3.1 Language and semantics of SVL

Let $\mathsf{Nom} = \{i,j,k,...\}$ be a countable set of nominals, and $\mathsf{Top} = \{p,q,r,...\}$ be a countable set of topics, such that $\mathsf{Nom} \cap \mathsf{Top} = \emptyset$ .

Definition 1. We define the well-formed formulas of the language of the static fragment of visibility logic $\mathbb{SVL}$ to be generated by the following grammar:

\[\varphi ::= p^+ \mid p^- \mid i \mid \neg \varphi \mid (\varphi \wedge \varphi) \mid \Diamond_{i:p} \varphi \mid \Diamond^{-1}_{i:p} \varphi \mid \blacklozenge \varphi \mid \blacklozenge^{-1} \varphi \mid @_i \varphi\]

where $p \in \mathsf{Top}$ and $i \in \mathsf{Nom}$ . We define propositional connectives like $\vee, \to$ and the formulas $\top, \perp$ as usual and the duals as standard $\Box := \neg \Diamond \neg$ , $\Box^{-1} := \neg \Diamond^{-1} \neg$ , $\blacksquare := \neg \blacklozenge \neg$ , and $\blacksquare^{-1} := \neg \blacklozenge^{-1} \neg$ .

Given a formula $\varphi \in \mathbb{SVL}$ , we can recursively define the modal depth of the formula $md(\varphi)$ in the following way: $md(p^+) = md(p^-) = md(i) = 0$ , $md(\lnot \varphi) = md(@_i \varphi) = md(\varphi)$ , $md(\varphi \land \psi) = \textrm{max}\{md(\varphi), md(\psi)\}$ , and $md(\Diamond_{i:p} \varphi) = md(\Diamond^{-1}_{i:p} \varphi) = md(\blacklozenge \varphi) = md(\blacklozenge^{-1} \varphi) = md(\varphi) + 1$ . The size of $\varphi$ , denoted $|\varphi|$ , is defined as follows: $|p^+| = |p^-| = |i| = 1$ , $|\lnot \varphi| = |\Diamond_{i:p} \varphi| = |\Diamond^{-1}_{i:p} \varphi| = |\blacklozenge \varphi| = |\blacklozenge^{-1} \varphi| = |@_i \varphi| = |\varphi| + 1$ , and $|\varphi \land \psi| = |\varphi| + |\psi| + 1$ .

In our language, similar to other approaches to logic-based analysis of social networks (see, e.g., Christoff et al. Reference Christoff, Hansen and Proietti2016), we distinguish three possible dispositions of an agent to a topic $p \in \mathsf{Top}$ . The agent may be pro p, which we express with $p^+$ , contra p, expressed by $p^-$ , or indifferent to p, if the agent is neither pro nor contra p. The agent cannot be both pro and contra p.

Constructs $\blacklozenge \varphi$ and $\blacklozenge^{-1} \varphi$ express that “the current agent follows an agent satisfying $\varphi$ ” and “the current agent is followed by an agent that satisfies $\varphi$ ,” respectively. Formulas $\Diamond_{i:p} \varphi$ and $\Diamond_{i:p}^{-1} \varphi$ mean that “there is an agent satisfying $\varphi$ who sees the (re)post by the current agent on topic p (originally posted by an agent named i)” and “there is an agent satisfying $\varphi$ whose (re)post on topic p (originally posted by an agent named i) is seen by the current one.”

Formulas of SVL are defined on relational visibility models.

Definition 2. A visibility model (or a model) M is a tuple $(A,F,+,-,V,R)$ , where

• • A is a nonempty set of agents,

• • $F: A \rightarrow 2^{A}$ is an irreflexive followership relation,

• • $+: A \rightarrow 2^{\mathsf{Top}}$ assigns to each agent a set of topics she is pro,

• • $-: A \rightarrow 2^{\mathsf{Top}}$ assigns to each agent a set of topics she is contra such that for all agents $a \in A$ , it holds that $+(a) \cap -(a) = \emptyset$ ,

• • $V: \mathsf{Nom} \rightarrow 2^{A}$ is a valuation such that for all $i \in \mathsf{Nom}$ : $|V(i)| = 1$ ,

• • $R:\mathsf{Top} \times A \rightarrow 2^{A \times A}$ is a visibility relation for each topic and each agent satisfying the following conditions, where $p \in \mathsf{Top}$ and $a, b, c \in A$ :

  1. (1) If $(a,b) \in R(p,c)$ , then $(a,a) \in R(p,c)$ .

  2. (2) If $(a,a) \in R(p,c)$ , then $(a,b) \in R(p,c)$ for all b such that $b \in F(a)$ .

  3. (3) If $(a,b) \in R(p,c)$ , $p \in +(b)$ , and $b \neq c$ , then $(b,b) \in R(p,c)$ and $b \in F(c)$ .

  4. (4) If $(a,b) \in R(p,c)$ , $p \in -(b)$ , and $a \neq b$ , then $(b,b) \not \in R(p,c)$ and $b \not \in F(a)$ .

  5. (5) If $(a,b) \in R(p,c)$ , $p \not \in +(b)$ , $p \not \in -(b)$ , and $a \neq b$ , then $(b,b) \not \in R(p,c)$ .

A pointed visibility model $M_{a}$ is a model M with a distinguished point $a \in A$ where evaluation takes place. If necessary, we refer to the elements of the tuple as $A_M$ , $F_M$ , $+_M$ , $-_M$ , $V_M$ , and $R_M$ . A visibility model such that for all $a \in A$ there is some $i \in \mathsf{Nom}$ such that $V(i) = \{a\}$ is called named. All models we will be dealing with in the paper are named. Let $Nom(a):=\{i \in \mathsf{Nom} \mid a \in V(i)\}$ be a set of all nomimals assigned to an agent, and $Top(a):= \{p \in \mathsf{Top} \mid R(p,a)\}$ be a set of all topics that an agent posted. A visibility model M is finite if all of A, $\bigcup \{+(a)\mid a \in A\}$ , $\bigcup \{-(a)\mid a \in A\}$ , $\bigcup \{Nom(a) \mid a \in A\}$ , and $\bigcup \{Top(a) \mid a \in A\}$ are finite.

