Problem 2: Coverability Graph and Place Unboundedness

Concurrency Theory(WS 2011/12) Out: Tue, Nov 29 Due: Mon, Dec 5

Exercise Sheet 7

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

Problem 1: Easy Lemma/Theorem Proofs

LetN = (S, T, W, M0). Prove the following statements:

(a) IfM₀[σiM then there is aσ-labeled pathM₀ −→^σ M_ω inCov(N)withM_ω ≥M. (b) MarkingM ∈N^S is coverable inN iff there isM_ω ∈Cov(N)withM_ω ≥M. (c) Places∈Sis unbounded iff there isMω ∈Cov(N)withMω(s) = ω.

Problem 2: Coverability Graph and Place Unboundedness

Construct the coverability graph for the following Petri net:

p₁ t₁ p₂

t₂

p₃

t₃ t₄

p₄

(a) Name the unbounded places in the net by specifying all nodes in the coverability graph which allow you to deduce their unboundedness.

(b) For each of the following markings of the Petri net:

(1 0 0 0)^T, (0 0 1 0)^T, (1 1 0 0)^T, (0 0 1 1)^T, (1 0 1 0)^T, (0 1 0 1)^T specify all the nodes in the coverability graph (if any) that cover them.

Problem 3: Proof of Lemma - t introduces new ω’s

LetN = (S, T, W, M₀)be a Petri net with coverability graph Cov(N) = (V, E, M₀) and let M0

−→σ M_ωⁿ−→^t M_ωⁿ⁺¹for someσ∈Tⁿandt ∈T.

Assume that for allk ∈Nthere existsM ∈R(N)such that (M(s)≥k ifs∈Ω(M_ωⁿ)

M(s) = M_ωⁿ(s) ifs∈S\Ω(M_ωⁿ).

Fixk₀ ∈Nand assume|Ω(M_ωⁿ⁺¹)|=|Ω(M_ωⁿ)|+ 1. Prove that there isM⁰ ∈R(N)such that (M⁰(s)≥k₀ ifs∈Ω(M_ωⁿ⁺¹)

M⁰(s) = M_ωⁿ⁺¹(s) ifs∈S\Ω(M_ωⁿ⁺¹).

Problem 4: Coverability Graph and Net Boundedness

Consider the Petri net below:

p₁

p2

t₂

t₁

t3

p₃

p4

2

(a) Use the Karp and Miller algorithm to construct the coverability graph.

(b) Describe in words an algorithm that takes a Petri netN = (S, T, W, M₀)and returns the optimal boundb∈Nωfor the token count on all places. Argument termination and correctness.

To be precise, the algorithm should

• returnb =ω, ifN is unbounded

• return the smallestb∈Nso thatM(p)≤bfor allM ∈R(N)and allp∈S, otherwise.