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Concurrency Theory (SS 2015) Out: Wed, May 13 Due: Tue, May 19

Exercise Sheet 4

Prof. Roland Meyer, Florian Furbach Technische Universit¨at Kaiserslautern

Problem 1: Invariants for Mutual Exclusion

Consider the following mutual exclusion protocol:

pw

1

t

1

pcs

1

t

2

sem = 1

sem = 0

t

01

pw

2

pcs

2

t

02

Notice that (0 1 0 1 2 1)

T

is an invariant used to prove no marking M with M (pcs

1

) = M (pcs

2

) = M (sem = 0) = 1

is reachable.

Which invariants can be used to prove no marking M with just M (pcs

1

) = M(pcs

2

) = 1 is reachable? Give a general description and prove mutual exclusion for such a concrete invariant.

Problem 2: Simple Mutual Exclusion

Consider the Petri net N given below:

p

1

t

2

p

4

t

3

t

1

p

2

t

4

p

5

t

5

p

3

(a) Compute the set of structural invariants.

(b) Determine the set of all traps.

(c) Using (a) and (b), prove that p

3

and p

4

are mutually exclusive.

(2)

Problem 3: S/T-Invariants for Petri Nets

Let N = (S, T, W ) be a Petri net.

(a) Prove that if I and J are structural (S-) invariants of N , so are I + J and k · I (∀k ∈ Z ).

(b) Prove that if I and J are transition (T-) invariants of N , so are I + J and k · I (∀k ∈ N ).

(c) Give a Petri net that is acyclic but has a transition (T-) invariant.

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