Let $M = (A,F,+,-,V,R)$ be a finite visibility model. The size of M equals to

In Definition 2 above, the first condition on R states that if agent b sees a post, which was originally posted by agent c on topic p, from agent a, then a herself can see the post. The second condition ensures that if an agent posts a post, all her followers can see the post. Condition number three specifies that if an agent sees a post on a topic she likes, she will reshare the post and follow the original poster. The fourth condition says that if an agent sees a post on a topic she dislikes, she does not reshare it and unfollows the agent from whom she has seen the post. Finally, the last condition stipulates that if an agent sees a post on a topic she is indifferent to, she does not reshare the post.

Note that our definition of R does not preclude situations where agents may have seen a post on a topic they dislike from an agent they do not follow. How such situations may come about will be the focus of the next section.

Definition 3. Let $M = (A,F,+,-,V,R)$ be a model, $a,b,c \in A$ , $p \in \mathsf{Top}$ , $i \in \mathsf{Nom}$ , and $\varphi, \psi \in \mathbb{SVL}$ . The semantics of SVL is recursively defined as follows:

\[\begin{array}{lll}M_a \models p^+ &\text{iff} &p \in +(a) \\M_a \models p^- &\text{iff} &p \in -(a) \\M_a \models i &\text{iff} &a \in V(i) \\M_a \models \lnot \varphi &\text{iff} &M_a \not \models \varphi\\M_a \models \varphi \land \psi &\text{iff} &M_a \models \varphi \text{ and } M_a \models \psi\\M_a \models \Diamond_{i:p} \varphi &\text{iff} &\exists b,c \in A: (a,b) \in R (p,c) \text{ and } V(i) = \{c\} \text{ and } M_b \models \varphi \\M_a \models \Diamond^{-1}_{i:p} \varphi &\text{iff} &\exists b,c \in A: (b,a) \in R (p,c) \text{ and } V(i) = \{c\} \text{ and } M_b \models \varphi \\M_a \models \blacklozenge \varphi &\text{iff} &\exists b \in A: a \in F(b) \text{ and } M_b \models \varphi\\M_a \models \blacklozenge^{-1} \varphi &\text{iff} &\exists b \in A: b \in F(a) \text{ and } M_b \models \varphi\\M_a \models @_i \varphi &\text{iff} &M_b \models \varphi \text{ and } \{b\} = V(i)\end{array}\]

Observe that if $M_a \not \models p^+$ then we have that either $p \in -(a)$ or not. This corresponds to the intuition that agent a is not pro p if she actively dislikes the topic (she is contra p), or if she is indifferent to it. Similarly, for $M_a \not \models p^-$ .

The valuation function V is such that for all $i \in \mathsf{Nom}: |V(i)| = 1$ . In other words, a name can only be true for one agent. However, note that one agent can have several names, that is, it can be the case that two nominals i and j are forced at the same agent a. Furthermore, we allow for two agents to post on the same topic, that is, regardless of whether two nominals i and j refer to the same agent, both $\Diamond_{i:p} \top$ and $\Diamond_{j:p} \top$ can be forced simultaneously.

Recall the example from Fig. 1. In the figure, we have that $M_a \models p^+$ , $M_c \models p^-$ , and $M_b \models \lnot p^+ \land \lnot p^-$ , meaning that agent a is pro topic p, agent b is indifferent toward the topic, and c is contra p. Moreover, we have, for example, that $M_c \models \Diamond^{-1}_{i:p} \top \land \blacksquare \lnot p^+$ , meaning that agent c has seen a post by the agent with name i on topic p, and that all agents that c follows are not pro p.

We can formalize some of the notions of visibility and reachability as formulas in SVL. Let $M = (A, F, +, -, V, R)$ be a model. Some quantitative amounts related to visibility that we can count in finite models are as follows:

• • How many followers the agent called i has: $|\{a \in A \mid M_{a} \models \blacklozenge i\}|$

• • How many agents have seen the agent called i’s post on p: $|\{a \in A \mid M_{a} \models \Diamond_{i:p}^{-1} \top\}|$

• • How many agents that are pro p have seen the agent called i’s post on p: $|\{a \in A \mid M_{a} \models p^{+} \wedge \Diamond_{i:p}^{-1} \top\}|$

We also present some formulas corresponding to qualitative properties of agents in the network. The following formulas are forced at an agent iff the property holds of that agent:

• • The current agent i is the original poster of a post on p: $i \wedge \Diamond_{i:p}\top$

• • The current agent has seen i’s post on p: $\Diamond^{-1}_{i:p} \top$

• • All the followers of the current agent i have shared i’s post on p: $i \wedge \blacksquare^{-1}\Diamond_{i:p} \top$

• • The current agent i shared a post to a follower j, but j also saw the post from another source: $i \wedge \blacklozenge^{-1}(j \wedge \Diamond^{^-1}_{i:p} i \wedge \Diamond^{^-1}_{i:p} (\neg i \wedge \neg j))$

• • The current agent i has gained a follower who is pro p, after i posted on p:

$i \wedge \Diamond_{i:p} \top \wedge \blacklozenge^{-1}(p^{+} \wedge \Diamond^{-1}_{i:p}i)$

• • The current agent i has reached the agent j with i’s post on p in no more than three steps: $i \wedge \Diamond_{i:p} \Diamond_{i:p} \Diamond_{i:p} j$

The following notion of bisimulation is based on hybrid bisimulation (Areces and ten Cate, Reference Areces, ten Cate, Blackburn, van Benthem and Wolter2007) and on bisimulation for logics with “backward-looking” modalities (see, e.g., Kurtonina and de Rijke Reference Kurtonina and de Rijke1997).

The following theorem is a standard result in hybrid modal logic (Areces and ten Cate, Reference Areces, ten Cate, Blackburn, van Benthem and Wolter2007).

