business processes modelling mpb (6 cfu,...
TRANSCRIPT
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Business Processes Modelling MPB (6 cfu, 295AA)
Roberto Bruni http://www.di.unipi.it/~bruni
09 - Petri nets properties
!1
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Object
2
Free Choice Nets (book, optional reading) https://www7.in.tum.de/~esparza/bookfc.html
We give a formal account of some key properties of Petri nets
N `
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Petri nets: behavioural properties
3
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Properties of Petri nets
4
We introduce some of the properties of Petri nets that can play an important role in the verification of
business processes
Liveness Deadlock-freedom
Boundedness Cyclicity (also Reversibility)
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Liveness, intuitively
5
A transition t is live if from any reachable marking M another
marking M’ can be reached where t is enabled
In other words: at any point in time of the computation, we cannot exclude that t will fire in the future
or, equivalently, at any point in time of the computation, it is still
possible to enable t in the future
A Petri net is live if all of its transitions are live
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Liveness illustrated
6
M0
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Liveness illustrated
6
M0 M
For any reachable marking M
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Liveness illustrated
6
M0 M
Can we find a way to enable t?
For any reachable marking M
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Liveness illustrated
6
M0 M
Can we find a way to enable t?
For any reachable marking M
M’
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Liveness illustrated
6
M0 M
Can we find a way to enable t?
For any reachable marking M
M’t
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Liveness, formally
7
(P, T, F,M0)
⌅t ⇤ T, ⌅M ⇤ [M0 ⌃, ⇧M � ⇤ [M ⌃, M � t�⇥
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Digression
8
Order of quantifier is important:
quantification of the same kind can be switched
the order of universal and existential quantification is important
8n. 9m. n < m 9m. 8n. n < m6⌘
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Example: Non Live
9
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Example: Live
10
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Liveness: pay attention!
11
Liveness should not be confused with the following property:
starting from the initial marking M0 it is possible to reach a marking M that enables t
(this property just ensures that t is not "dead" at M0)
9M 2 [M0i. Mt�!
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Example: non live net (but t1 non dead!)
12
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Dead transition
13
Given a marking M
A transition t is dead at M if t will never be enabled in the future
(i.e., t is not enabled at any marking reachable from M)
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Liveness: example
14
Which transitions are live? Which are not?
Which are dead? Is the net live?
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Liveness on the occurrence graph
16
A transition t is live iff
From any node of the occurrence graph we can reach a node with an outgoing arc labelled by t
A transition t is dead (at M0) iff
There is no t-labelled arc in the occurrence graph
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Marked place
17
Given a marking M
We say that a place p is marked (at M) if M(p) > 0
(i.e., there is a token in p in the marking M)
We say that p is unmarked if M(p) = 0
(i.e., there is no token in p in the marking M)
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Place-liveness, intuitively
18
A place p is live if every time it becomes unmarked
there is still the possibility to be marked in the future (or if it always stays marked)
A Petri net is place-live if all of its places are live
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Live place
19
Definition: Let (P, T, F,M0) be a net system.
A place p � P is live if ⇥M � [M0 ⌅. ⇤M � � [M ⌅.M �(p) > 0
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Place-liveness, formally
20
(P, T, F,M0)
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Dead nodes, intuitively
21
Given a marking M
A transition t is dead at M if t will never be enabled in the future
(i.e., t is not enabled in any marking reachable from M)
A place p is dead at M if p will never be marked in the future
(i.e., p is unmarked in any marking reachable from M)
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Dead nodes
22
Definition: Let (P, T, F ) be a net system.
A transition t ⇤ T is dead at M if ⇧M � ⇤ [M ⌃.M � ⌅ t�⇥
A place p ⇤ P is dead at M if ⇧M � ⇤ [M ⌃.M �(p) = 0
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Non-live vs Dead
23
If a transition is dead at some reachable marking M then it is non-live
If a place is dead at some reachable marking M then it is non-live
being non-live implies possibly becoming dead (but not necessarily in the current marking)
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Some obvious facts
24
If a system is not live, it must have a transition dead at some reachable marking
If a system is not place-live, it must have a place dead at some reachable marking
If a place / transition is dead at M, then it remains dead at any marking reachable from M
(the set of dead nodes can only increase during a run)
Every transition in the pre- or post-set of a dead place is also dead
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True or false?
