  In this paper, based on the contributions of Tucker (1983) and Seb{\H{o}}
(1992), we generalize the concept of a sequential coloring of a graph to a
framework in which the algorithm may use a coloring rule-base obtained from
suitable forcing structures. In this regard, we introduce the {\it weak} and
{\it strong sequential defining numbers} for such colorings and as the main
results, after proving some basic properties, we show that these two parameters
are intrinsically different and their spectra are nontrivial. Also, we consider
the natural problems related to the complexity of computing such parameters and
we show that in a variety of cases these problems are ${\bf NP}$-complete. We
conjecture that this result does not depend on the rule-base for all nontrivial
cases.
