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We demonstrate how a single-celled organism could undertake associative learning. Although to date only one previous study has found experimental evidence for such learning, there is no reason in principle why it should not occur. We propose a gene regulatory network that is capable of associative learning between any pre-specified set of chemical signals, in a Hebbian manner, within a single cell. A mathematical model is developed, and simulations show a clear learned response. A preliminary design for implementing this model using plasmids within Escherichia coli is presented, along with an alternative approach, based on double-phosphorylated protein kinases.

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Received (?? ?? 2011) Revised (Day Month Year)Purpose-In recent years Monte-Carlo sampling methods, such as Monte Carlo tree search, have achieved tremendous success in model free reinforcement learning. A combination of the so called upper confidence bounds policy to preserve the "exploration vs. exploitation" balance to select actions for sample evaluations together with massive computing power to store and to update dynamically a rather large pre-evaluated game tree lead to the development of software that has beaten the top human player in the game of Go on a 9 by 9 board. Much effort in the current research is devoted to widening the range of applicability of the Monte-Carlo sampling methodology to partially observable Markov decision processes with non-immediate payoffs. The main challenge introduced by randomness and incomplete information is to deal with the action evaluation at the chance nodes due to drastic differences in the possible payoffs the same action could lead to. The aim of this article is to establish a version of a theorem that originated from population genetics and has been later adopted in evolutionary computation theory that will lead to novel Monte-Carlo sampling algorithms that provably increase the AI potential. Due to space limitations the actual algorithms themselves will be presented in the sequel papers, however, the current paper provides a solid mathematical foundation for the development of such algorithms and explains why they are so promising. Design/Methodology/Approach-In the current paper we set up a mathematical framework, state and prove a version of a Geiringer-like theorem that is very well-suited for the development of Mote-Carlo sampling algorithms to cope with randomness and incomplete information to make decisions. From the framework it will be clear that such algorithm increase what seems like a limited sample of rollouts exponentially in size by exploiting the symmetry within the state space at little or no additional * EPSRC EP/D003/05/1 "Amorphous Computing" and EPSRC EP/I009809/1 "Evolutionary Approximation Algorithms for Optimization: Algorithm Design and Complexity Analysis" Grants. computational cost. Appropriate notions of recombination (or crossover) and schemata are introduced to stay inline with the traditional evolutionary computation terminology. The main theorem is proved using the methodology developed in the PhD thesis of the first author, however the general case of nonhomologous recombination presents additional challenges that have been overcome thanks to a lovely application of the classical and elementary tool known as the "Markov inequality" together with the lumping quotients of Markov chains techniques developed and successfully applied by the authors in the previous research for different purposes. This methodology will be mildly extended to establish the main result of the current article. In addition to establishing the Geiringer-like theorem for Monte Carlo sampling, which is the central objective of this paper, we also strengthen the...