We consider now binary aggregation structures with constraints \cite{Grandi_2013}: \(\mathcal{A}=\left\langle N,{\bf P},{\bf C}\right\rangle\) where \({\bf C}\subseteq\mathcal{L}\) is a finite set of propositional formulas over \({\bf P}\). Intuitively, such formulas make explicit the logical interdependencies among the issues in \({\bf P}\). {example} The binary aggregation structure (with constraints) of the discoursive paradox is: \(N=\left\{1,2,3\right\}\), \({\bf P}=\left\{p,q,r\right\}\) and \({\bf C}=\left\{p\leftrightarrow(q\land r)\right\}\) Given a BA structure with constraints, an opinion \(O:{\bf P}\to\{0,1\}\) is now an assignment of truth values to the set of issues \({\bf P}\) which satisfies \({\bf C}\). That is, individual opinions are assumed to satisfy the constraints of the BA structure. As above, an opinion profile \({\bf O}=(O_{1},\dots,O_{n})\) records the opinion, on the given set of issues, of every individual in \(N\).
BDPs on aggregation structures with constraints may lead individuals to update with logically inconsistent opinions. The following processes are simple adaptations of BDPs where agents update their opinions only if the opinions of their gurus, on the respective issues, are consistent with the constraints.
Fix an opinion profile \({\bf O}\), an influence profile \({\bf G}\), and a set of constraints \({\bf C}\). Consider the stream \({\bf O}^{0},{\bf O}^{1},\ldots,{\bf O}^{n},\ldots\) of opinion profiles recursively defined as follows:
Base: \({\bf O}_{0}:={\bf O}\)
Step: for all \(i\in N\), \(p\in{\bf P}\), \(O_{i}^{n+1}(p):=\left\{\begin{array}{ll}O^{n}_{R_{p}(i)}(p)&\mbox{if }\bigwedge_{p\in{\bf P}}O^{n}_{R_{p}(i)}(p)\wedge\bigwedge_{\varphi\in{\bf C}}\mbox{ is consistent}\\ O_{i}^{n}(p)&\mbox{otherwise}\end{array}\right.\).
We call processes defined by the above dynamics resistant BDPs.
Resistant BDPs converge in some cases in which BDPs do not. There are cases in which there is disagreement in the cycles but the process still converges, because of the safeguard built in the dynamics.
Consider the following example. Let \({\bf C}=\{p\leftrightarrow\neg q\}\) and let \({\bf O}\) and \({\bf G}\) be as illustrated by the figure below: