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

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## 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.

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