allow us to circumvent this limitation.

Given a set of communication rules $R$, we say that $k'\in K$ is reachable from $k\in K$ via $R$ if $k'$ is the result of applying all rules in $R$ to $k$ at most once; i.e. $k'=f_R(k)$ where

$f_R: K \longrightarrow K$ such that $$f_R(k)=k\cup \left\{ (p_a,v) : \begin{array}{l}
(p_b,v) \in k, k_a \subseteq k, \mbox{ and } \\
(p_b,v,p_a,k_a) \in R,\\
\mbox{for some } p_b\in P \mbox{ and } k_a\in K \end{array} \right\}.$$ \[fR-def\]

A state of knowledge $k_n$ is reachable in the context of a knowledge flow $(R,k_0)$ if $k_n=f_R^n(k_0)$. The [*maximal state of knowledge*]{} $f^*_R(k_0)$ is the limit of $k_n=f_R^n(k_0)$ as $n\rightarrow \infty$. A state of knowledge $f^*_{R_\kappa}(\kappa)$ is [*valid*]{} for a knowledge flow $(R,k_0)$ if $R_\kappa \subseteq R$ and $\kappa\subseteq k_0$. Since $f_R(k_0)$ is monotonically increasing[^2] in $R$ and $k_0$, any valid state of knowledge is a subset of the maximal state of knowledge. Hence, the maximal state of knowledge is also the smallest fixed point of $f_R$ which subsumes $k_0$.

Initial Knowledge
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For each value $v$, $Source(v)=\{p: (p,v)\in k_0\}$ defines the set of principals who draw $v$. In the knowledge flow framwork, a principal $p$ outside of $Source(v)$ 