Theorie van Concurrency najaar 2011 - - PowerPoint PPT Presentation

Theorie van Concurrency

najaar 2011 http://www.liacs.nl/home/rvvliet/tvc/ derde college: 13 september 2011 4.4 Concurrency 4.5 Fundamental Situations eerste werkcollege: 15 september 2011

• pgaven bij 4. EN Systems

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Theorie van Concurrency — najaar 2011 http://www.liacs.nl/home/rvvliet/tvc/

• hoorcollege/werkgroep ∼ 2/1

Gecorrigeerde data: dinsdag 6 september - 25 oktober, zaal 403, 11.15–13.00 donderdag 8 september - 27 oktober, zaal 403, 11.15–13.00 donderdag 3 november - 8 december, zaal 403, 10.00–13.00

• dictaat + survey paper
• opgavenbundel + oplossingen + oude tentamens

Samen voor EUR 10,50

• modelleertoets, donderdag 17 november 2011, 10:00–13:00

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Definition 13. Let M = (P, T, F, Cin) be an EN system. (1) Let U ⊆ T. U is a disjoint set of transitions, notation disj(U), if 1. U = ∅ and 2. for all transitions t1 = t2 ∈ U: nbh(t1) ∩ nbh(t2) = ∅. (2) Let U ⊆ T and let C ⊆ P. Then U has concession in C (or U can be fired in C, or U is enabled in C) if

• 1. disj(U), 2. •U ⊆ C, and 3. U• ∩ C = ∅.

Notation: U con C. (3) Let U ⊆ T and let C, D ⊆ P. Then U fires from C to D, written as C[UD, if

• 1. U con C and 2. D = (C − •U) ∪ U•.

If #U ≥ 2, then U is a concurrent step from C to D.

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Lemma 15. Let M = (P, T, F, Cin) be an EN system. Let C ⊆ P and let U ⊆ T with U = ∅. Then U con C iff (1) t con C for all t ∈ U, and (2) for all t1 = t2 ∈ U, •t1 ∩ •t2 = ∅ and t1• ∩ t2• = ∅.

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Lemma 16. Let M = (P, T, F, Cin) be an EN system. Let C, D ⊆ P, and let U ⊆ T. Let {U1, U2} be a partition of U.∗ If C[UD, then there is E1 ⊆ P such that C[U1E1 and E1[U2D.

∗ U = U1 ∪ U2, U1 ∩ U2 = ∅ and U1, U2 = ∅

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D U1

• U1

U U2• E2 U2

• U1

C U1 U2 E1 U1• U1• U2•

• U2
• U2
• Fig. 17. A diamond.

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p2bc1 p2c2 p1c2 p2bc2 p1bc2 p2c1 p1c1 p c c c p c pc p1bc1 p p pe pc e f f e fc

• Fig. 18. A configuration graph.

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Lemma 17. Let M = (P, T, F, Cin) be an EN system. Let C, D ⊆ P and let U ⊆ T. If C[UD, then C[t1 · · · tnD for each ordering (t1, . . . , tn) of the elements of U.

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p1bc1 p2bc1 p2c2 p1c2 p2bc2 p1bc2 p2c1 p1c1 p c c p c p p c e e f f

• Fig. 16. A sequential configuration graph.

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p2bc1 p2c2 p1c2 p2bc2 p1bc2 p2c1 p1c1 p c c c p c pc p1bc1 p p pe pc e f f e fc

• Fig. 18. A configuration graph.

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Lemma 19. Let M = (P, T, F, Cin) be an EN system. Let C ⊆ P and let s, t ∈ T. If st con C and t con C, then {s, t} con C.

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Lemma 15. Let M = (P, T, F, Cin) be an EN system. Let C ⊆ P and let U ⊆ T with U = ∅. Then U con C iff (1) t con C for all t ∈ U, and (2) for all t1 = t2 ∈ U, •t1 ∩ •t2 = ∅ and t1• ∩ t2• = ∅.

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Lemma 19.5 Let M = (P, T, F, Cin) be an EN system. Let C ⊆ P and let U ⊆ T. If ti con C for every ti ∈ U and t1t2 . . . tn con C for some order of the elements of U = {t1, t2, . . . , tn}, then U con C.

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Theorem 20. Let M = (P, T, F, Cin) be an EN system. Let C, D ⊆ P and let U ⊆ T with U = ∅. Then (1) U con C iff t1 · · · tn con C for every ordering (t1, . . . , tn) of the elements of U, and (2) C[UD iff C[t1 · · · tnD for every ordering (t1, . . . , tn) of the elements of U.

