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On the constructive orbit problem

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Abstract

Symmetry reduction techniques aim to combat the state-space explosion problem for model checking by restricting search to representative states from equivalence classes with respect to a group of symmetries. The standard approach to representative computation involves converting a state to its minimal image under a permutation group G, before storing the state. This is known as the constructive orbit problem (COP), and is \({\mathit{NP}}\) hard. It may be possible to solve the COP efficiently if G is known to have certain structural properties: in particular if G is isomorphic to a full symmetry group, or G is a disjoint/wreath product of subgroups. We extend existing results on solving the COP efficiently for fully symmetric groups, and investigate the problem of automatically classifying an arbitrary permutation group as a disjoint/wreath product of subgroups. We also present an approximate COP strategy based on local search, and some computational group-theoretic optimisations to improve the basic approach of solving the COP by symmetry group enumeration. Experimental results using the TopSPIN symmetry reduction package, which interfaces with the computational group-theoretic system GAP, illustrate the effectiveness of our techniques.

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Authors and Affiliations

  1. Oxford University Computing Laboratory, Oxford, UK

    Alastair F. Donaldson

  2. Department of Computing Science, University of Glasgow, Glasgow, UK

    Alice Miller

Authors
  1. Alastair F. Donaldson
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  2. Alice Miller
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Correspondence to Alastair F. Donaldson.

Additional information

Alastair F. Donaldson is supported by EPSRC grant EP/G051100. Alice Miller is supported by EPSRC grant EP/E032354.

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Donaldson, A.F., Miller, A. On the constructive orbit problem. Ann Math Artif Intell 57, 1–35 (2009). https://doi.org/10.1007/s10472-009-9171-4

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