Optimised Calculation of Symmetries for State Space Reduction

Optimised Calculation ofSymmetries for State SpaceReduction

HARRO WIMMEL

Universität Rostock, Institut für Informatik

Overview

What are Symmetries?DefinitionCalculating Symmetries in LoLA

Optimised Calculation of SymmetriesInheritance of Dead BranchesProducts instead of PowersPartial Orthogonalisation

Experimental Results

What are Symmetries?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.

What are Symmetries?An Example

1 - 12 -

1 - 12 - 4

1 - 12 - 4, b - d

1 - 12 - 4, b - d3 - 5

1 - 12 - 4, b - d3 - 5, c - e

1 - 12 - 4, b - d3 - 5, c - e2 - 6

1 - 12 -

1 - 1, a - a2 - 3, b - c

1 - 1, a - a2 - 3, b - c3 - 2, c - b

1 - 1, a - a2 - 3, b - c3 - 2, c - b4 - 5, d - e5 - 6, e - f6 - 4, f - d

Written as a permutation:(1)(a)(2 3)(b c)(4 5 6)(d e f )

The mapping of transitions is usuallyforced, that is, (2 3)(4 5 6) suffices asa full representation.

Caclulating Symmetries in LoLAThe 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

Caclulating Symmetries in LoLAThe 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

Calculating Symmetries in LoLABuilding 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.

Calculating Symmetries in LoLAPartial Symmetries and Dead Branches

The brute-force approach recursively completes partial symmetries (starting withthe empty set) to full symmetries by assigning images to objects.

The examples assignment 1- 1, 2- 4 cannot be completed whatever wechoose 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, and6 to exclude the existence of a symmetry, i.e. it has to search this dead branchcompletely.

Calculating Symmetries in LoLAGenerators

Symmetries can be built from a set of generators. For n objects (places) we needfrom each class S i

j (1 ≤ i ≤ j ≤ n) of symmetries with

• k - k for all k < i and

• i - j

exactly one symmetry %ij ∈ S i

j , then we can write any symmetry σ as

σ = %nmn◦ . . . ◦ %2

m2◦ %1

with m1 = σ(1), m2 = (%1m1

)−1(σ(2)), m3 = (%2m2

)−1((%1m1

)−1(σ(3))) etc.We obtain a „small“ set of generators (compared to the number of all symmetries).

Optimisation: Inheritance of dead branches

PrerequisitesThe brute-force approach has shown a dead branch for some S i

j , i.e. S ij = ∅.

There is a symmetry in some S jm.

TheoremThen, S i

m = ∅ and needs not be computed by the brute-force approach.

ProofFrom %i

m ∈ S im and %j

m ∈ S jm we conclude %i

m ◦ (%jm)−1 ∈ S i

j . A contradiction.

Optimisation: Products instead of Powers

Instead of computing only the powers σ1, σ2, σ3, . . . and checking whether thesebelong to a so far empty class S i

j , we can use all symmetries found up until now:

Building ProductsFor each newly found symmetry σ ∈ S i

j and each formerly known symmetry% ∈ Sk

m with k ≥ i we can check in O(1) if we already know of a symmetry inS i

%(j). If not, σ ◦ % ∈ S i%(j) is one.

Optimisation: Products instead of PowersRough Complexity

If n is the number of objects (places) and m is the size of a set of generators for thesymmetries, we need for all symmetries

• for the calcuation of powers:

• n powers (up to the „identity“ S ii ) 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 OrthogonalisationAn Example

Generators (e.g.):• %1

1 = (2 3)(4 6 5),• %2

2 = id, %23 = (2 3)(4 5 6)

• %44 = id, %4

5 = (4 5 6), %46 = (4 6 5)

Reduction:• (%1

1)6 = id ∈ S1

1 ,• %2

2 = id, (%23)

3 = (2 3) ∈ S23• %4

4 = id, %45 = (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 sameclass S i

• Can be infeasible, for an optimal power eliminate all prime factors of |GC |from |OC |

ResultPartial orthogonalisation reduces the size of the representation and partially allowsto commute symmetries, i.e. such symmetries do not influence each other.

Experimental ResultsDead Branches

Let Ni be the Petri Net with i cycles of lengths from 1 to i (N3 being the examplenet). Many dead branches occur. A comparison of LoLA without and with theproposed optimisation yields:

Ni #gen LoLA-old LoLA-newN10 45 <0.1s <0.1sN15 105 0.4s 0.2sN20 190 2.4s 1.3sN25 300 8.2s 4.4sN30 435 25s 13sN35 595 66s 34s

#gen is the size of the generator set.

Experimental ResultsProducts instead of Powers

The optimisation of using products instead of powers can be shown with a grid ofcommunication agents in d dimensions with n agents in any direction (ECHO d/n).

ECHO #symm #gen LoLA-old LoLA-new3/3 48 10 0.2s <0.1s3/5 48 10 4.6s 2.9s3/7 48 10 29s 16s4/3 384 21 3.5s 1.6s4/5 384 21 244s 96s5/3 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, HumboldtUniversität 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!

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