Problem 2: Words are wqo

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Concurrency Theory(WS 2016) Out: Sat, 26 Nov Due: Wed, 30 Nov

Exercise Sheet 5

D’Osualdo, Lederer, Schneider Technische Universit¨at Kaiserslautern

Problem 1: Minimal elements

Let(Q,≤)be a qo. ForA ⊆Q, we sayx∈Ais minimal inAif there is nox⁰ ∈Awithx⁰ < x.

The upward closure ofA⊆Qis the setA↑:={x∈Q|x≥x⁰for somex⁰ ∈A}.

Consider the following statements:

1 Every strictly decreasing sequence overQis finite and every antichain is finite.

2 For everyA⊆Qthere is a finite setB ⊆Aof elements minimal inAsuch thatA⊆B↑.

3 (Q,≤)is wqo.

From the lecture we know that³ =⇒, so prove the three statements equivalent by proving¹ 1 =⇒2 and² =⇒.3

Problem 2: Words are wqo

Let (Q,≤)be a wqo. For u = u₁· · ·u_m, v = v₁· · ·v_n ∈ Q^∗, we write u ≤^∗ v if there are 1≤i₁ <· · ·< i_m ≤nwithu_j ≤v_i_j for allj = 1, . . . , m.

Derive that(Q^∗,≤^∗)is a wqo as a corollary of the lemmas proved in the lecture.

Problem 3: Multisets are wqo

A (finite) multiset overXis a functionm: X →Nsuch that the setbmc:={x∈X |m(x)>0}

is finite. We denote by M(X) the set of such multisets. Let(X,≤_X)be a quasi order and m₁, m₂ ∈ M(X), anembeddingfromm₁ tom₂ is an injective functionφ: bm₁c → bm₂csuch that x ≤_X φ(x)andm₁(x) ≤ m₂(φ(x))for allx ∈ bm₁c. We definem₁ ≤M(X) m₂ to hold when there exists an embedding fromm₁ tom₂.

Prove that(M(X),≤M(X))is a wqo if(X,≤_X)is a wqo.

[Hint:adapt the proof seen in the lecture for finite sets]

[Bonus:there is a shorter proof,]