SLIDE 1 Theoretical Foundations of the UML
Lecture 14: Realising Local-Choice MSGs Joost-Pieter Katoen
Lehrstuhl für Informatik 2 Software Modeling and Verification Group
moves.rwth-aachen.de/teaching/ss-20/fuml/
June 9, 2020
Joost-Pieter Katoen Theoretical Foundations of the UML 1/14
SLIDE 2 Outline
1
Introduction
2
Local Choice MSGs
3
A Realisation Algorithm for MSGs
Joost-Pieter Katoen Theoretical Foundations of the UML 2/14
SLIDE 3 Overview
1
Introduction
2
Local Choice MSGs
3
A Realisation Algorithm for MSGs
Joost-Pieter Katoen Theoretical Foundations of the UML 3/14
SLIDE 4 Today’s topic
Today’s lecture
An algorithm to realise local-choice MSGs using CFMs with synchronisation messages.
Results:
1 An algorithm that generates a CFM from local-choice MSGs. Joost-Pieter Katoen Theoretical Foundations of the UML 4/14
SLIDE 5 Overview
1
Introduction
2
Local Choice MSGs
3
A Realisation Algorithm for MSGs
Joost-Pieter Katoen Theoretical Foundations of the UML 5/14
SLIDE 6 Non-local choice
p q a
msc
p q b
msc
G: v1 v2 Inconsistency if process p behaves according to vertex v1 and process q behaves according to vertex v2 = ⇒ realisation by a CFM may yield a deadlock
Problem:
Subsequent behavior in G is determined by distinct processes. When several processes independently decide to initiate behavior, they might start executing different successor MSCs (= vertices). This is called a non-local choice.
Joost-Pieter Katoen Theoretical Foundations of the UML 6/14
SLIDE 7 Local choice property
Definition (Local choice)
Let MSG G = (V, →, v0, F, λ). MSG G is local choice if for every branching vertex v ∈ V it holds: ∃process p.
- ∀π ∈ Paths(v). | min(π′)| = 1 ∧ min(π′) ⊆ Ep
- where for π = vv1v2 . . . vn we have π′ = v1v2 . . . vn.
Joost-Pieter Katoen Theoretical Foundations of the UML 7/14
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SLIDE 8 Local choice property
Definition (Local choice)
Let MSG G = (V, →, v0, F, λ). MSG G is local choice if for every branching vertex v ∈ V it holds: ∃process p.
- ∀π ∈ Paths(v). | min(π′)| = 1 ∧ min(π′) ⊆ Ep
- where for π = vv1v2 . . . vn we have π′ = v1v2 . . . vn.
Intuition:
There is a single process that initiates behavior along every path from the branching vertex v. This process decides how to proceed. In a realisation by a CFM, it can inform the other processes how to proceed.
Local choice or not?
Deciding whether MSG G is local choice or not is in P.
Joost-Pieter Katoen Theoretical Foundations of the UML 7/14
SLIDE 9 Overview
1
Introduction
2
Local Choice MSGs
3
A Realisation Algorithm for MSGs
Joost-Pieter Katoen Theoretical Foundations of the UML 8/14
SLIDE 10 Local choice MSGs
An example local-choice MSG on black board.
Joost-Pieter Katoen Theoretical Foundations of the UML 9/14
SLIDE 11 I 2
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SLIDE 12 Realising local choice (C)MSGs
Theorem
[Genest et al., 2005]
Any local-choice MSG G is safely realisable by a CFM with synchronisation data (which is of size linear in G).
Proof
As MSG G is local choice, at every branch v of G there is a unique process, p(v), say, such that on every path from v the unique minimal event occur at p(v). Then:
1 Process p(v) determines the successor vertex of v. 2 Process p(v) informs all other processes about its decision by
adding synchronisation data to the exchanged messages.
3 Synchronisation data is the successor vertex (in G) from v chosen
by p(v).