25
Every place in the pre- or post-set of a dead transition is also dead
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Liveness implies place-liveness
27
Proposition: Live systems are also place-live
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Liveness implies place-liveness
27
Proposition: Live systems are also place-live
Take any p and any t ⇤ •p ⌅ p•
Let M ⇤ [M0 ⇧
By liveness: there is M �,M �� ⇤ [M ⇧ s.t. M � t�⇥ M ��
Then M �(p) > 0 or M ��(p) > 0
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Place liveness on the occurrence graph
28
A place p is live iff
From any node of the occurrence graph we can reach a node with a token in p
A place p is dead (at M0) iff
All the nodes of the occurrence graph have no token in p
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Exercise
29
Draw a net that is place-live but not live
(if you can)
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Deadlock-freedom
30
A Petri net is deadlock free, if every reachable marking enables some transition
In other words, we are guaranteed that at any point in time of the computation, some
transition can be fired
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Deadlock-freedom illustrated
31
M0
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Deadlock-freedom illustrated
31
M0 M
For any reachable marking M
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Deadlock-freedom illustrated
31
M0 M
Can we fire some transition?
For any reachable marking M
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Deadlock-freedom illustrated
31
M0 M
Can we fire some transition?
For any reachable marking M
M’t
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Deadlock freedom, formally
32
(P, T, F,M0)
⌅M ⇤ [M0 ⌃, ⇧t ⇤ T, Mt�⇥
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Deadlock-freedom: example
33
Is the net deadlock-free?
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Deadlock freedom on the occurrence graph
35
A net is deadlock free iff
Every node of the occurrence graph has an outgoing arc
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Question time
36
Does liveness imply deadlock-freedom? (Can you exhibit a live Petri net that is not deadlock-free?)
Does deadlock-freedom imply liveness? (Can you exhibit a deadlock-free net that is not live?)
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Liveness implies deadlock freedom
38
Lemma If (P, T, F,M0) is live, then it is deadlock-free
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Liveness implies deadlock freedom
38
Lemma If (P, T, F,M0) is live, then it is deadlock-free
By contradiction, let M ⇤ [M0 ⌃, with M ⌅⇥
Let t ⇤ T (T cannot be empty).
By liveness, ⇧M � ⇤ [M ⌃ with M � t�⇥ .
Since M is dead, [M ⌃ = {M }.Therefore M = M � t�⇥, which is absurd.
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Digression
39
Contraposition
P ) Q ⌘ (¬Q) ) ¬P
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Exercises
40
Prove each of the following properties or give some counterexamples
If a system is not place-live, then it is not live
If a system is not live, then it is not place-live
If a system is place-live, then it is deadlock-free
If a system is deadlock-free, then it is place-live
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k-Boundedness
41
Let k be a natural number
A place p is k-bounded if no reachable marking has more than k tokens in place p
A net is k-bounded if all of its places are k-bounded
In other words, if a net is k-bounded, then k is a capacity constraint that can be imposed over places
without any risk of causing “overflow”
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Safe nets
42
A place p is safe if it is 1-bounded
A net is safe if all of its places are safe
In other words, if the net is safe, then we know that, in any reachable marking, each place
contains one token at most
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Boundedness
43
A place p is bounded if it is k-bounded for some natural number k
A net is bounded if all of its places are bounded
A net is unbounded if it is not bounded (i.e., there is at least one place in which any
number of tokens can appear)
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Boundedness, formally
44
(P, T, F,M0)
⌅k ⇥ N, ⇤M ⇥ [M0 ⇧, ⇤p ⇥ P, M(p) � k
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Boundedness: example
45
Which places are bounded? Is the net bounded?
Which places are safe? Is the net safe?
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A puzzle about reachability
47
Theorem: If a system is... then its reachability graph is finite
Theorem: A system is... iff its reachability graph is finite
(fill the dots and the proofs)
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A puzzle about reachability
48
Theorem: If a system is bounded then its reachability graph is finite
Theorem: A system is bounded iff its reachability graph is finite
(fill the dots and the proofs)
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Boundedness implies finiteness
49
Theorem: If a system is bounded then its reachability graph is finite
Proof: if the system is bounded there exists k such that each place contains at most k tokens.
If there are n places it means that there are at most (k+1)n reachable markings.
