transition system (Γ, γ₀,→) and a relation≤ ⊆Γ×Γ. Prove that≤is a simulation if and only ifpre(I) is upward-closed for every upward-closed setI ⊆Γ.

b) LetTS = (Γ, γ₀,→,≤) be a well-structured transition system where

• γ ≤γ⁰ is decidable for allγ, γ⁰ ∈Γ and

• for all γ ∈Γ, the set post(γ) :={γ⁰∈Γ|γ →γ⁰} is finite and computable.

Prove that termination is decidable forTS.

Note: A transition system isterminating, if there are no infinite runs.

Exercise 5.4 (Petri nets)

Consider the following definition of Petri nets and their firing relation:

• A Petri net is a triple N = (P, T, W) where P is a set of places, T is a set of transitions, and W : (P×T)∪(T ×P)→Nis aweight function.

• A marking M ∈N^|P| of N is a function that maps places to natural numbers.

• A transitiont∈T is enabled inM, ifM ≥W(−, t).

W(−, t) is the vector (W(p1, t), . . . , W(p|P|, t)) andW(t,−) is defined analogously.

• Iftis enabled inM₁, it transforms the marking into a new markingM₂by removing W(−, t) fromM and adding W(t,−).

Formally, thefiring relation [i ⊆N^|P|×T×N^|P| contains the triple (M1, t, M2) (or M₁[tiM₂) ift is enabled inM₁ and M₂ = (M₁−W(−, t)) +W(t,−).

Given an initial marking M0, prove that the transition system (N^|P|, M0,[i,≤^|P|) is a well-structured transition system.

Delivery until 27.05.2014 at 12h into the box next to 34-401.4