Optimised Calculation of Symmetries for State Space Reduction

Optimised Calculation of
Symmetries for State Space
Reduction
HARRO WIMMEL
Universität Rostock, Institut für Informatik
8. Oktober 2010 c 2010 UNIVERSITÄT ROSTOCK | FAKULTÄT FÜR INFORMATIK & ELEKTROTECHNIK, INSTITUT FÜR INFORMATIK 1 / 17

Overview
What are Symmetries?
Deﬁnition
Calculating Symmetries in LoLA
Optimised Calculation of Symmetries
Inheritance of Dead Branches
Products instead of Powers
Partial Orthogonalisation
Experimental Results

Symmetries are Automorphisms
A symmetry maps a Petri Net onto itself:
• Places to places
• Transitions to transitions
• Edges to edges
• Tokens to tokens, if applicable
• bijectively, i.e. each object appears exactly once as image.

An Example
1
a
2 b
3c
4 d 5
e6f

An Example
1
a
2 b
3c
4 d 5
e6f
1 -

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1
2 -

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1
2 - 4

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1
2 - 4, b - d

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1
2 - 4, b - d
3 - 5

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1
2 - 4, b - d
3 - 5, c - e

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1
2 - 4, b - d
3 - 5, c - e
2 - 6

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1, a - a
2 - 3, b - c

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1, a - a
2 - 3, b - c
3 - 2, c - b

An Example
1
a
2 b
3c
4 d 5
e6f
1 - 1, a - a
2 - 3, b - c
3 - 2, c - b
4 - 5, d - e
5 - 6, e - f
6 - 4, f - d
Written as a permutation:
(1)(a)(2 3)(b c)(4 5 6)(d e f )
The mapping of transitions is usually
forced, that is, (2 3)(4 5 6) sufﬁces as
a full representation.

Caclulating Symmetries in LoLA
The Implementation so far
1. Brute-force-method to calculate symmetries (see the example)
• high complexity (solves the graph isomorphism problem, unknown if polynomial)
• needs to be done at least for the ﬁrst symmetry
2. Calculation of powers of symmetries
• in linear time (per power)
• needs a symmetry as input
3. Combination of these methods

Building Powers of Symmetries
For our example σ = (2 3)(4 5 6) we calculate
• σ1
= (2 3)(4 5 6),
• σ2
= (4 6 5),
• σ3
= (2 3),
• σ4
= (4 5 6),
• σ5
= (2 3)(4 6 5),
• σ6
= id (identity).
By chance, these are all the symmetries there are.

Partial Symmetries and Dead Branches
The brute-force approach recursively completes partial symmetries (starting with
the empty set) to full symmetries by assigning images to objects.
The examples assignment 1- 1, 2- 4 cannot be completed whatever we
choose to be the images of the places 3, 4, 5, and 6.
But: the brute-force algorithm has to check all possible assignments to 3, 4, 5, and
6 to exclude the existence of a symmetry, i.e. it has to search this dead branch
completely.

Generators
Symmetries can be built from a set of generators. For n objects (places) we need
from each class Si
j (1 ≤ i ≤ j ≤ n) of symmetries with
• k - k for all k < i and
• i - j
exactly one symmetry i
j ∈ Si
j , then we can write any symmetry σ as
σ = n
mn
◦ . . . ◦ 2
m2
◦ 1
m1
with m1 = σ(1), m2 = ( 1
m1
)−1
(σ(2)), m3 = ( 2
m2
)−1
(( 1
m1
)−1
(σ(3))) etc.
We obtain a „small“ set of generators (compared to the number of all symmetries).

Optimisation: Inheritance of dead branches
Prerequisites
The brute-force approach has shown a dead branch for some Si
j , i.e. Si
j = ∅.
There is a symmetry in some Sj
m.
Theorem
Then, Si
m = ∅ and needs not be computed by the brute-force approach.
Proof
From i
m ∈ Si
m and j
m ∈ Sj
m we conclude i
m ◦ ( j
m)−1
∈ Si
j . A contradiction.

