# Problem solution sustenance in XCS: Markov chain analysis of niche support distributions and the impact on computational complexity

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## Abstract

Michigan-style learning classifier systems iteratively evolve a distributed solution to a problem in the form of potentially overlapping subsolutions. Each problem *niche* is covered by subsolutions that are represented by a set of predictive rules, termed classifiers. The genetic algorithm is designed to evolve classifier structures that together cover the whole problem space and represent a complete problem solution. An obvious challenge for such an online evolving, distributed knowledge representation is to continuously sustain all problem subsolutions covering all problem niches, that is, to ensure *niche support*. Effective niche support depends both on the probability of reproduction and on the probability of deletion of classifiers in a niche. In XCS, reproduction is occurrence-based whereas deletion is support-based. In combination, niche support is assured effectively. In this paper we present a Markov chain analysis of the niche support in XCS, which we validate experimentally. Evaluations in diverse Boolean function settings, which require non-overlapping and overlapping solution structures, support the theoretical derivations. We also consider the effects of mutation and crossover on niche support. With respect to computational complexity, the paper shows that XCS is able to maintain (partially overlapping) niches with a computational effort that is linear in the inverse of the niche occurrence frequency.

## Keywords

Learning classifier systems LCS XCS Niching Markov chain analysis Solution sustenance Mutation## Notes

### Acknowledgments

We are grateful to Xavier Llorà, Kei Onishi, Martin Pelikan, Stewart Wilson, Tian-Li Yu, and the whole IlliGAL lab for their help and the useful discussions.

This work was supported by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant FA9550-06-1-0096, the National Science Foundation under ITR grant DMR-03-25939 (at Materials Computation Center, UIUC), The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. Butz's contribution received additional funding from the European commission contract no. FP6-511931.

The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of any of the organizations mentioned above

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