Modular Processings
based on Unfoldings
Eric Fabre & Agnes Madalinski
DistribCom Team
Irisa/Inria
UFO workshop

June 26, 2007
Assembling Petri nets
products, pullbacks, unfoldings and trellises
Modular computations
on a constraint graph : an abstract viewpoint
Application 1: modular diagnosis
or modular computation of a minimal product covering
Application 2: modular prefixes
or how to compute a FCP directly in factorized form
Conclusion
Outline
Nets as Products of Automata
Caution
: in this talk, for simplicity
we limit ourselves to safe Petri nets,
although most results extend to ½ weighted nets
we represent safe nets in “complemented” form,
i.e. their number of tokens remains constant
Building bloc: a
site
or
variable V
= labeled automaton
labeling of transitions
V
= (
S
,
T
,s
0
,
,
¸
,
¤
)
¸
: T
¤
Nets as Products of Automata
(2)
Composition of variables by product :
disjoint union
of places
transitions with shared labels are “glued”
transitions with private labels don’t change
S =
V
1
£
V
2
£
V
3
This product yields a safe (labeled) nets,
and extends to safe nets
Interest of Product Forms
The 1
st
interests are
a natural construction method starting from modules
the compactness of the product form
on this example, the expanded product contains
m*n
transitions, instead of
m+n
in the factorized form
Composition by Pullback
Generalizes the product
allows interactions of nets by an interface (sub

net)
outside the interface, interactions are still by shared labels
S =
V
1
£
V
2
£
V
3
= (
V
1
£
V
2
)
Æ
(
V
2
£
V
3
)
Main property
Graph of a Product Net
Interaction graph of a net
shared labels define the local interactions…
… but it’s better to re

express interactions under the form of
shared variables (or sub

nets).
S =
V
1
£
…
£
V
n
V
1
£
V
2
£
V
3
= (
V
1
£
V
2
)
Æ
(
V
2
£
V
3
)
= S
1
Æ
S
2
Translation in terms of
pullbacks
define
components
S
i
in order to “cover” the shared labels
Unfoldings in Factorized Form
The key =
Universal Property
of the unfolding of S
Let denote the unfolding of S,
and its associated folding (labeling)
8
O,
8
Á
:O
S,
9
!
Ã
:O
U
(S),
Á
= f
S
±
Ã
f
S
:
U
(S)
S
U
(S)
Consequences:
functor has a left adjoint,
and thus preserves products, pullbacks, …
U
U
(S) =
U
(S
1
)
£
O
…
£
O
U
(S
n
)
S = S
1
£
…
£
S
n
)
U
(S) =
U
(S
1
)
Æ
O
…
Æ
O
U
(S
m
)
S = S
1
Æ
…
Æ
S
m
)
Unfoldings in Factorized Form
(2)
Example:
U
(S) =
U
(
V
1
)
£
O
U
(
V
2
)
£
O
U
(
V
3
)
S =
V
1
£
V
2
£
V
3
)
Important properties
The category theory approach naturally provides
an expression for operators (and )
recursive procedures
to compute them (as for unfoldings)
notions of
projections
associated to products/pullbacks:
£
O
Æ
O
¦
S
i
:
U
(S)
U
(S
i
)
Important properties
The category theory approach naturally provides
an expression for operators (and )
recursive procedures
to compute them (as for unfoldings)
notions of
projections
associated to products/pullbacks:
£
O
Æ
O
¦
S
i
:
U
(S)
U
(S
i
)
Important properties
The category theory approach naturally provides
an expression for operators (and )
recursive procedures
to compute them (as for unfoldings)
notions of
projections
associated to products/pullbacks:
£
O
Æ
O
¦
S
i
:
U
(S)
U
(S
i
)
Important properties
The category theory approach naturally provides
an expression for operators (and )
recursive procedures
to compute them (as for unfoldings)
notions of
projections
associated to products/pullbacks:
£
O
Æ
O
¦
S
i
:
U
(S)
U
(S
i
)
Important properties
(2)
Thm
let O
i
be an occ. net of component S
i
,
then is an occ. net of
define then
and this is the
minimal product covering
of O
O=O
1
£
O
…
£
O
O
n
S=S
1
£
…
£
S
n
O’
i
=
¦
S
i
(O)
v
O
i
O=O’
1
£
O
…
£
O
O’
n
The
reduced
occurrence nets
represent the behaviors of component S
i
that remain
once S
i
is inserted in the global system S
or the local view in each component S
i
of the behaviors of the
global system S
are interesting objects !
O’
i
v
O
i
Factorized forms of unfoldings are often more compact…
…but they can however contain useless parts.
Trellises in Factorized Form
The trellis of net S is
obtained by merging conditions of with identical height
a close cousin of merged processes
(Khomenko
et al.,
2005)
T
(S)
U
(S)
time is counted
independently in each V
i
for S = V
1
£
…
£
V
n
Trellises in Factorized Form
The trellis of net S is
obtained by merging conditions of with identical height
a close cousin of merged processes
(Khomenko
et al.,
2005)
enjoys exactly the same factorization properties as unfoldings
T
(S) =
T
(S
1
)
£
T
…
£
T
T
(S
n
)
S = S
1
£
…
£
S
n
)
T
(S) =
T
(S
1
)
Æ
T
…
Æ
T
T
(S
m
)
S = S
1
Æ
…
Æ
S
m
)
T
(S)
U
(S)
Assembling Petri nets
products, pullbacks, unfoldings and trellises
Modular computations
on a constraint graph : an abstract viewpoint
Application 1: modular diagnosis
or modular computation of a minimal product covering
Application 2: modular prefixes
or how to compute a FCP directly in factorized form
Conclusion
Outline
S
2
S
3
S
4
“Abstract” Constraint Reduction
Ingredients :
variables
“systems” or “components” S
i
defined by (local) constraints on
V
max
= {
V
1
,
V
2
,…}
V
i
µ
{
V
1
,…,V
n
}
S
1
V
1
V
5
V
3
V
2
V
7
V
6
V
4
V
8
S = S
1
Æ
S
2
a composition operator (conjunction)
“Abstract” Constraint Reduction
(2)
Reductions:
for , reduces constraints of S to variables V
reductions are projections
V
µ
V
max
¦
V
(S)
¦
V
1
±
¦
V
2
=
¦
V
1
Å
V
2
Central axiom
:
S
1
operates on V
1
, S
2
operates on V
2
let then
i.e.
all interactions go through shared variables
V
3
¶
V
1
Å
V
2
¦
V
3
(S
1
Æ
S
2
) =
¦
V
3
(S
1
)
Æ
¦
V
3
(S
2
)
Modular reduction algorithms
Problem :
Given where S
i
operates on V
i
compute the reduced components
i.e.
how does S
i
change once inserted into the global S ?
S = S
1
Æ
…
Æ
S
n
S’
i
=
¦
V
i
(S)
This can be solved by
Message Passing Algorithms
(MPA)
always converges
only involves local computations
exact if the graph of S is a (hyper