One of the classic results in standard modal logic is a restricted version of Theorem 7 stating that n-bisimulation implies satisfaction of the same modal formulas up to modal depth n (Goranko and Otto, Reference Goranko, Otto, Blackburn, van Benthem and Wolter2007). This, however, is not the case for hybrid modal logic, since operator $@_i$ allows reaching an agent named i no matter “how far” the agent is from the current agent. On the other hand, the result holds for hybrid modal logic if we only concentrate on formulas that do not include nominals that are used in models at hand. Granted this is an even more restrictive case, the fact will be useful later in the paper.

The proof of the theorem is standard noting that all nominals appearing in $\varphi$ have the empty denotation.

3.2 A Sound and complete axiomatization of SVL

In this section, we present a sound and complete axiomatization of SVL. The main direction of our proofs follows the strategy of establishing completeness of hybrid logic that can be found in works by Blackburn and ten Cate (Reference Blackburn and ten Cate2006) and Blackburn et al. (Reference Blackburn, de Rijke and Venema2001). We omit proofs of lemmas if they can be adopted from the cited literature with minimal changes.

Definition 10 The proof system of SVL, $\mathbf{SVL}$ , comprises axioms and rules of inference from Tables 1 and 2.

Table 1. Hybrid and bidirectional axioms

In rule SUB, σ is a substitution that uniformly replaces nominals by nominals, and topics with formulas.

Table 2. Followership and visibility axioms

The axiomatization of SVL combines the axiomatizations of hybrid logic (Blackburn et al., Reference Blackburn, de Rijke and Venema2001, Section 7.3), hybrid tense logic (Blackburn and ten Cate, Reference Blackburn and ten Cate2006), and additional novel axioms and rules of inference for followership and visibility.

Let us consider Table 1 first. Axioms K are standard modal distributivity axioms. Axiom schema Self-Dual states that hybrid operator $@_i$ is its own dual. Ref states that an agent named i actually satisfies nominal i. According to Agree, accessing an agent in two $@$ -steps is the same as accessing the same agent in a single $@$ -step. Axiom Intro allows us to put an arbitrary true formula under the scope of $@$ . Interactions between $@$ and all diamonds in our language are captured by the set of Back axiom schemata. Axioms T are standard axioms of tense logic ensuring that our models are bidirectional. The interaction between $@$ and the fact that our followership and visibility relations are bidirectional is captured by axioms HT. Finally, we have standard rules of inference like modus ponens MP, necessitations Nec, and substitution Sub. Rule Name states that if we can prove that $\varphi$ holds at an arbitrary agent i, then we can prove $\varphi$ .

Now let us turn to the axiom schemata in Table 2. Irref captures the irreflexivity of the followership relation, and Cons states that an agent’s views on a topic are consistent. Axiom Foll states that if the current agent sees a post from some other agent on a topic she likes, she starts following the original poster and reposts the post. Alternatively, Unfoll specifies that if the current agent sees a post on the topic she does not like, she unfollows the agent she the post from. Finally, according to Indiff, if the current agent is indifferent to the topic of the post she sees, then she does nothing.

That $\mathbf{SVL}$ is sound can be shown by a direct application of the definition of semantics.

As an example of an $\mathbf{SVL}$ derivation, consider a set of theorems, where $\heartsuit \in \{\Diamond_{i : p}, \Diamond_{i:p}^{-1}, \blacklozenge, \blacklozenge^{-1}\}$ :

We show how to derive Bridge $_{\blacklozenge}$ :

Now we turn to the proof of the completeness of $\mathbf{SVL}$ . In our proof sketch, we follow Blackburn et al. (Reference Blackburn, de Rijke and Venema2001, Section 7.3) and Blackburn and ten Cate (Reference Blackburn and ten Cate2006) and omit proofs of lemmas that can be taken “as is” from the cited literature with straightforward changes.

As usual in the completeness proofs that employ the canonical model construction, we commence with the notion of a maximal consistent set (MCS).

It is straightforward to show that MCSs contain all the instances of the axioms of $\mathbf{SVL}$ and are closed under MP. From the hybrid perspective, each MCS itself contains a collection of named MCSs with the properties defined in the following lemma.

Before we continue, we need a set of additional rules of inference that are called , where $\heartsuit \in \{\Diamond_{i : p}, \Diamond_{i:p}^{-1}, \blacklozenge, \blacklozenge^{-1}\}$ .

These rules are derivable from $\mathbf{SVL}$ in the presence of axioms HT (Blackburn and ten Cate, Reference Blackburn and ten Cate2006). Rules may not look entirely straightforward. However, for the purposes at hand, we need these rules just for the next definition, and more on the intuition behind Paste rules can be found in Blackburn et al. (Reference Blackburn, de Rijke and Venema2001, Section 7.3).

The next lemma is a hybrid version of Lindenbaum Lemma, and its proof relies on rules .

Now we have all the ingredients for the construction of the canonical model.

There are several properties that we need to check in order to ensure that $\mathfrak{M}^\Gamma$ is indeed a visibility model. First, observe that by items (1) and (2) of Lemma 14, each nominal names a unique agent in $\mathfrak{A}^\Gamma$ . Next, no agent is pro and contra the same topic. This follows from the fact that each agent $\mathfrak{a} \in \mathfrak{A}^\Gamma$ satisfies all the instances of axiom Cons.

The followership and visibility relations of visibility models are bidirectional, which is manifested by the presence of converse modalities in $\mathbb{SVL}$ . In the next two lemmas, we argue that the corresponding relations of the canonical model $\mathfrak{M}^\Gamma$ are also bidirectional.

Finally, we show that the followership and visibility relations of $\mathfrak{M}^\Gamma$ satisfy the properties from Definition 2, that is, that $\mathfrak{M}^\Gamma$ is indeed a visibility model. But before we delve into the proof per se, we also mention that the following schema is a theorem of $\mathbf{SVL}$ :

Elim can be derived from the contraposition of Intro using Self-Dual.

