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On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

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Genetic Programming (EuroGP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9594))

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Abstract

This work presents a first step towards a systematic time and space complexity analysis of genetic programming (GP) for evolving functions with desired input/output behaviour. Two simple GP algorithms, called (1+1) GP and (1+1) GP*, equipped with minimal function (F) and terminal (L) sets are considered for evolving two standard classes of Boolean functions. It is rigorously proved that both algorithms are efficient for the easy problem of evolving conjunctions of Boolean variables with the minimal sets. However, if an extra function (i.e. NOT) is added to F, then the algorithms require at least exponential time to evolve the conjunction of n variables. On the other hand, it is proved that both algorithms fail at evolving the difficult parity function in polynomial time with probability at least exponentially close to 1. Concerning generalisation, it is shown how the quality of the evolved conjunctions depends on the size of the training set s while the evolved exclusive disjunctions generalize equally badly independent of s.

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Notes

  1. 1.

    We assume the SUB and DEL of an empty tree return an empty tree.

  2. 2.

    Simplification is a conceptual tool used for the proofs. The actual tree contains all the variables (i.e., the algorithm does not simplify the trees).

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Acknowledgements

The authors would like to thank Alberto Moraglio for constructive discussions which initialised this work. Further preliminary discussions occurred at Dagstuhl Seminar N.15211. This work was supported by EPSRC under Grant n. EP/M004252/1.

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Authors and Affiliations

  1. University of Sheffield, Sheffield, UK

    Andrea Mambrini & Pietro S. Oliveto

Authors
  1. Andrea Mambrini
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  2. Pietro S. Oliveto
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Corresponding author

Correspondence to Pietro S. Oliveto .

Editor information

Editors and Affiliations

  1. Dalhousie University, Halifax, Nova Scotia, Canada

    Malcolm I. Heywood

  2. University College Dublin, Dublin, Ireland

    James McDermott

  3. Universidade Nova de Lisboa, Lisboa, Portugal

    Mauro Castelli

  4. CIUSC Dept Informatics Engg, University of Coimbra, Coimbra, Portugal

    Ernesto Costa

  5. Edinburgh Napier University, Edinburgh, United Kingdom

    Kevin Sim

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Mambrini, A., Oliveto, P.S. (2016). On the Analysis of Simple Genetic Programming for Evolving Boolean Functions. In: Heywood, M., McDermott, J., Castelli, M., Costa, E., Sim, K. (eds) Genetic Programming. EuroGP 2016. Lecture Notes in Computer Science(), vol 9594. Springer, Cham. https://doi.org/10.1007/978-3-319-30668-1_7

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  • DOI: https://doi.org/10.1007/978-3-319-30668-1_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30667-4

  • Online ISBN: 978-3-319-30668-1

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