Optimisation: Products instead of Powers
Instead of computing only the powers σ1
, σ2
, σ3
, . . . and checking whether these
belong to a so far empty class Si
j , we can use all symmetries found up until now:
Building Products
For each newly found symmetry σ ∈ Si
j and each formerly known symmetry
∈ Sk
m with k ≥ i we can check in O(1) if we already know of a symmetry in
Si
(j). If not, σ ◦ ∈ Si
(j) is one.

Optimisation: Products instead of Powers
Rough Complexity
If n is the number of objects (places) and m is the size of a set of generators for the
symmetries, we need for all symmetries
• for the calcuation of powers:
• n powers (up to the „identity“ Si
i ) for m symmetries
• O(n) per power, altogether O(m ∗ n2
).
• for the calculation of products:
• m products, m < n or m > n possible, for m symmetries
• O(1) per test, if no new symmetry is obtained
• O(n) per new symmetry, altogether O(m2
+ m ∗ n)
Typically, products lead faster to more symmetries than powers.

Optimisation: Partial Orthogonalisation
An Example
1
a
2 b
3c
4 d 5
e6f
Generators (e.g.):
• 1
1 = (2 3)(4 6 5),
• 2
2 = id, 2
3 = (2 3)(4 5 6)
• 4
4 = id, 4
5 = (4 5 6), 4
6 = (4 6 5)
Reduction:
• ( 1
1)6
= id ∈ S1
1 ,
• 2
2 = id, ( 2
3)3
= (2 3) ∈ S2
3
• 4
4 = id, 4
5 = (4 5 6), 4
6 = (4 6 5)

Optimisation: Partial Orthogonalisation
• Generating cycle GC contains index (for 2
3 = (2 3)(4 5 6) it is
(2 3) = GC)
• All powers σ|GC|k+1
(k ∈ N) of σ contain GC
• All powers σ|OC|
( ∈ N) do not contain some other cycle OC
• Solution of |GC|k + 1 = |OC| eliminates OC and remains in the same
class Si
j
• Can be infeasible, for an optimal power eliminate all prime factors of |GC|
from |OC|
Result
Partial orthogonalisation reduces the size of the representation and partially allows
to commute symmetries, i.e. such symmetries do not inﬂuence each other.

Dead Branches
Let Ni be the Petri Net with i cycles of lengths from 1 to i (N3 being the example
net). Many dead branches occur. A comparison of LoLA without and with the
proposed optimisation yields:
Ni #gen LoLA-old LoLA-new
N10 45 <0.1s <0.1s
N15 105 0.4s 0.2s
N20 190 2.4s 1.3s
N25 300 8.2s 4.4s
N30 435 25s 13s
N35 595 66s 34s
#gen is the size of the generator set.

Products instead of Powers
The optimisation of using products instead of powers can be shown with a grid of
communication agents in d dimensions with n agents in any direction (ECHO d/n).
ECHO #symm #gen LoLA-old LoLA-new
3/3 48 10 0.2s <0.1s
3/5 48 10 4.6s 2.9s
3/7 48 10 29s 16s
4/3 384 21 3.5s 1.6s
4/5 384 21 244s 96s
5/3 3840 41 82s 23s
#symm is the number of all symmetries, #gen the size of the generator set.

K. Schmidt: Explicit State Space Veriﬁcation, Habilitation Thesis, Humboldt
Universität zu Berlin, 2002.
K. Wolf: LoLA – A low level analyzer,
http://www.informatik.uni-rostock.de/∼nl/wiki/tools/lola, 2010.
W. Reisig: Elements of Distributed Algorithms, Springer Verlag, 1998.

Thanks For Your Attention!

Optimised Calculation of Symmetries for State Space Reduction

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Optimised Calculation of Symmetries for State Space Reduction