1. (1). We need to show that if $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , then $(\mathfrak{a},\mathfrak{a}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , where $\mathfrak{a},\mathfrak{b},\mathfrak{c} \in \mathfrak{A}^\Gamma$ . Thus, assume that $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , and let $\mathfrak{V}^\Gamma (i) = \{\mathfrak{a}\}$ and $\mathfrak{V}^\Gamma (j) = \{\mathfrak{c}\}$ . Since a is an MCS, it contains all instances of Ag-See, and in particular $@_i(\Diamond_{j:p} \top \to \Diamond_{j:p}i) \in \mathfrak{a}$ . By Elim and MP, we further have that $\Diamond_{j:p} \top \to \Diamond_{j:p}i \in \mathfrak{a}$ . From the fact that $\top \in \mathfrak{b}$ and $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , we have $\Diamond_{j:p} \top \in \mathfrak{a}$ . The latter implies that $\Diamond_{j:p}i \in \mathfrak{a}$ by MP. From $i \in \mathfrak{a}$ and $\Diamond_{j:p} i \in \mathfrak{a}$ , we conclude, by Lemma 19, that $(\mathfrak{a},\mathfrak{a}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ .

2. (2). We show that if $(\mathfrak{a},\mathfrak{a}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , then $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ for all $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{a})$ . Let $\mathfrak{V}^\Gamma (i) = \{\mathfrak{a}\}$ , $\mathfrak{V}^\Gamma (j) = \{\mathfrak{c}\}$ , and $\mathfrak{b} \in \mathfrak{A}^\Gamma$ be an arbitrary agent such that $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{a})$ . Moreover, let us assume $(\mathfrak{a},\mathfrak{a}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ . Since $\mathfrak{a}$ is an MCS, it contains the following instance of Fol-See: $@_i(\Diamond_{j:p} i \rightarrow \blacksquare^{-1}\Diamond^{-1}_{j:p} i)$ . We can use to get $\Diamond_{j:p} i \rightarrow \blacksquare^{-1}\Diamond^{-1}_{j:p} i \in \mathfrak{a}$ . By the third item of Lemma 19, $(\mathfrak{a},\mathfrak{a}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ is equivalent to the fact that if $i \in \mathfrak{a}$ , then $\Diamond_{j:p} i \in \mathfrak{a}$ . We obtain the latter from $i \in \mathfrak{a}$ and MP. Furthermore, from $\Diamond_{j:p} i \rightarrow \blacksquare^{-1}\Diamond^{-1}_{j:p} i \in \mathfrak{a}$ and $\Diamond_{j:p} i \in \mathfrak{a}$ , we derive $\blacksquare^{-1}\Diamond^{-1}_{j:p} i \in \mathfrak{a}$ . From $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{a})$ and $\blacksquare^{-1}\Diamond^{-1}_{j:p} i \in \mathfrak{a}$ , we have, by item (2) of Lemma 18, $\Diamond^{-1}_{j:p} i \in \mathfrak{b}$ . Finally, the fact that $i \in \mathfrak{a}$ and $\Diamond^{-1}_{j:p} i \in \mathfrak{b}$ is equivalent to $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ by item (4) of Lemma 19.

3. (3). We need to show that if $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , $p \in +^\Gamma (\mathfrak{b})$ , and $\mathfrak{b} \neq \mathfrak{c}$ , then $(\mathfrak{b},\mathfrak{b})\in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ and $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{c})$ . Assume that it holds that $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , $p \in +^\Gamma (\mathfrak{b})$ , and $\mathfrak{b} \neq \mathfrak{c}$ , and let $\mathfrak{V}^\Gamma (i) = \{\mathfrak{b}\}$ and $\mathfrak{V}^\Gamma (j) = \{\mathfrak{c}\}$ . As $\mathfrak{b} \in \mathfrak{A}^\Gamma$ and thus is an MCS, it contains the following instance of Foll: $@_{i}((p^{+} \wedge \Diamond^{-1}_{j:p} \top \wedge \neg j) \to (\Diamond_{j:p}i \wedge \blacklozenge j))$ . From $i \in b$ by Elim and MP, we obtain $(p^{+} \wedge \Diamond^{-1}_{j:p} \top \wedge \neg j) \to (\Diamond_{j:p}i \wedge \blacklozenge j) \in \mathfrak{b}$ . The truth of the antecedent follows from the assumptions that $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , $p \in +^\Gamma (\mathfrak{b})$ , $\mathfrak{b} \neq \mathfrak{c}$ , and $\mathfrak{V}^\Gamma (j) = \{\mathfrak{c}\}$ . Hence, $\Diamond_{j:p}i \wedge \blacklozenge j \in \mathfrak{b}$ , or, equivalently, $\Diamond_{j:p}i \in \mathfrak{b}$ and $\blacklozenge j \in \mathfrak{b}$ . From $\Diamond_{j:p}i \in \mathfrak{b}$ and $i \in \mathfrak{b}$ by item (4) of Lemma 19, we have that $(\mathfrak{b},\mathfrak{b})\in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ . Similarly, from $\blacklozenge j \in \mathfrak{b}$ and $j \in \mathfrak{c}$ by item (3) of Lemma 18, we obtain $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{c})$ .

