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

(1)

Concurrency Theory (WS 2011/12) Out: Tue, Nov 15 Due: Mon, Nov 21

Exercise Sheet 5

Jun.-Prof. Roland Meyer, Georgel C˘alin 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 colinear 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 feasable and evs infeasible?

The exercise is a bit of work but demonstrates the power of the analysis technique. The example is taken from J. Esparza, S. Melzer: Verification of Safety Properties Using Integer Program- ming: Beyond the State Equation. Formal Methods in System Design 16(2): 159-189 (2000).

Problem 2: Conflict vs. Causality vs. Concurrency

Prove that for occurrence net (B, E, G) and distinct x, y ∈ B ∪E, precisely one of the following holds:

• x and y are causally related (i.e. x < y or x > y)

• x and y are in conflict (i.e. x ] y)

• x and y are concurrent (i.e. x co y)

(2)

Problem 3: Configurations and Firing Sequences

Let C = {e

₁

, . . . , e

_n

} be a configuration with normal event ordering: i < j if e

_i

< e

_j

. Prove that e

₁

. . . e

_n

is a possible firing sequence in the unfolding, producing the marking ({e

_⊥

}

^•

∪ C

^•

)\

^•

C.

Problem 4: Unfoldings, Configurations, and Cuts

Consider the Petri net depicted below:

(a) unfold the Petri net and outline one of its prefixes;

(b) describe/list all the configurations and cuts of your unfolding.

p

1

p

2

t

₁

t

₂

p

3

t

₃

p

4

p

5

t

₄

p

₆

p

₇

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

Concurrency Theory (WS 2011/12) Out: Tue, Nov 15 Due: Mon, Nov 21

Exercise Sheet 5

Jun.-Prof. Roland Meyer, Georgel C˘alin 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

idle

req

nid

id

af ter you

await

id

idle

cs

req

(a) set up the colinear 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 feasable and evs infeasible?

The exercise is a bit of work but demonstrates the power of the analysis technique. The example is taken from J. Esparza, S. Melzer: Verification of Safety Properties Using Integer Program- ming: Beyond the State Equation. Formal Methods in System Design 16(2): 159-189 (2000).

Problem 2: Conflict vs. Causality vs. Concurrency

Prove that for occurrence net (B, E, G) and distinct x, y ∈ B ∪E, precisely one of the following holds:

• x and y are causally related (i.e. x < y or x > y)

• x and y are in conflict (i.e. x ] y)

• x and y are concurrent (i.e. x co y)

Problem 3: Configurations and Firing Sequences

Let C = {e

, . . . , e

} be a configuration with normal event ordering: i < j if e

< e

. Prove that e

. . . e

is a possible firing sequence in the unfolding, producing the marking ({e

}

∪ C

)\

C.

Problem 4: Unfoldings, Configurations, and Cuts

Consider the Petri net depicted below:

(a) unfold the Petri net and outline one of its prefixes;

(b) describe/list all the configurations and cuts of your unfolding.

p

p

t

t

p

t

p

p

t

p

p

Why is your unfolding finite?