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Crossover can be constructive when computing unique input–output sequences

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Abstract

Unique input–output (UIO) sequences have important applications in conformance testing of finite state machines (FSMs). Previous experimental and theoretical research has shown that evolutionary algorithms (EAs) can compute UIOs efficiently on many FSM instance classes, but fail on others. However, it has been unclear how and to what degree EA parameter settings influence the runtime on the UIO problem. This paper investigates the choice of acceptance criterion in the (1 + 1) EA and the use of crossover in the \((\mu+1)\) Steady State Genetic Algorithm. It is rigorously proved that changing these parameters can reduce the runtime from exponential to polynomial for some instance classes of the UIO problem.

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Acknowledgments

The authors would like to thank Pietro Oliveto for useful comments. This work was supported by EPSRC under grant no. EP/D052785/1.

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Authors and Affiliations

  1. The Centre of Excellence for Research in Computational Intelligence and Applications (CERCIA), School of Computer Science, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    Per Kristian Lehre & Xin Yao

Authors
  1. Per Kristian Lehre
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  2. Xin Yao
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Corresponding author

Correspondence to Per Kristian Lehre.

Additional information

A preliminary version of this paper appeared in Proceedings of the 7th International Conference on Simulated Evolution and Learning (SEAL’08) (Lehre and Yao 2008).

Appendix

Appendix

The appendix states a result obtained elsewhere which was used and cited in runtime analysis of EAs in this paper.

Theorem 6 [Simple drift theorem (Oliveto and Witt 2008)]

Let \(X_t, t\geq 0,\) be the random variables describing a Markov process over the state space \(S:=\left\{0,1,\ldots,N\right\},\) and denote \(\Updelta_t(i) := (X_{t+1}-X_t\mid X_t=i)\) for \(i\in S\) and \(t\geq 0.\) Suppose there exists an interval \([a,b]\) of the state space and three constants \(\beta,\delta,r>0\) such that for all \(t\geq 0\)

  1. 1.

    \({\mathbf{E}}\left[{\Updelta_t(i)}\right]\geq \beta\) for \(a<i<b,\) and

  2. 2.

    \({\mathbf{Pr}}\left[{\Updelta_t(i)=-j}\right] \leq 1/(1+\delta)^{j-r}\) for \(i>a\) and \(j\geq 1,\) then there is a constant \(c^*>0\) such that for

    $$ T^* := \min\left\{t\geq 0:X_t\leq a\mid X_0\geq b\right\} $$

    it holds that \({\mathbf{Pr}}\left[{T^*\leq 2^{c^*(b-a)}}\right]=2^{-\Omega(b-a)}.\)

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Lehre, P.K., Yao, X. Crossover can be constructive when computing unique input–output sequences. Soft Comput 15, 1675–1687 (2011). https://doi.org/10.1007/s00500-010-0610-2

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