4. (4). Assume that for some $\mathfrak{a},\mathfrak{b},\mathfrak{c} \in \mathfrak{A}^\Gamma$ , we have that $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , $p \in -^\Gamma (\mathfrak{b})$ , $\mathfrak{a} \neq \mathfrak{b}$ , and $\mathfrak{V}^\Gamma (i) = \{\mathfrak{b}\}$ , $\mathfrak{V}^\Gamma (j) = \{\mathfrak{c}\}$ , and $\mathfrak{V}^\Gamma (k) = \{\mathfrak{a}\}$ . We need to show that all of these imply $(\mathfrak{b},\mathfrak{b}) \not \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ and $\mathfrak{b} \not \in \mathfrak{F}^\Gamma (\mathfrak{a})$ . Assume toward a contradiction that $(\mathfrak{b},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ or $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{a})$ . First, let $(\mathfrak{b},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ . Since $\mathfrak{b}$ is an MCS, it contains the following instance of Unfoll: $@_i( (p^- \land \Diamond^{-1}_{j:p} (\lnot i \land k)) \rightarrow (\lnot \Diamond_{j:p} \top \land \lnot \blacklozenge k))$ . From $i \in \mathfrak{b}$ and Elim, we have $ (p^- \land \Diamond^{-1}_{j:p} (\lnot i \land k)) \rightarrow (\lnot \Diamond_{j:p} \top \land \lnot \blacklozenge k) \in \mathfrak{b}$ . Our assumptions imply the truth of the antecedent. Hence, $\lnot \Diamond_{j:p} \top \land \lnot \blacklozenge k \in \mathfrak{b}$ . At the same time, $(\mathfrak{b},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ is equivalent to $\top \in \mathfrak{b}$ implies $\Diamond_{j:p} \top \in \mathfrak{b}$ , by item (3) of Lemma 19. Since $\mathfrak{b}$ is an MCS, we have that $\top \in \mathfrak{b}$ , and hence $\Diamond_{j:p} \top \in \mathfrak{b}$ , which contradicts $\lnot \Diamond_{j:p} \top \in \mathfrak{b}$ . Now, let $\mathfrak{b} \in \mathfrak{F}^\Gamma (\mathfrak{a})$ . From $k \in \mathfrak{a}$ by item (3) of Lemma 18, we conclude that $\blacklozenge k \in \mathfrak{b}$ , which contradicts $\lnot \blacklozenge k \in \mathfrak{b}$ .

5. (5). Assume that for some $\mathfrak{a},\mathfrak{b},\mathfrak{c} \in \mathfrak{A}^\Gamma$ , we have that $(\mathfrak{a},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ , $p \in +^\Gamma (\mathfrak{b})$ , $p \in -^\Gamma (\mathfrak{b})$ , $\mathfrak{a} \neq \mathfrak{b}$ , and $\mathfrak{V}^\Gamma (i) = \{\mathfrak{b}\}$ , $\mathfrak{V}^\Gamma (j) = \{\mathfrak{c}\}$ , and $\mathfrak{V}^\Gamma (k) = \{\mathfrak{a}\}$ . We demonstrate that $(\mathfrak{b},\mathfrak{b}) \not \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ . Assume toward a contradiction that $(\mathfrak{b},\mathfrak{b}) \in \mathfrak{R}^\Gamma (p,\mathfrak{c})$ . Since $\mathfrak{b}$ is an MCS, $\top \in \mathfrak{b}$ , and by item (3) of Lemma 19, we thus have $\Diamond_{j:p} \top \in \mathfrak{b}$ . At the same time, b contains all the instances of Indiff, and in particular, $@_i ((\lnot p^- \land \lnot p^+ \land \Diamond^{-1}_{j:p} \lnot i) \rightarrow \lnot \Diamond_{j:p} \top) \in \mathfrak{b}$ . By $i \in \mathfrak{b}$ and Elim, we have $(\lnot p^- \land \lnot p^+ \land \Diamond^{-1}_{j:p} \lnot i) \rightarrow \lnot \Diamond_{j:p} \top \in \mathfrak{b}$ . The antecedent follows from our assumption. Hence, $\lnot \Diamond_{j:p} \top \in \mathfrak{b}$ , which contradicts the aforeshown $\Diamond_{j:p} \top \in \mathfrak{b}$ .

Lemma 20 thus establishes that our canonical model $\mathfrak{M}^\Gamma$ is indeed a visibility model.

With the next lemma, usually called in the literature The Truth Lemma, we establish the equivalence between membership of a formula in some $\mathfrak{a} \in \mathfrak{A}^\Gamma$ and truth of the formula at agent $\mathfrak{a}$ of the canonical model.

Case $\varphi = \blacklozenge \psi$ . From left to right. Let $\blacklozenge \psi \in \mathfrak{a}$ . By item (3) of Lemma 21, this implies that there is a $\mathfrak{b} \in \mathfrak{A}^\Gamma$ such that $\mathfrak{a} \in \mathfrak{F} (\mathfrak{b})$ and $\psi \in \mathfrak{b}$ . By the Induction Hypothesis, the latter is equivalent to the fact that there is a $\mathfrak{b} \in \mathfrak{A}^\Gamma$ such that $\mathfrak{a} \in \mathfrak{F} (\mathfrak{b})$ and $\mathfrak{M}^\Gamma_{\mathfrak{b}} \models \psi$ , which, in turn, is equivalent to $\mathfrak{M}^\Gamma_{\mathfrak{a}} \models \blacklozenge \psi$ by the definition of semantics.

From right to left. Assume that $\mathfrak{M}^\Gamma_{\mathfrak{a}} \models \blacklozenge \psi$ . By the definition of semantics, this is equivalent to the fact that there is a $\mathfrak{b} \in \mathfrak{A}^\Gamma$ such that $\mathfrak{a} \in \mathfrak{F} (\mathfrak{b})$ and $\mathfrak{M}^\Gamma_{\mathfrak{b}} \models \psi$ . By the Induction Hypothesis, the latter is equivalent to $\mathfrak{a} \in \mathfrak{F} (\mathfrak{b})$ and $\psi \in \mathfrak{b}$ , which implies $\blacklozenge \psi \in \mathfrak{a}$ by item (3) of Lemma 18.

Case $\varphi = @_i \psi$ . Assume that $\mathfrak{M}^\Gamma_{\mathfrak{a}} \models @_i \psi$ . By the definition of semantics, this is equivalent to $\mathfrak{M}^\Gamma_{\mathfrak{b}} \models \psi$ for a $\mathfrak{b} \in \mathfrak{A}^\Gamma$ such that $\mathfrak{V}^\Gamma (i) = \{\mathfrak{b}\}$ . By the Induction Hypothesis, $\mathfrak{M}^\Gamma_{\mathfrak{b}} \models \psi$ if and only if $\psi \in \mathfrak{b}$ , which implies $@_i \psi \in \mathfrak{b}$ by Intro and MP. By item (3) of Lemma 14, we have that $@_i \psi \in \mathfrak{a}$ . The other direction is similar by using Elim instead of Intro.

