Use the ERV algorithm given in class with McMillan’s adequate order to compute the finite and complete prefix of the unfolding of the following Petri net.

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Concurrency Theory (WS 2010/11) Out: Thu, Nov 25 Due: Mon, Nov 29

Exercise Sheet 5

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

Problem 1: Unfolding Prefix

Use the ERV algorithm given in class with McMillan’s adequate order to compute the finite and complete prefix of the unfolding of the following Petri net.

p ₁ p ₂

p ₃

p ₄ p ₅ t ₁

t ₂

t ₃

t ₄

Provide the set of possible extensions and cut-offs at each iteration. Note: The initial configu- ration of an unfolding can be seen as the local configuration [e ⊥ ] caused by an event e ⊥ .

Problem 2: Adequate Orderings

Establish adequacy of the following orderings:

(a) ≺ _m : [e] ≺ [e ⁰ ] iff. |[e]| < |[e ⁰ ]|;

(b) ≺ _lex : [e] ≺ _lex [e ⁰ ] iff. |[e]| < | [e ⁰ ]| or, |[e]| = |[e ⁰ ]| and order([e]) < _lex order([e ⁰ ]).

Assume the set of transitions is totally ordered by the transitions’ indices. Define order(C) :=

t ^# ₁

^t¹

^(C) . . . t ^# n

^tn

^(C) , where # t (C) denotes the number of events e ∈ C labelled by t. For example, if C = {e ₂ , e ₄ , e ₃ , e ₁ } is labelled by h(e ₂ ) = h(e ₄ ) = h(e ₃ ) = t ₁ , h(e ₁ ) = t ₃ then order(E) :=

t ³ ₁ t ⁰ ₂ t ¹ ₃ t ⁰ ₄ = t ₁ t ₁ t ₁ t ₃ . Hence, for any two configurations C ₁ and C ₂ , the strings order(C ₁ ) and order(C 2 ) can be ordered lexicographically by < lex (e.g. t 1 t 1 t 1 t 3 < lex t 1 t 2 ).

Problem 3: Yet Another Unfolding Prefix

Consider the following Petri net:

p ₁

p ₂

p ₃ t ₁

t ₂

t ₃

t ₄

Use ≺ _m and ≺ _lex from the previous problem to construct the net’s unfolding. Provide the set of

possible extensions and cut-offs at each iteration. What do you observe?

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Problem 4: SAT-Based Verification

Consider the Petri net depicted below.

p ₁ t ₁

p 2

p ₃

t ₂

t ₃

p ₄ t 4

(a) Compute the finite complete prefix of the Petri net’s unfolding under McMillan’s (≺ m ) order and express the reachability of the marking (0 0 0 1) ^T as violation constraint V.

(b) Find a general formula M characterizing reachability of a marking in a complete prefix starting from the C ∧ V formula given in class.

Hint: Use extra Boolean variables x _b for every b ∈ B of the unfolding (O, h) = (B, E, G).

Use the ERV algorithm given in class with McMillan’s adequate order to compute the finite and complete prefix of the unfolding of the following Petri net.

Concurrency Theory (WS 2010/11) Out: Thu, Nov 25 Due: Mon, Nov 29

Exercise Sheet 5

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

Problem 1: Unfolding Prefix

Use the ERV algorithm given in class with McMillan’s adequate order to compute the finite and complete prefix of the unfolding of the following Petri net.

p 1 p 2

p 3

p 4 p 5 t 1

t 2

t 3

t 4

Provide the set of possible extensions and cut-offs at each iteration. Note: The initial configu- ration of an unfolding can be seen as the local configuration [e ⊥ ] caused by an event e ⊥ .

Problem 2: Adequate Orderings

Establish adequacy of the following orderings:

(a) ≺ m : [e] ≺ [e 0 ] iff. |[e]| < |[e 0 ]|;

(b) ≺ lex : [e] ≺ lex [e 0 ] iff. |[e]| < | [e 0 ]| or, |[e]| = |[e 0 ]| and order([e]) < lex order([e 0 ]).

Assume the set of transitions is totally ordered by the transitions’ indices. Define order(C) :=

t # 1

(C) . . . t # n

(C) , where # t (C) denotes the number of events e ∈ C labelled by t. For example, if C = {e 2 , e 4 , e 3 , e 1 } is labelled by h(e 2 ) = h(e 4 ) = h(e 3 ) = t 1 , h(e 1 ) = t 3 then order(E) :=

t 3 1 t 0 2 t 1 3 t 0 4 = t 1 t 1 t 1 t 3 . Hence, for any two configurations C 1 and C 2 , the strings order(C 1 ) and order(C 2 ) can be ordered lexicographically by < lex (e.g. t 1 t 1 t 1 t 3 < lex t 1 t 2 ).

Problem 3: Yet Another Unfolding Prefix

Consider the following Petri net:

p 1

p 2

p 3 t 1

t 2

t 3

t 4

Use ≺ m and ≺ lex from the previous problem to construct the net’s unfolding. Provide the set of

possible extensions and cut-offs at each iteration. What do you observe?

Problem 4: SAT-Based Verification

Consider the Petri net depicted below.

p 1 t 1

p 2

p 3

t 2

t 3

p 4 t 4

(a) Compute the finite complete prefix of the Petri net’s unfolding under McMillan’s (≺ m ) order and express the reachability of the marking (0 0 0 1) T as violation constraint V.

(b) Find a general formula M characterizing reachability of a marking in a complete prefix starting from the C ∧ V formula given in class.

Hint: Use extra Boolean variables x b for every b ∈ B of the unfolding (O, h) = (B, E, G).

p ₁ p ₂

p ₃

p ₄ p ₅ t ₁

t ₂

t ₃

t ₄

(a) ≺ _m : [e] ≺ [e ⁰ ] iff. |[e]| < |[e ⁰ ]|;

(b) ≺ _lex : [e] ≺ _lex [e ⁰ ] iff. |[e]| < | [e ⁰ ]| or, |[e]| = |[e ⁰ ]| and order([e]) < _lex order([e ⁰ ]).

t ^# ₁

^(C) . . . t ^# n

^(C) , where # t (C) denotes the number of events e ∈ C labelled by t. For example, if C = {e ₂ , e ₄ , e ₃ , e ₁ } is labelled by h(e ₂ ) = h(e ₄ ) = h(e ₃ ) = t ₁ , h(e ₁ ) = t ₃ then order(E) :=

t ³ ₁ t ⁰ ₂ t ¹ ₃ t ⁰ ₄ = t ₁ t ₁ t ₁ t ₃ . Hence, for any two configurations C ₁ and C ₂ , the strings order(C ₁ ) and order(C 2 ) can be ordered lexicographically by < lex (e.g. t 1 t 1 t 1 t 3 < lex t 1 t 2 ).

p ₁

p ₂

p ₃ t ₁

t ₂

t ₃

t ₄

Use ≺ _m and ≺ _lex from the previous problem to construct the net’s unfolding. Provide the set of

p ₁ t ₁

p ₃

t ₂

t ₃

p ₄ t 4

(a) Compute the finite complete prefix of the Petri net’s unfolding under McMillan’s (≺ m ) order and express the reachability of the marking (0 0 0 1) ^T as violation constraint V.

Hint: Use extra Boolean variables x _b for every b ∈ B of the unfolding (O, h) = (B, E, G).