  The generating functional method is employed to investigate the synchronous
dynamics of Boolean networks, providing an exact result for the system dynamics
via a set of macroscopic order parameters. The topology of the networks studied
and its constituent Boolean functions represent the system's quenched disorder
and are sampled from a given distribution. The framework accommodates a variety
of topologies and Boolean function distributions and can be used to study both
the noisy and noiseless regimes; it enables one to calculate correlation
functions at different times that are inaccessible via commonly used
approximations. It is also used to determine conditions for the annealed
approximation to be valid, explore phases of the system under different levels
of noise and obtain results for models with strong memory effects, where
existing approximations break down. Links between BN and general Boolean
formulas are identified and common results to both system types are
highlighted.
