Factored Planning: From Automata to Petri Nets
Loïg Jezequel1 , Eric Fabre2 , Victor Khomenko3
ACSD, July 9, 2013
1
2
3
ENS Cachan Bretagne
INRIA Rennes
Newcastle University
Introduction
From automata to Petri nets
Experimental results
Conclusions
Planning
Factored planning
Previous results
Why Petri nets?
Goal
Find a
: a sequence of actions (with minimal cost) moving the
system from its initial state to one of its goal states
plan
2 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Planning
Factored planning
Previous results
Why Petri nets?
Each component is a planning
problem with its own resources and
actions
Goal
Find a set of
global plan
compatible local plans: they can be interleaved into a
3 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Planning
Factored planning
Previous results
Why Petri nets?
Each component is a planning
problem with its own resources and
actions
The components interact by
resources and/or actions
Goal
Find a set of
global plan
compatible local plans: they can be interleaved into a
3 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Components ⇒ Automata
Plans ⇒ Words
Interaction ⇒ Synchronous product
Planning
Factored planning
Previous results
Why Petri nets?
New goal
Given A = A1 || . . . ||A , nd a
in A by local computations
n
word
4 / 16
Introduction
Planning
From automata to Petri nets
Factored planning
Experimental results
Previous results
Conclusions
Why Petri nets?
Components ⇒ Automata
Plans ⇒ Words
Interaction ⇒ Synchronous product
A possibility [Fabre et al. 10]
New goal
Given A = A1 || . . . ||A , nd a
in A by local computations
n
word
i
Compute A0 = ΠΣ (A) for each without computing A
i
i
Why?
any word
any word
1
2
w of A can be projected into a word w of ΠΣ (A)
w of ΠΣ (A) is the projection of a word w of A
i
i
i
i
⇒ Easy extraction of a word from A by local searches
4 / 16
Introduction
Planning
From automata to Petri nets
Factored planning
Experimental results
Previous results
Conclusions
A possibility [Fabre et al. 10]
Why Petri nets?
i
Compute A0 = ΠΣ (A) for each without computing A
i
i
How? Conditional independence like property
ΠΣ
1
∩Σ2 (A1
× A2 ) ≡L ΠΣ
1
∩Σ2 (A1 )
× ΠΣ
1
∩Σ2 (A2 )
Application:
Σ1 ∩ Σ2
A1
ΠΣ (A) =
1
Σ2 ∩ Σ3
A2
A3
ΠΣ (A1 × A2 × A3 )
1
≡L
ΠΣ (A1 ) × ΠΣ (A2 × A3 )
≡L
A1 × Π Σ
∩Σ2 (A2
× A3 )
≡L
A1 × Π Σ
∩Σ2 (A2
× ΠΣ
1
1
1
1
2
∩Σ3 (A3 ))
5 / 16
Introduction
Planning
From automata to Petri nets
Factored planning
Experimental results
Previous results
Conclusions
A possibility [Fabre et al. 10]
Why Petri nets?
i
Compute A0 = ΠΣ (A) for each without computing A
i
i
How? Conditional independence like property
ΠΣ
1
∩Σ2 (A1
× A2 ) ≡L ΠΣ
1
∩Σ2 (A1 )
× ΠΣ
1
∩Σ2 (A2 )
Application:
A1
Σ1 ∩ Σ2
A2
×
Σ2 ∩ Σ3
×
ΠΣ1 ∩Σ2
ΠΣ (A) =
1
A3
ΠΣ2 ∩Σ3
ΠΣ (A1 × A2 × A3 )
1
≡L
ΠΣ (A1 ) × ΠΣ (A2 × A3 )
≡L
A1 × Π Σ
∩Σ2 (A2
× A3 )
≡L
A1 × Π Σ
∩Σ2 (A2
× ΠΣ
1
1
1
1
2
∩Σ3 (A3 ))
5 / 16
Introduction
Planning
From automata to Petri nets
Factored planning
Experimental results
Previous results
Conclusions
A possibility [Fabre et al. 10]
Why Petri nets?
i
Compute A0 = ΠΣ (A) for each without computing A
i
i
How? Generalization
Message passing algorithms : proceed by successive renements
A1
ΠΣ
1
×
A2
×
ΠΣ
ΠΣ
A3
×
A4
2
ΠΣ
3
A5
2
messages from root to leaves
messages from leaves to root
A1
ΠΣ
2
ΠΣ
3
×
A3
×
A2
ΠΣ
4
×
A4
ΠΣ
5
A5
6 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Planning
Factored planning
Previous results
Why Petri nets?
Concurrency in factored planning problems
Global concurrency: between components (private actions)
Local concurrency: internal to a component
Remark: local concurrency is not anecdotal
p1
f4
p1 , f4
f1
f1 , p4
p4
p2
p2 , f3
f3
f2
p3
f2 , p3
7 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Planning
Factored planning
Previous results
Why Petri nets?
Concurrency in factored planning problems
Global concurrency: between components (private actions)
Local concurrency: internal to a component
Concurrency in networks of automata
Global concurrency: taken into account
Local concurrency: ignored!
7 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Planning
Factored planning
Previous results
Why Petri nets?