We finally have all the ingredients in order to demonstrate that the axiom system $\mathbf{SVL}$ is complete for the class of visibility models.

4.1 Example: Taking the advantage to be seen by many

In some networks, the best tactic for exposing more agents to a controversial opinion is to first post on a popular topic. Consider the follower-network M in Fig. 2. For simplicity, we do not include nominals in the figure. This network consists of six agents named in alphabetical order from a to f. Agent a has two followers: b and c. Agent b has three followers: d, e, and f. We assume that agents d, e, and f might have some followers that we do not have information about, noted in the figure with dots. Furthermore, agent a is positive in favor of vaccination (abbreviated v in the figure) which is a controversial topic among the agents: all agents have an opinion about vaccines and three of the agents a, c, and f are pro vaccination, whereas b, d, and e are contra vaccination. The topic of dogs (abbreviated d), on the other hand, is widely liked. All agents like dogs, except a who is indifferent: $d \not \in +(a)$ and $d \not \in -(a)$ .

Figure 2. A follower-network M where vaccination is a controversial topic.

Imagine that agent a wants to post on vaccines and wants as many as possible of the other agents in the network to see the post. We show that the best tactic for agent a is to first post on dogs, even though a is indifferent about dogs, and then later post on vaccines. Consider first the scenario in Fig. 3 where agent a posts v from the outset. An update happens in two steps. First, we add visibility arrows corresponding to posting and resharing. In the second step, we update the followership relation based on whether the agents who have seen a post are pro or contra the post. The resulting update is $M^{a:v}$ in Fig. 3, where agents b and c have seen a’s vaccine post, and only c remains as a’s follower.

Figure 3. Update $M^{a:v}$ after agent a posts in favor of vaccines.

Then, consider instead the situation where agent a posts on dogs in the update $M^{a:d}$ in Fig. 4 before posting on vaccines in the update $M^{a:d,a:v}$ in Fig. 5. Note that to make the situation easier to read, we omit the followership arrows in the visibility update and the visibility arrows in the followership update in both figures. After $M^{a:d}$ , all agents have seen agent a’s post on dogs. Since they like dogs, all agents also follow a after the update.

Figure 4. Update $M^{a:d}$ after agent a posts on dogs.

Figure 5. Update $M^{a:d,a:v}$ after agent a posts on vaccination after first posting on dogs.

In $M^{a:d,a:v}$ , we see the results after agent a first posted on dogs and then vaccines. All the agents have now seen a’s vaccine post. Most of them did not like it and unfollowed a, but only after they were exposed to the post. Interestingly, we also notice that agent f, who was not originally a follower of a in the initial network outset, now follows a and has shared the vaccine post to their followers.

There are two tactical reasons for agent a to post on dogs before their more controversial post on vaccines. Firstly, a larger portion of the agents now saw a’s post on vaccines since they followed agent a after the dog post. Second, a has been able to reach out and expand their network: agent f who is also pro vaccines has shared the vaccination post to their, for us unknown, followers.

The reason behind a phenomenon such as this is directly connected to an underlying notion of trust between agents in the network. In our setting, agents follow other agents when the former is exposed to content that they like by the latter. In the example, we imagine agents likely followed a because they wanted to see more dog-friendly content. Agent a misuses the trust of their followers by pretending to be interested in dogs before posting on vaccination.

What becomes clear in this example is that in our simplified setting of posting and sharing in a social network, the interests of an agent’s followers matter a lot to what information is shared and seen. Second, the system is vulnerable to exploitation by a potentially malicious agent: there are opportunities to tactically post on popular topics to later expose more agents to a controversial opinion. To reason about dynamic situations such as these, we introduce VL.

4.2 Language, semantics, and logical properties of VL

The language of VL is an extension of the language of SVL.

Definition 24. Syntax. The language of visibility logic $\mathbb{VL}$ is defined recursively by the following BNF:

\begin{align*}\begin{array}{lllll}\varphi &::= &p^+ \mid p^- \mid i \mid \neg \varphi \mid (\varphi \wedge \varphi) \mid \Diamond_{i:p} \varphi \mid \Diamond^{-1}_{i:p} \varphi \mid \blacklozenge \varphi \mid \blacklozenge^{-1} \varphi \mid @_i \varphi \mid [\pi]\varphi\\\pi &::= &p \mid (\pi \cup \pi)\end{array}\end{align*}

where $[\pi]\varphi$ is read “after the current agent executes action $\pi$ , $\varphi$ holds.”

Union of actions $(\pi \cup \tau)$ was inherited by DELs from propositional dynamic logic (PDL) (Fischer and Ladner, Reference Fischer and Ladner1979), and in the context of visibility formulas, $[p \cup q] \varphi$ mean “whichever topic the current agent posts on, p or q, $\varphi$ will be true (in both cases).”

Given a formula $\varphi \in \mathbb{VL}$ , we define modal depth and the size of the formula similarly to the corresponding definitions for $\mathbb{SVL}$ with the following additional cases: $md([\pi]\varphi) = md(\varphi)$ , and $|[\pi]\varphi| = |\pi| + |\varphi| + 1$ , where $|\pi \cup \tau| = |\pi| + |\tau| + 1$ .

