Learning capabilities of Boolean networks

Boolean networks can learn 10 perform specific tasks. Here, some cases are described where a network is trained on a given problem, namely, a sub-set of examples of this task is provided to the system. This procedure results in a final network which is able to perform correctly on instances that differ from those in the examples, thus showing a capability to recover the general rule underlying the task. The training phase amounts to an optimization process which is carried out using simulated annealing, a search strategy derived from statistical physics. Depending on problem complexity, network size, and number of examples used in the training, different learning regimes occur. For small networks an exact analysis of the statistical mechanics of the system shows that learning takes place as a phase transition. Indeed, thermodynamical quantities related to this transition (critical temperature, specific heat) are related to the complexity of the problem. We also show that a problem can be efficiently learned when it can be implemented in many different ways, ie, generalization is an entropy effect. We also discuss a comparison between different learning schemes and optimization algorithms.