On the conversion of multivalued to Boolean dynamics

Abstract: Results and tools on discrete interaction networks are often concerned with Boolean variables, whereas considering more than two levels is sometimes useful. Multivalued networks can be converted to partial Boolean maps, in a way that preserves the asynchronous dynamics. We investigate the problem of extending these maps to non-admissible states, i.e. states that do not have a multivalued counterpart. We observe that attractors are preserved if a stepwise version of the original function is considered for conversion. Different extensions of the Boolean conversion affect the structure of the interaction graphs in different ways. A particular technique for extending the partial Boolean conversion is identified, that ensures that feedback cycles are preserved. This property, combined with the conservation of the asymptotic behaviour, can prove useful for the application of results and analyses defined in the Boolean setting to multivalued networks, and vice versa. As a first application, by considering the conversion of a known example for the discrete multivalued case, we create a Boolean map showing that the existence of a cyclic attractor and the absence of fixed points are compatible with the absence of local negative cycles. We then state a multivalued version of a result connecting mirror states and local feedback cycles.