Definition 25. (Semantics). Let $M = (A,F,+,-,V,R)$ be a visibility model, $a \in A$ , and $p,q \in \mathsf{Top}$ . The semantics of VL is the same as in Definition 3 with the following additions:

\begin{align*}\begin{array}{lll}M_a \models [p]\varphi &\text{iff} &M_a^{a:p} \models \varphi\\M_a \models [\pi \cup \tau] \varphi &\text{iff} &M_a \models [\pi]\varphi \text{ and } M_a \models [\tau]\varphi\\M_a \models \langle \pi \cup \tau \rangle \varphi &\text{iff} &M_a \models \langle \pi \rangle \varphi \text{ or } M_a \models \langle \tau \rangle \varphi\end{array}\end{align*}

where $M_a^{a:p}$ is defined in the following two steps. First, let $M^\ast = (A$ , F, $+$ , $-$ , V, $R^\ast)$ , where $R^\ast (p,a)$ is the least fixed point of function $f: 2^{A \times A} \rightarrow 2^{A \times A}$ defined as:

$$\begin{aligned}f(X) = &X \cup \{(a,a)\} \cup \{(b,c) \mid (b,b) \in X \text{ and } c \in F(b)\} \\&{}\cup\{(c,c) \mid p \in +(c) \text{ and } \exists b: (b,c) \in X\}.\end{aligned}$$

Informally, intermediate model $M^\ast$ differs from M only in R in such a way that $R^\ast$ now contains the fact that a has posted on p, that her post has reached all her followers, and that all followers who are pro p reshare the post further to their followers. Second, we construct $M^{a:p}$ out of $M^\ast$ by updating F:

• (1) $F^{a:p}(a) = F(a) \cup \{b\}$ , if $a\neq b$ , $p \in +(b)$ , and $\exists c: (c,b) \in R^\ast(p,a)$ ,

• (2) $F^{a:p}(b) = F(b) \setminus \{c\}$ , if $p \in -(b)$ and $(c,b) \in R^\ast(p,a)$ .

Intuitively, agent b will follow the original poster a if she has seen the post, maybe not even from a, and if she is pro the topic. Agent c will stop following anyone from whom she has seen a post on a topic she dislikes.

Returning to the example in the beginning of this section, we can use the new operators to reason about the situation with formulas that hold in the models given in the example. For simplicity, we name the agents with nominals corresponding to their labels in the model. That is, $M_{a} \vDash a$ , $M_{b} \vDash b$ and so on, where $a, b \in \mathsf{Nom}$ . To avoid confusion with the agent named d, we denote the topics “vaccines” and “dogs” with vacc and dogs instead of v and d as used in the example. The following formulas, with their respective intuitive readings, hold in the initial model M in Fig. 2:

• • $M_{a} \vDash [vacc] \blacksquare^{-1}c$

“After posting on vaccines, a will have only one follower, c.”

• • $M_{a} \vDash [vacc]( \Diamond_{a:vacc} b \wedge \Diamond_{a:vacc} c \wedge \Box_{a:vacc}(a \vee b \vee c))$

“After posting on vaccines, only b and c, except for a, will have seen a’s post.”

• • $M_{a} \vDash [dogs] (\blacklozenge^{-1} b \wedge \blacklozenge^{-1} c \wedge \blacklozenge^{-1} d \wedge \blacklozenge^{-1} e \wedge \blacklozenge^{-1} f)$

“After posting on dogs, all agents from b to f will follow a.”

• • $M_{a} \vDash [dogs][vacc] (\Diamond_{a:vacc} b \wedge \Diamond_{a:vacc} c \wedge \Diamond_{a:vacc} d \wedge \Diamond_{a:vacc} e \wedge \Diamond_{a:vacc} f)$

“After a first posts on dogs, and then on vaccines, all agents from a to f will have seen a’s post on vaccines.”

To give a further taste of VL, let us provide some properties that are valid or not valid on visibility models. All the validities can be shown by an application of the definition of the semantics.

The first formula states that the operator of posting on a topic is its own dual. The second property shows how to eliminate nondeterministic choice. The third item claims that once an agent has seen a post of an agent with the name i on topic p, no further post can revoke this. The fact that formulas four and five are not valid indicates that consecutive posting on the same topic yields different results. A counterexample showing this would include the current agent with the name i posting on p and gaining new followers. Additional posts on the same topic by the same agent will add new p-arrows to those new followers, thus resulting in a different updated model that is not guaranteed to satisfy $\varphi$ .

4.3 Expressivity and model checking

Now, we state that VL is more expressive than its static fragment SVL. This result is quite interesting, since many of DELs, for example, public announcement logic (Plaza, Reference Plaza2007), arrow update logic (Kooi and Renne, Reference Kooi and Renne2011), and action model logic (van Ditmarsch et al., Reference van Ditmarsch, van der Hoek and Kooi2008, Chapter 6), are equally expressive as the static logic they are built upon. Those expressivity results are usually obtained with the use of so-called reduction axioms that allow one to equivalently rewrite formulas of dynamic extensions to formulas of the static fragment. Thus, the fact that VL is more expressive than SVL also entails that no reduction axioms for VL are possible.

Consider models M and N in Fig. 6. The models are chains of length $n+2$ that start with agent a and with each next agent following the previous one. The only difference between the models is that the last agent in the chain in model M is pro topic p, and the last agent in the chain in model N is neither pro nor contra topic p.

Figure 6. Models M and N.

Now we will argue that $[p]\blacklozenge^{-1} \blacksquare^{-1} \bot$ distinguishes $M_a$ and $N_a$ . In particular, $M_a \models [p]\blacklozenge^{-1} \blacksquare^{-1} \bot$ and $N_a \not \models [p]\blacklozenge^{-1} \blacksquare^{-1} \bot$ . Indeed, agent a posting on topic p results in the updated visibility model $M_a^{a:p}$ presented in Fig. 7. In the updated model, it holds that $M_{b_{n+1}}^{a:p} \models \blacklozenge i$ , that is, that agent $b_{n+1}$ follows agent a. Moreover, agent $b_{n+1}$ does not have any followers so $M_{b_{n+1}}^{a:p} \models \blacksquare^{-1} \bot$ is vacuously true. Hence, $M_a \models [p]\blacklozenge^{-1} \blacksquare^{-1} \bot$ . To see that $N_a \not \models [p]\blacklozenge^{-1} \blacksquare^{-1} \bot$ , it is enough to notice that agent $b_{n+1}$ is not pro topic p, and thus they do not follow agent a in the updated model. Updated model $N_a^{a:p}$ is depicted in Fig. 7 on the right.

Figure 7. Models $M^{a:p}$ and $N^{a:p}$ . Reflexive $p_a$ -arrows and followership arrows from $b_k$ to a for $k \in \{1, ..., n\}$ are omitted for readability.