Hence the occurrence graph has a finite number of nodes
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Finiteness implies boundedness
50
Theorem: A system is bounded if its reachability graph is finite
Proof: for each node M we take kM be the maximum number of tokens in the same place.
Then we let k be the largest among all kM k = max {kM | M is a node of the graph}
(k exists because the reachability graph is finite)
Clearly the system is k-bounded and thus bounded
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Cyclicity (aka Reversibility)
51
A marking M is a home marking if it can be reached from every reachable marking
A net is cyclic (or reversible) if its initial marking is a home marking
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Orthogonal properties
52
Liveness, boundedness and cyclicity are independent of each other
In other words, you can find nets that satisfy any arbitrary combination of the above three properties
(and not the others)
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Exercises
53
For each of the following nets, say if they are live, deadlock-free, bounded, safe, cyclic
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Exercises
54
For each of the following nets, say if they are live, deadlock-free, bounded, safe, cyclic
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Petri nets: structural properties
55
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Structural properties
56
All the properties we have seen so far are behavioural (or dynamic)
(i.e. they depend on the initial marking and firing rules)
It is sometimes interesting to connect them to structural properties
(i.e. the shape of the graph representing the net)
This way we can give structural characterization of behavioural properties for a class of nets (computationally less expensive to check)
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A matter of terminology
57
To better reflect the above distinction, it is frequent:
to use the term net system for denoting a Petri net with a given initial marking
(we study behavioural properties of systems)
to use the term net for denoting a Petri net without specifying any initial marking (we study structural properties of nets)
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Paths and circuits
58
A path of a net (P, T, F ) is a non-empty sequence x1x2...xk such that
(xi, xi+1) 2 F for every 1 i < k
(and we say that it leades from x1 to xk)
A path from x to y is called a circuit if:no element occurs more than once in it and (y, x) 2 F(since for any x we have (x, x) 62 F , hence a circuit involves at least two nodes)
s
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Connectedness
59
A net (P,T,F) is weakly connected iff it does not fall into (two or more) unconnected parts
(i.e. no two subnets (P1,T1,F1) and (P2,T2,F2) with disjoint and non-empty sets of elements can be
found that partition (P,T,F))
A weakly connected net is strongly connected iff for every arc (x,y) there is a path from y to x
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Connectedness, formally
60
A net (P, T, F ) is weakly connected if every two nodes x, y satisfy
(x, y) � (F ⇥ F�1)⇥
(i.e. if there is an undirected path from x to y)
It is strongly connected if (x, y) � F ⇥
(here the * denotes the reflexive and transitive closure of a binary relation)
(here the * denotes the reflexive and transitive closure of a binary relation)
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A note
61
In the following we will consider (implicitly) weakly connected nets only
(if they are not, then we can study each of their subsystems separately)
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Question time
62
Is the net strongly connected?
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Interference of conflicts and synch
63
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Typical situation:
initially t1 and t2 are not in conflict
but when t3 fires they are in conflict (the firing of t3 is not controllable)
How to rule this situation out?
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S-systems / S-nets
64
A Petri net is called S-system if every transition has one input place and one output place
(S comes from Stellen, the German word for place)
This way any synchronization is ruled out
The theory of S-systems is very simple
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T-systems / T-nets
65
A Petri net is called T-system if every place has one input transition and one output transition
This way all choices/conflicts are ruled out
T-systems are concurrent but essentially deterministic
T-systems have been studied extensively since the early Seventies
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Free-choice nets
66
The aim is to avoid that a choice between transitions is influenced by the rest of the system
Easiest way: keep places with more than one output transition apart
from transitions with more than one input place
In other words, if (p,t) is an arc, then it means that t is the only output transition of p (no conflict)
OR p is the only input place of t (no synch)
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Free-choice systems / nets
67
But we can study a slightly more general class of nets by requiring a weaker constraint
A Petri net is free-choice if for any pair of transitions
their pre-sets are either disjoint or equal
or, equivalently, if for any pair of places
their post-sets are either disjoint or equal
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Question time
68
Is the net an S-net, a T-net, free-choice?
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Free-choice
Exercises
!69
Prove that every S-net is free-choice
Prove that every T-net is free-choice
Show a free-choice net that is neither an S-net nor a T-net
S-nets T-nets