) tree
Modular reduction algorithms
Problem :
Given where S
i
operates on V
i
compute the reduced components
i.e.
how does S
i
change once inserted into the global S ?
S = S
1
Æ
…
Æ
S
n
S’
i
=
¦
V
i
(S)
This can be solved by
Message Passing Algorithms
(MPA)
always converges
only involves local computations
exact if the graph of S is a (hyper

) tree
Modular reduction algorithms
Problem :
Given where S
i
operates on V
i
compute the reduced components
i.e.
how does S
i
change once inserted into the global S ?
S = S
1
Æ
…
Æ
S
n
S’
i
=
¦
V
i
(S)
This can be solved by
Message Passing Algorithms
(MPA)
always converges
only involves local computations
exact if the graph of S is a (hyper

) tree
Modular reduction algorithms
Problem :
Given where S
i
operates on V
i
compute the reduced components
i.e.
how does S
i
change once inserted into the global S ?
S = S
1
Æ
…
Æ
S
n
S’
i
=
¦
V
i
(S)
This can be solved by
Message Passing Algorithms
(MPA)
always converges
only involves local computations
exact if the graph of S is a (hyper

) tree
Modular reduction algorithms
Problem :
Given where S
i
operates on V
i
compute the reduced components
i.e.
how does S
i
change once inserted into the global S ?
S = S
1
Æ
…
Æ
S
n
S’
i
=
¦
V
i
(S)
This can be solved by
Message Passing Algorithms
(MPA)
always converges
only involves local computations
exact if the graph of S is a (hyper

) tree
Modular reduction algorithms
Problem :
Given where S
i
operates on V
i
compute the reduced components
i.e.
how does S
i
change once inserted into the global S ?
S = S
1
Æ
…
Æ
S
n
S’
i
=
¦
V
i
(S)
This can be solved by
Message Passing Algorithms
(MPA)
always converges
only involves local computations
exact if the graph of S is a (hyper

