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\item $\Atoms = \set{p_1,\dots,p_m}$ is a finite set of issues, each represented by a propositional atom.  \end{itemize}  We denote with $\L$ the propositional language constructed by closing $\Atoms$ under a functionally complete set of Boolean connectives (e.g., $\set{\neg, \wedge}$).  We denote with $\D= \{B \mid B: \I \to \{0,1\}\}$ the set of all possible assignments of truth values to the set of issues $\Atoms$ and call an element $O\in\D$ an \emph{opinion}. Thus, $O(p)=0$ (respectively, \mbox{$O(p)=1$}) indicates that opinion $O$ rejects (respectively, accepts) the issue $p$. Syntactically, the two opinions correspond to the truth of the literals $p$ or $\neg p$. For $p \in \Atoms$ we write $\pm p$ to denote one element from $\set{p, \neg p}$. An \emph{opinion profile} $\O=(O_1,\dots,O_n)$ records the opinion, on the given set of issues, of every individual in $\N$. Given a profile $\O$ the i$^{\mathit{th}}$ $i^{\mathit{th}}$  projection $\O_i$ denotes the opinion of agent $i$ in profile $\O$. We also denote by $\O(p)=\{i \in \N \mid \O_{i}(p)=1\}$ the set of agents accepting issue $p$ in profile $\O$.  \subsection{Binary Aggregation and Binary Influence}  In \cite{DeGroot}, DeGroot proposes a simple model of step by step opinion change under social influence. The model combines two types of matrices. Assuming a group of $n$ agents, a first $n\times n$ matrix represents the weighted influence network (who influences whom and how much), and a second $n \times m$ matrix represents the probability assigned by agents each agent  to each of the  $m$ different alternatives. Both the agents' opinion and the influence weights are taken within $[0,1]$ and are (row) stochastic (each row sums up to $1$). Given an opinion and an influence matrix, the opinion of each agent in the next time step is obtained through linear averaging. In this paper we focus on the Boolean extreme of a DeGroot process. Opinions are defined over a BA structure, and hence are taken to be binary. Similarly, we take influence to be modeled by the binary case of an influence matrix. Influence is of an ``all-or-nothing'' type and each agent is therefore taken to be influenced by exactly one agent, possibly herself. The graph induced by such a binary influence matrix (called \emph{influence graph}) is therefore a structure $G = \tuple{\N, R}$ where $R \subseteq \N^2$ is a binary relation where $i R j$ is taken here to denote that ``$i$ is influenced by $j$''. Such relation is serial ($\forall i\in \N, \exists j \in \N: i R j$) and functional ($\forall i,j,k \in \N$ if $i R j$ and $i R k$ then $j = k$). So each agent $i$ has exactly one successor (influencer or `guru'), possibly itself, which we denote $R(i)$.