  To model biological systems using networks, it is desirable to allow more
than two levels of expression for the nodes and to allow the introduction of
parameters. Various modeling and simulation methods addressing these needs
using Boolean models, both synchronous and asynchronous, have been proposed in
the literature. However, analytical study of these more general Boolean
networks models is lagging. This paper aims to develop a concise theory for
these different Boolean logic based modeling methods. Boolean models for
networks where each node can have more than two levels of expression and
Boolean models with parameters are defined algebraically with examples
provided. Certain classes of random asynchronous Boolean networks and
deterministic moduli asynchronous Boolean networks are investigated in detail
using the setting introduced in this paper. The derived theorems provide a
clear picture for the attractor structures of these asynchronous Boolean
networks.