) tree
What about systems with loops ?
Message passing algorithms
converge to a unique fix point
(independent of message scheduling)
that gives an upper approximation:
How good are their results ?
Local extendibility to any tree around each component.
¦
V
i
(S)
v
S’
i
v
S
i
What about systems with loops ?
Message passing algorithms
converge to a unique fix point
(independent of message scheduling)
that gives an upper approximation:
How good are their results ?
Local extendibility to any tree around each component.
¦
V
i
(S)
v
S’
i
v
S
i
Assembling Petri nets
products, pullbacks, unfoldings and trellises
Modular computations
on a constraint graph : an abstract viewpoint
Application 1: modular diagnosis
or modular computation of a minimal product covering
Application 2: modular prefixes
or how to compute a FCP directly in factorized form
Conclusion
Outline
centralized supervizor
Distributed system monitoring…
ab c
b b
a b
c
aa
distributed supervision
Consider the net
and move to trajectory sets
(unfolding or trellis)
In the category of occurrence nets
(for ex.),
we have
a composition operator, the pullback
trajectories of S are in factorized form
we have projection operators on occ. nets,
where V
i
are the variables of S
i
Thm:
projections and pullback satisfy the central axiom
(here we cheat a little however…)
We are already equipped for that !
Æ
O
S = S
1
Æ
…
Æ
S
m
U
(S) =
U
(S
1
)
Æ
O
…
Æ
O
U
(S
m
)
¦
V
i
A computation example
A computation example
A computation example
A computation example
A computation example
A computation example
Assembling Petri nets
products, pullbacks, unfoldings and trellises
Modular computations
on a constraint graph : an abstract viewpoint
Application 1: modular diagnosis
or modular computation of a minimal product covering
Application 2: modular prefixes
or how to compute a FCP directly in factorized form
Conclusion
Outline
Objective
Given
compute a finite complete prefix of in factorized form
Obvious solution:
compute a FCP of
then compute its minimal pullback covering
where
S = S
1
Æ
…
Æ
S
m
U
(S)
U
s
(S)
U
(S)
U
s
(S)
v
U
’(S
1
)
Æ
…
Æ U
’(S
m
)
U
’(S
i
) =
¦
V
i
(
U
s
(S))
but this imposes to work on the global unfolding…
… we rather want to
obtain directly the factorized form
Local canonical prefixes don’t work
Canonical prefix
defined by a
cutting context
Θ = ( ~ ,
⊲
, {
κ
e
}
e
E
)
~
equivalence relation on
Conf
set of reachable markings
⊲
adequate order on
Conf
partial order on
Conf
refining inclusion
{
κ
e
}
e
E
a subset of
Conf
,
configurations used for cut

off identification
cut

off
event
Extended canonical prefix
Toy example :
two components, elementary interface (=automaton)
S =
A
£
C
£
B
= (
A
£
C
)
Æ
(
C
£
B
)
= S
A
Æ
S
B
interface
Extended canonical prefix
(2)
extended prefix of w.r.t. its interface
C
restriction of the cutting context
Θ
C
= (~,
⊲
, {
κ
e
}
e
E
)
to
particular configurations
κ
e
e cut

off event, corresponding event e’ :
κ
e
~
κ
e’
and
κ
e’
⊲
κ
e
where usually
κ
e
=[e]
if e
is a
private event, then
P
C
(
κ
e
∆
κ
e’
)=Ø
if e is an interface event, then e’ is also an interface event
S
A
where
∆
is the symmetric set difference
Extended cut

off event
e : extended cut

off
e’ : interface event
Summary net
Summary net =
behaviors allowed by an extended prefix on the interface:
obtained by projecting the extended prefix on the interface,
and refolding matching markings
merge
Distributed computations
augmented
prefixes
Distributed computations
extract
summary nets
Distributed computations
exchange
summary nets
Distributed computations
build
pullbacks
Distributed computations
construct
prefixes
Distributed computations
Killed in the pullback
Local factors are a little too conservative
(not the minimal pullback covering of the FCP)
Assembling Petri nets
products, pullbacks, unfoldings and trellises
Modular computations
on a constraint graph : an abstract viewpoint
Application 1: modular diagnosis
or modular computation of a minimal product covering
Application 2: modular prefixes
or how to compute a FCP directly in factorized form
Conclusion
Outline
A few lessons…
Factorized forms of unfoldings are generally more compact.
One can work directly on them, in an efficient modular
manner, without ever having to compute anything global.
Optimal when component graphs are trees.
Sub

optimal, but provide “good” upper approximations
otherwise.
…and some questions
Finite complete prefixes in factorized form:
we need to understand better how to compute them,
and provide complexity results.
Can this be useful for model checking?
Can this be useful for distributed optimal planning?
(see last talk today)
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