Concurrency in factored planning problems
Global concurrency: between components (private actions)
Local concurrency: internal to a component
Concurrency in networks of automata
Global concurrency: taken into account
Local concurrency: ignored!
Networks of Petri nets
Global concurrency: taken into account
Local concurrency: taken into account
7 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Languages, automata, Petri nets
Product of Petri nets
Projection of Petri nets
The real purpose of automata
Implementation of product and projection of regular languages with
nite objects
Our goal
More ecient implementation by taking local concurrency into
account:
product of languages ⇒ product of Petri nets
projection of languages ⇒ projection of Petri nets
8 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
p1
q1
p2
a a
q2
a
N1
Languages, automata, Petri nets
Product of Petri nets
Projection of Petri nets
p1
b a
N2
q1
a
p2
N1 × N2
b
q2
9 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
N
ΠΣ ( ): a two step procedure
1
2
Languages, automata, Petri nets
Product of Petri nets
Projection of Petri nets
silent transitions
Replace the transitions with label not in Σ by
Remove silent transitions (optimisation purpose)
10 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
N
ΠΣ ( ): a two step procedure
1
2
Languages, automata, Petri nets
Product of Petri nets
Projection of Petri nets
silent transitions
Replace the transitions with label not in Σ by
Remove silent transitions (optimisation purpose)
How to remove silent transitions
Use the reachability graph: no more concurrency
Preservation of concurrency: for restricted class of nets only
[Wimmel 04]
Transition contraction: ecient in practice
[André 82] [Vogler and Kangsah 07]
10 / 16
Introduction
Languages, automata, Petri nets
From automata to Petri nets
Product of Petri nets
Experimental results
Projection of Petri nets
Conclusions
t
Contraction of a silent transition , only when
p1
q1
( 1,
b
ε
p2
p p2 )
q2
t ∩ t• = ∅
•
q p2)
( 2,
b
p q1)
( 1,
q q1)
( 2,
11 / 16
Introduction
Languages, automata, Petri nets
From automata to Petri nets
Product of Petri nets
Experimental results
Projection of Petri nets
Conclusions
t
Contraction of a silent transition , only when
p1
q1
( 1,
b
ε
p2
p p2 )
t ∩ t• = ∅
•
q2
q p2)
( 2,
b
p q1)
q q1)
( 1,
Language and safeness preserving contraction of
( 2,
t
t = 1, t = {t } and M p) = 0 with t • = {p}
or |•t | = 1, •(t • ) = {t } and ∀p ∈ t • , M 0 (p ) = 0
or |•t | = 1 and (•t )• = {t }
|
•|
•( • )
0(
11 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Experimental setting
Negative results
Positive results
Benchmark selection
From Corbett96
Scale well (number of components vs. size of components)
Tree shape (manually obtained)
Benchmark set
Dining philosophers
Dining philosophers with a dictionary
Divide and conquer
Milner's cyclic scheduler
Token-ring mutual exclusion protocol
What we compare
Times spent to compute updated automata/Petri nets
12 / 16
Introduction
Experimental setting
From automata to Petri nets
Negative results
Experimental results
Positive results
Conclusions
Divide and conquer
Dining philosophers
10000
Automata
Petri nets
1000
1000
100
100
time (s)
time (s)
10000
10
Automata
Petri nets
10
1
1
0.1
0.1
0.01
0.01
20
40
60
80
100 120 140
number of tasks
160
180
200
20
40
60
80
100 120 140
number of philosophers
160
180
200
13 / 16
Introduction
Experimental setting
From automata to Petri nets
Negative results
Experimental results
Positive results
Conclusions
Token-ring mutual exclusion protocol
10000
Automata
Petri nets
1000
1000
100
100
time (s)
time (s)
10000
10
10
1
1
0.1
0.1
0.01
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
number of users
Petri nets
0.01
20
40
60
80
100 120 140
number of users
160
180
200
14 / 16
Introduction
Experimental setting
From automata to Petri nets
Negative results
Experimental results
Positive results
Conclusions
Dining philosophers with a dictionary
10000
Automata
Petri nets
1000
1000
100
100
time (s)
time (s)
10000
10
10
1
1
0.1
0.1
0.01
3
4
5
6
7
0.01
8 9 10 11 12 13 14 15 16 17 18 19
number of philosophers
Petri nets
20
40
60
80
100 120 140
number of philosophers
160
180
200
180
200
Milner's cyclic scheduler
10000
Automata
Petri nets
1000
1000
100
100
time (s)
time (s)
10000
10
10
1
1
0.1
0.1
0.01
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
number of schedulers
Petri nets
0.01
20
40
60
80
100 120 140
number of schedulers
160
15 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Contribution
Networks of automata ⇒ networks of Petri nets for planning
Experimental comparison: Petri nets can bring an important
eciency gain by handling local concurrency
Extension to weighted systems (in the paper)
16 / 16
Introduction
From automata to Petri nets
Experimental results
Conclusions
Contribution
Networks of automata ⇒ networks of Petri nets for planning
Experimental comparison: Petri nets can bring an important
eciency gain by handling local concurrency
Extension to weighted systems (in the paper)
Possible future work
Compare transition contraction without and with weights
Relax the conditions for transition contraction with weights
16 / 16