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Lemma 17. Let M = (P, T, F, Cin) be an EN system. Let C, D ⊆ P and let U ⊆ T. If C[UD, then C[t1 · · · tnD for each ordering (t1, . . . , tn) of the elements of U.

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Definition 10. Let G1 = (V1, Γ1, Σ1, v1) and G2 = (V2, Γ2, Σ2, v2) be edge-labelled graphs. Then G1 and G2 are isomorphic, denoted by G1 ≡ G2, if there exist two bijections α : V1 → V2 and β : Σ1 → Σ2 such that α(v1) = v2 and, for all v, w ∈ V1 and all U ⊆ Σ1, (v, U, w) ∈ Γ1 iff (α(v), β(U), α(w)) ∈ Γ2.

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Theorem 21. For EN systems M and M′, SCG(M) ≡ SCG(M′) iff CG(M) ≡ CG(M′).

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t1 t2 t2 t1

• Fig. 19, 20. Causality.

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t1 t2

• Fig. 21. Concurrency.

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t1 t2 P C

• t1
• t2

t2• t1• P − C

• Fig. 22. Concurrency, the complete picture.

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Causality: t1t2 con C, but not t2 con C. Concurrency: t1t2 con C, and t2 con C (Lemma 17 and Lemma 19). Hence, if t1t2 con C, then either causality or concurrency.

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Definition: Transitions t1 and t2 are in conflict in configuration C, if t1 con C and t2 con C, but not {t1, t2} con C.

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w1 w2 c1 p c2 d1 d2 r2 r1 component 1 component 2 in1 in2

• ut1
• ut2
• Fig. 5. The mutual exclusion problem.

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• t1

t1 t2

• t2
• Fig. 23. Input-conflict.

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t1 t2 t1• t2•

• Fig. 24. Output-conflict.

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Concurrency: {t1, t2} con C, hence t1, t2 con C (Lemma 15). Conflict: t1, t2 con C, but not {t1, t2} con C. Hence, if t1, t2 con C, then either concurrency or conflict.

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Definition 23. Let M = (P, T, F, Cin) be an EN system. Let C ∈ CM, and let t ∈ T be such that t con C. Then cfl(t, C) = {t′ ∈ T | t′ con C and ¬ {t, t′} con C} is the conflict set of t in C.

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Definition 22. An EN system M = (P, T, F, Cin) is conflict-free if, for every C ∈ CM and all transitions t1, t2 ∈ T: {t1, t2} con C whenever t1 con C and t2 con C.

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p1 c2 c1 p b c f e p2

• Fig. 12. Conflict-free.

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p1 p3 p5 t2 p2 p4 p6 t1 t3

• Fig. 25. A conflict-increasing confusion.

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Definition 24. Let M = (P, T, F, Cin) be an EN system. Let C ∈ CM, and let t1, t2 ∈ T. The triple (C, t1, t2) is called a confusion (in C) if

• 1. t1 = t2,
• 2. {t1, t2} con C, and
• 3. cfl(t1, C) = cfl(t1, D), where C[t2D.

M is confused in C if there is a confusion in C.

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p5 t1 t3 p3 p1 p2 p4 t2

• Fig. 26. A conflict-decreasing confusion.

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Definition 25 Let M = (P, T, F, Cin) be an EN system. Let C ∈ CM and t1, t2 ∈ T. Let γ = (C, t1, t2) be a confusion and C[t2D. (1) γ is a conflict-increasing confusion, ci confusion for short, if

cfl(t1, D) cfl(t1, C).

(2) γ is a conflict-decreasing confusion, cd confusion for short, if cfl(t1, D) cfl(t1, C).

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p5 t3 p7 p1 t1 t4 t2 p6 p4 p2 p3

• Fig. 27. A confusion which is neither ci nor cd.

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p3 p1 t1 t4 p4 t3 p2 p5 p6 t2

• Fig. 28. A symmetric confusion.

Definition 26. Let M = (P, T, F, Cin) be an EN system. Let C ∈ CM and t1, t2 ∈ T. Let γ = (C, t1, t2) be a confusion. γ is symmetric if (C, t2, t1) is also a confusion, otherwise γ is asymmetric.

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Consider the EN system Mutex (Figure 5). Give CG(Mutex) and determine all confusions (C, t1, t2) with C ∈ CMutex. Give - if possible - examples of confusions which are conflict-increasing, conflict-decreasing, neither and in addition (a)symmetric. Prove: every confusion which is not ci is symmetric.

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