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Consider the Petri net below, describing Lamport’s 1-bit mutual exclusion algorithm.

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Concurrency Theory (SS 2015) Out: Wed, May 20 Due: Tue, May 26

Exercise Sheet 5

Prof. Meyer, Furbach, D’Osualdo Technische Universit¨at Kaiserslautern

Problem 1: Lamport’s Mutual Exclusion Algorithm

Consider the Petri net below, describing Lamport’s 1-bit mutual exclusion algorithm.

cs

1

idle

1

req

1

nid

1

id

1

af ter you

2

await

2

id

2

idle

2

cs

2

req

2

(a) Set up the co-linear property one would want the mutex to satisfy and determine the connectivity and trap matrices of the given Petri net.

(b) Prove that the basic verification system is feasible.

(c) Prove that the enhanced verification system is infeasible.

How do you interpret the fact that bvs is feasible and evs infeasible?

Problem 2: Minimal Traps vs Generating Traps

(a) Give a Petri net where the minimal traps are a family of generating traps and describe that family.

(b) Give a Petri net where the set of minimal traps are not generating and describe both the minimal traps and a family of generating traps.

(c) Give a Petri net where the only trap is the empty set.

(2)

Problem 3: Family of Generating Traps

Add arcs to the Petri net N below so that its family of generating traps contains exponentially (in N ’s size) many traps. Once added, describe N = (S, T, W ) formally and prove that the family of generating traps is exponential in N ’s size.

t

1,2

p

1

p

−1

p

2

p

−2

· · · · · · · · · t

n−1,n

p

n

p

−n

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