# Geiringer theorems: from population genetics to computational intelligence, memory evolutive systems and Hebbian learning

## Abstract

The classical Geiringer theorem addresses the limiting frequency of occurrence of various alleles after repeated application of crossover. It has been adopted to the setting of evolutionary algorithms and, a lot more recently, reinforcement learning and Monte-Carlo tree search methodology to cope with a rather challenging question of action evaluation at the chance nodes. The theorem motivates novel dynamic parallel algorithms that are explicitly described in the current paper for the first time. The algorithms involve independent agents traversing a dynamically constructed directed graph that possibly has loops and multiple edges. A rather elegant and profound category-theoretic model of cognition in biological neural networks developed by a well-known French mathematician, professor Andree Ehresmann jointly with a neurosurgeon, Jan Paul Vanbremeersch over the last thirty years provides a hint at the connection between such algorithms and Hebbian learning.

### Keywords

Geiringer theorems Partially observable Markov decision processes Monte-Carlo tree search Reinforcement learning Memory evolutive systems Hebbian learning## Notes

### Acknowledgments

This work has been supported by the EPSRC EP/I009809/1 “Evolutionary Approximation Algorithms for Optimization: Algorithm Design and Complexity Analysis” Grant.

### References

- Agrawal R (1995) Sample mean based index policies with
*o*(log*n*) regret for the multi-armed bandit problem. Adv Appl Probab 27:1054–1078CrossRefMATHGoogle Scholar - Auer P (2002) Using confidence bounds for exploration–exploitation trade-offs. J Mach Learn Res 3:397–422MathSciNetGoogle Scholar
- Auger A, Doerr B (2011) Theory of randomized search heuristics. Series on theoretical computer science. Elsevier, AmsterdamGoogle Scholar
- Barr M, Wells C (1998) Category theory for computing science. Prentice Hall, Upper Saddle RiverGoogle Scholar
- Ehresmann A, Smeonov P (2012) Wlimes: towards a theoretical framework for wandering logic intelligence memory evolutive systems. In: Simeonov PL, Smith LS, Ehresmann AC (eds) Integral biomathics: tracing the road to reality. Springer, HeidelbergGoogle Scholar
- Ehresmann A, Vanbremeersch JP (2006) The memory evolutive systems as a model of Rosens organisms. Axiomathes 16:165–214CrossRefGoogle Scholar
- Ehresmann A, Vanbremeersch JP (2007) Memory evolutive systems: hierarchy, emergence, cognition, studies in multidisciplinarity, vol 4. Elsevier, AmsterdamGoogle Scholar
- Ehresmann A, Baas N, Vanbremeersch JP (2004) Hyperstructures and memory evolutive systems. Int J Gen Syst 33(5):553–568CrossRefMATHGoogle Scholar
- Geiringer H (1944) On the probability of linkage in mendelian heredity. Ann Math Stat 15:25–57MathSciNetCrossRefMATHGoogle Scholar
- Geiringer H (1948) On the mathematics of random mating in case of different recombination values for males and females. Genetics 33:548–564Google Scholar
- Geiringer H (1949) Chromatid segregation of tetraploids and hexaploids. Genetics 34:665–684Google Scholar
- Hebb DO (1949) The organization of behavior. Wiley, New YorkGoogle Scholar
- Mc Lane S (1971) Categories for the working mathematician. Springer, New YorkCrossRefGoogle Scholar
- Mitavskiy B, Cannings C (2006) Exploiting quotients of Markov chains to derive properties of the stationary distribution of the Markov chain associated to an evolutionary algorithm. In: Simulated evolution and learning (SEAL-2006), HefeiGoogle Scholar
- Mitavskiy B, Cannings C (2009) Estimating the ratios of the stationary distributions of Markov chains modeling evolutionary algorithms using the quotient construction method. Evol Comput 17(3):343–377CrossRefGoogle Scholar
- Mitavskiy B, He J (2013) A further generalization of the finite-population Geiringer-like theorem for POMDPs to allow recombination over arbitrary set covers. In: Foundations of genetic algorithms 12 (FOGA-2013). ACM Press, New YorkGoogle Scholar
- Mitavskiy B, Rowe J (2005) A schema-based version of Geiringer theorem for nonlinear genetic programming with homologous crossover. In: Foundations of genetic algorithms 8 (FOGA-2005). Lecture notes in computer science, vol 3469. Springer, Heidelberg, pp 156–175Google Scholar
- Mitavskiy B, Rowe J (2006) An extension of Geiringer theorem for a wide class of evolutionary algorithms. Evol Comput 14(1):87–118MathSciNetGoogle Scholar
- Mitavskiy B, Rowe J, Wright A, Schmitt L (2007) An improvement of the quotient construction method and further asymptotic results on the stationary distribution of the Markov chains modeling evolutionary algorithms. In: IEEE congress on evolutionary computation (CEC-2007), SingaporeGoogle Scholar
- Mitavskiy B, Rowe J, Wright A, Schmitt L (2008) Quotients of Markov chains and asymptotic properties of the stationary distribution of the Markov chain associated to an evolutionary algorithm. Genet Program Evol Mach 17(3):109–123CrossRefGoogle Scholar
- Mitavskiy B, Rowe J, Cannings C (2012) A version of Geiringer-like theorem for decision making in the environments with randomness and incomplete information. Int J Intell Comput Cybern 5(1):36–90MathSciNetCrossRefGoogle Scholar
- Muhlenbein H (1991) Parallel genetic algorithms, population genetics, and combinatorial optimization. In: Parallelism, learning, evolution, Neubiberg, pp 398–406Google Scholar
- Poli R, Stephens C, Wright A, Rowe J (2002) A schema-theory-based extension of Geiringer’s theorem for linear GP and variable-length GAs under homologous crossover. In: Foundations of genetic algorithms (FOGA 2002), Torremolinos, pp 45–62Google Scholar
- Rabani Y, Rabinovich Y, Sinclair A (1995) A computational view of population genetics. In: Annual ACM symposium on the theory of computing. ACM Press, New York, pp 83–92Google Scholar
- Weng J (2012) Natural and artificial intelligence. BMI Press, OkemosGoogle Scholar