Joost-Pieter Katoen Theoretical Foundations of the UML 10/14
SLIDE 13 Structure of the CFM of local choice MSG G
Let MSG G = (V, →, v0, F, λ) be local choice. Define the CFM AG = (((Sp, ∆p))p∈P, D, sinit, F ′) with:
1 Local automaton Ap = (Sp, ∆p) as defined on next slides Joost-Pieter Katoen Theoretical Foundations of the UML 11/14 I
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SLIDE 14 Structure of the CFM of local choice MSG G
Let MSG G = (V, →, v0, F, λ) be local choice. Define the CFM AG = (((Sp, ∆p))p∈P, D, sinit, F ′) with:
1 Local automaton Ap = (Sp, ∆p) as defined on next slides 2 D = V
synchronisation data = vertices in the MSG
3 sinit = { (v0, ∅) }n where n = |P|
each local automaton Ap starts in initial state (v0, ∅), i.e., in initial vertex v0 while no events of p have been performed
Joost-Pieter Katoen Theoretical Foundations of the UML 11/14
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SLIDE 15 Structure of the CFM of local choice MSG G
Let MSG G = (V, →, v0, F, λ) be local choice. Define the CFM AG = (((Sp, ∆p))p∈P, D, sinit, F ′) with:
1 Local automaton Ap = (Sp, ∆p) as defined on next slides 2 D = V
synchronisation data = vertices in the MSG
3 sinit = { (v0, ∅) }n where n = |P|
each local automaton Ap starts in initial state (v0, ∅), i.e., in initial vertex v0 while no events of p have been performed
4 s ∈ F ′ iff for all p ∈ P, local state s[p] = (v, E) with E ⊆ Ep and: 1
v ∈ F and E contains a maximal event wrt. <p in MSC λ(v), or
2
v ∈ F and π = v . . . w is a path in G with w ∈ F and E contains a maximal event wrt. <p in MSC M(π).
Joost-Pieter Katoen Theoretical Foundations of the UML 11/14
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SLIDE 16 State space of local automaton Ap
Sp = V × Ep such that for any s = (v, E) ∈ Sp: ∀e, e′ ∈ λ(v).
implies e ∈ E
- that is, E is downward-closed with respect to <p in MSC λ(v)
Joost-Pieter Katoen Theoretical Foundations of the UML 12/14
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SLIDE 17 State space of local automaton Ap
Sp = V × Ep such that for any s = (v, E) ∈ Sp: ∀e, e′ ∈ λ(v).
implies e ∈ E
- that is, E is downward-closed with respect to <p in MSC λ(v)
Intuition: a state (v, E) means that process p is currently in vertex v of MSG G and has already performed the events E of λ(v) Initial state of Ap is (v0, ∅)
Joost-Pieter Katoen Theoretical Foundations of the UML 12/14
SLIDE 18 Transition relation of local automaton Ap
Executing events within a vertex of the MSG G: e ∈ Ep ∩ λ(v) and e ∈ E (v, E)
l(e),v
− − − − →p (v, E ∪ { e }) Note: since E ∪ {e} is downward-closed wrt. <p, e is enabled
Joost-Pieter Katoen Theoretical Foundations of the UML 13/14
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SLIDE 19 Transition relation of local automaton Ap
Executing events within a vertex of the MSG G: e ∈ Ep ∩ λ(v) and e ∈ E (v, E)
l(e),v
− − − − →p (v, E ∪ { e }) Note: since E ∪ {e} is downward-closed wrt. <p, e is enabled Taking an edge (possibly a self-loop) of the MSG G: E = Ep ∩ λ(v) and e ∈ Ep ∩ λ(w) and vu0 . . . unw ∈ V ∗ with p not active in u0 . . . un (v, E)
l(e),w
− − − − →p (w, {e}) Note: vertex w is the first successor vertex of v on which p is active
Joost-Pieter Katoen Theoretical Foundations of the UML 13/14
SLIDE 20 ¥
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SLIDE 21 Examples
On the black board.
Joost-Pieter Katoen Theoretical Foundations of the UML 14/14
SLIDE 22 I 2
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SLIDE 23 Local
automata
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SLIDE 24 local
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SLIDE 25 Second
example
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SLIDE 26 local
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