Group-theoretic approach as a general framework for sensors, neural networks, fuzzy control, and genetic boolean networks

When describing a system of interacting genes, a useful approximation is provided by a Boolean network model, in which each gene is either switched on or off- i.e., its state is described by a Boolean variable. Recent papers by I. Shmulevich et al. show that although in principle, arbitrarily complex Boolean functions are possible, in reality, the corresponding Boolean networks can be well described by Boolean functions from one of the so-called Post classes - classes that are closed under composition. These classes were originally described by E. Post. It is known that the Boolean model is only an approximate description of the real-life gene interaction. In reality, the interaction may be more complex. How can we extend these results to more realistic continuous models of gene interaction? In this paper, we show that the Post class approach can be viewed as a particular case of a general group-theoretic framework that has already led to a successful justification of empirical formulas from such areas of signal processing as sensor analysis, neural networks, fuzzy techniques, etc. Because of this relation, we suggest group-theoretic approach as a framework for describing gene interaction in a more realistic way.