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.
We assume the SUB and DEL of an empty tree return an empty tree.
- 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).
References
Auger, A., Doerr, B.: Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific, Singapore (2011)
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276(1–2), 51–81 (2002)
Kötzing, T., Sutton, A.M., Neumann, F., O’Reilly, U.M.: The max problem revisited: the importance of mutation in genetic programming. Theor. Comp. Sci. 545, 94–107 (2014)
Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge (1992)
Koza, J.R.: Genetic Programming II: Automatic Discovery of Reusable Programs. MIT Press, Cambridge (1994)
Langdon, W.B., Poli, R.: Foundations of Genetic Programming. Springer, Heidelberg (2002)
Luke, S.: Genetic programming produced competitive soccer softbot teams for RoboCup97. In: Proceedings of the Third Annual Conference on Genetic Programming 1998, pp. 214–222. Morgan Kaufmann (1998)
Mitzenmacher, M., Upfal, E.: Probability and Computing. Cambridge University Press, Cambridge (2005)
Moraglio, A., Mambrini, A., Manzoni, L.: Runtime analysis of mutation-based geometric semantic genetic programming on boolean functions. In: Proceedings of FOGA XII, pp. 119–132. ACM (2013)
Neumann, F., O’Reilly, U.M., Wagner, M.: Computational complexity analysis of genetic programming - initial results and future directions. In: Riolo, R., Vladislavleva, E., Moore, J.H. (eds.) Genetic Programming Theory and Practice IX. Genetic and Evolutionary Computation, pp. 113–128. Springer, New York (2011)
Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity. Natural Computing Series. Springer, Heidelberg (2010)
Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica 59(3), 369–386 (2011)
O’Neill, M., Vanneschi, L., Gustafson, S., Banzhaf, W.: Open issues in genetic programming. Genet. Program. Evolvable Mach. 11(3–4), 339–363 (2010)
O’Reilly, U.-M., Oppacher, F.: Program search with a hierarchical variable length representation: genetic programming, simulated annealing and hill climbing. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) PPSN 1994. LNCS, vol. 866, pp. 397–406. Springer, Heidelberg (1994)
Poli, R., Langdon, W.B., McPhee, N.F.: A field guide to genetic programming (2008). http://lulu.com
Poli, R., McPhee, N.F., Rowe, J.E.: Exact schema theory and Markov chain models for genetic programming and variable-length genetic algorithms with homologous crossover. Genet. Program. Evolvable Mach. 5(1), 31–70 (2004)
Spector, L., Barnum, H., Bernstein, H.J., Swamy, N.: Quantum computing applications of genetic programming. In: Spector, L., Langdon, W.B., O’Reilly, U.M., Angeline, P.J. (eds.) Advances in Genetic Programming 3, pp. 135–160. MIT Press, Cambridge (1999)
Valiant, L.G.: Evolvability. J. ACM 56(1), 3:1–3:21 (2009)
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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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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