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On the constructive orbit problem. (English) Zbl 1205.68220
Summary: 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 \(NP\) 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.

68Q60 Specification and verification (program logics, model checking, etc.)
68N30 Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
68U07 Computer science aspects of computer-aided design
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20E22 Extensions, wreath products, and other compositions of groups
Full Text: DOI
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