Download Nested Nets for Adaptive Workflows

History-Dependent Petri Nets
Kees van Hee, Alexander Serebrenik,
Natalia Sidorova, Wil van der Aalst
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Motivation
 Choices made in processes often depend on
the previous events – process history
Process history is often available from logs
 Encoding history-based decisions into the net
structure can be
• complex,
• error-prone, and
• decreasing the readability
We do not model the whole state of a system, but only
some aspects of it!
a
a'
b
b'
Goal
History logic
History: a partially ordered set of labeled events
E, ≤ ,λ, where λ: E → Σ
Operations on history:
 extension with an event
 union of histories
Given a set of labels Σ and an event variable x, we define
a formula φ, a term q and a label expression l :
Examples of history guards
Short-hand notations:
#S – the number of events with a label from set S,
x: φ ≡ (#x: φ)>0
x: φ ≡ (#x: φ) =(#x: true)
Examples:
#{a} < #{b}
x: (λ(x) = a  y: (λ(y) = b  x ≤ y))
Token history nets
Tokens are colored with histories.
Transitions have guards evaluated on the union of
the histories of consumed tokens.
Tokens produced by a transition firing are colored
with the union of histories of the consumed tokens
extended by the firing event.
b
a
p
d
c
q
#{a}=1
Global history nets
History is available to all transitions.
Guards are evaluated on this history.
Transition firings extend the history with
a corresponding event.
Global history nets are Turing complete:
Take a net where transitions have unique labels.
Then guards of the type
allow to simulate inhibitor arcs.
Token history vs. global history
b
a
Suppose the marking of the token history net is
Suppose the global history is aabc.
[(p,HWhat
1)] + [(p,H
2)] + [(q,H
3)] + [(q,H4)] where
are token
histories?
H1 =Token
{e1,eon
≤ eq2},
{(e1,a),(e
p 1and
might
share2,b)},
the event
2}, {e
a,4 but
H2 =with
{e4label
,e5}, {e
≤ e5not
}, {(enecessarily.
4,a),(e5,b)},
p
d
c
q
#{a}=1
H3 = {e1,e3}, {e1 ≤ e3}, {(e1,a),(e3,c)},
Corollary: token history cannot be reconstructed
H4 = {e4,e6}, {e4 ≤ e6}, {(e4,a),(e6,c)}.
from the global history.
What is the global history?
It might be abcabc, or aabbcc, or abacbc, etc.
Corollary: the global history cannot be
reconstructed from the token histories.
However we can simulate global history by token history
Each transition t is connected to
p
extra place p.
Initially with the empty history:
the token history of p is the
global history
t
Counting Formulae Guards
A counting formula φ is defined by
Nets with counting formulas are Turing complete
(remember:
)
So we provide a translation of this class into bisimilar inhibitor
nets.
Guards (#A) ρ (#B + k)
g(t)
...
a
A
t
...
b
B
(a) Transition with guard g(t)
...
a'
...
...
a
A’
s'
t
s
invariant:
max{0, #A-#B-k -1}
...
b'
...
B’
(b) g(t) = #A ≤#B+k removed
s
initial:
1
...
B
...
A
invariant:
max{0, #A-#B+1}
invariant:
max{0,#B-#A+k +1}
b
a
A’
A
initial:
k +1
a'
b'
k+1
t
s’
invariant:
max{0,#B-#A -1}
b
...
B’
B
(c) g(t) = #A ≥ #B+k removed
Construction for compound formulae
•Consider a guard g(t) for transition t in disjunctive normal form
• ¬φ is trivial
• φ1  φ2: test arcs for each formulae to t
• φ1  φ2: duplicate t into t and t’ each with one test arc
Bounded synchronization distance
Let (N,m) be a Petri net with n transitions and y  Zn.
The y-distance of (N,m) is
D((N,m),y)) = yT ·
sup 
 
where  is the set of all executable finite firing sequences.
Let A and B be two label sets and y(A,B) is a vector with 1 for
transitions labeled by an A label, -1 for B labels and 0 for the rest.
The y(A,B) distance is the max difference in the number
of the firings of A-transitions and the number of the firings
of B-transitions. There is an efficient algorithm to determine if the distance is finite!
When the guards are of the form #A  #B + k and y(A,B) < , we can construct a
Petri net bisimilar to the global history net.
Similarly #A  #B + k and y(B,A) < 
LTL guards without Next
• LTL formulae can be translated into BÜchi automata
• LTL-guards are evaluated on finite traces (history!)  they
can be translated to finite deterministic automata.
• Synchronous product of the Petri net with these automata
yields a classical Petri net.
Conclusion
 Incorporating history into a model allows to separate
process information from safety constraints.
 History nets are Turing complete.
 Still, some subclasses can be translated to classical
PNs.
 For some other subclasses a translation to inhibitor nets
is proposed.
Future Work
 The use of tokens nets for component-based systems.
• Need for additional operations like projection and
intersection of histories
 Transformation of other subclasses to PNs.
 What about Presburger logic?
symbols and constructs
           
 

* 
           
   min(t) max(t)
a* t p:  t p: •
1(o) 2(o)