In order to show that $M_a \models \psi$ if and only if $N_a \models \psi$ , observe that $M_a$ and $N_a$ are n-bisimilar (but not $(n+1)$ -bisimilar), which implies that the pointed models are $(\mathsf{Nom} \setminus Q)$ -n-bisimilar. From the fact that $md(\psi) = n$ and Theorem 9, it follows that $M_a \models \psi$ if and only if $N_a \models \psi$ . Hence, we have a contradiction with an earlier assumption that $\psi$ is equivalent to $[p]\blacklozenge^{-1} \blacksquare^{-1} \bot$ .

Before we turn to the model checking problem for VL, we mention that the complexity of the model checking problem of SVL is in P. This result follows trivially from the fact that model checking hybrid tense logic with universal modality is in P (Franceschet and de Rijke, Reference Franceschet and de Rijke2006).

Not only is VL more expressive than SVL, but its model checking problem is also more computationally demanding. We show this by providing a model checking algorithm for VL that runs in polynomial space. For hardness, we use the classic reduction from quantified Boolean formulas.

Algorithm 1 An algorithm for model checking VL

The algorithm follows the semantics, and its correctness can be shown via induction on $\varphi$ . Now we argue that the algorithm requires at most polynomial space. The interesting case here is $\varphi = [p] \psi$ . Without giving an explicit algorithm for constructing $M^{a:p}$ , we note that the size of $M^{a:p}$ is bounded by $\mathcal{O}(|M|^2)$ (the worst-case scenario of R(p,a) and F being universal). Since there are at most $|\varphi|$ symbols in $\varphi$ , the total space required by the algorithm is bounded by $\mathcal{O}(|\varphi| \cdot |M|^2)$ .

To show hardness of the model checking problem, we use the classic reduction from the satisfiability of quantified Boolean formulas: given a QBF $\Psi := Q_1 p_1 \ldots Q_n p_n \psi(p_1, \ldots, p_n)$ , where $Q_i \in \{\forall, \exists\}$ , determine whether $\Psi$ is true. To reduce the satisfiability of QBF $\Psi$ to the model checking of VL, we construct a model $M_a$ and a formula $\Psi^\prime$ of $\mathbb{VL}$ such that $\Psi$ is true if and only if $M_a \models \Psi^\prime$ .

More specifically, given a QBF $Q_1 p_1 \ldots Q_n p_n \psi(p_1, \ldots, p_n)$ , we construct a visibility model $M = (A,F,+,-,V,R)$ , where $A = \{a_0, \ldots, a_n\}$ , $F(a_i) = \{a_0\}$ for all $i \neq 0$ , $+(a_i) = p_i$ for all $i \neq 0$ , $-(a_i) = \emptyset$ for all i, $V(i_j) = \{a_j\}$ , and $R(p,a) = \emptyset$ for all $p \in \mathsf{Top}$ and $a \in A$ . Additionally, we assume that there is a topic q that no agent is either pro or contra. Intuitively, M is a model consisting of $n+1$ agents, where everyone follows agent $a_0$ , who follows no one. Each agent, apart from $a_0$ , is pro exactly one topic, and no one is contra anything. Finally, the translation of the QBF is done recursively as follows:

\begin{align*}\psi_0^\prime &:= \psi( \Diamond_{i_0:p_1} (i_1 \land \Diamond_{i_0:p_1} \top), \ldots, \Diamond_{i_0:p_n} (i_n \land \Diamond_{i_0:p_n} \top))\\ \psi_k^\prime &:=\begin{cases} [p_k \cup q] \psi_{k-1}^\prime &\text{if } Q_k = \forall\\ \lnot [p_k \cup q] \lnot \psi_{k-1}^\prime &\text{if } Q_k = \exists\\\end{cases}\\\psi^\prime &:= \psi^\prime_{n}.\end{align*}

We need to show that

\begin{align*}Q_1 x_1 \ldots Q_n x_n \psi(p_1, \ldots, p_n)\, \text{is satisfiable iff}\, M_a \models \psi^\prime.\end{align*}

Agent $a_0$ posting on topic $p_i$ means that the truth value of $p_i$ has been set to 1. If agent $a_0$ posts on topic q, this means that the truth value of the corresponding $p_i$ has been set to 0. Since there are no two agents that are pro the same topic, the choice of truth values is unambiguous.

We use nondeterministic choice to model quantifiers. The universal quantifier $\forall p_k$ is emulated with $[p_k \cup q] \psi^\prime_{k-1}$ meaning that no matter what agent $a_0$ chooses to post on, $p_k$ or q, formula $\psi^\prime_{k-1}$ will be true. Similarly, the existential quantifier $\exists p_k$ is emulated with $\lnot [ p_k \cup q ] \lnot \psi^\prime_{k-1}$ meaning that agent $a_0$ can post on a topic, either $p_k$ or q, to make $\psi^\prime_{k-1}$ true. Finally, propositional variable $p_j$ is translated into the formula $\Diamond_{i_0:p_j} (i_j \land \Diamond_{i_0:p_j} \top)$ that is true if and only if there has been a post on $p_j$ , and the corresponding agent $a_j$ , who is pro $p_j$ , has reposted it. For all other agents $a_k$ , the formula will not hold. Posting on q instead of $p_j$ results in the fact that $\Diamond_{i_0:p_j} (i_j \land \Diamond_{i_0:p_j} \top)$ is not satisfied anywhere in the model, thus corresponding to setting $p_j$ to 0.

As an example, consider a QBF $\forall p_1 \exists p_2 (p_1 \rightarrow p_2)$ . The formula is first translated into a formula of $\mathbb{VL}$ : $[p_1 \cup q] \lnot [p_2 \cup q] \lnot (\Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top) \rightarrow \Diamond_{i_a:p_2} (i_c \land \Diamond_{i_a:p_2} \top))$ . The corresponding model M is depicted in Fig. 8.

Figure 8. Models M, $M^{a:p_1}$ , $M^{a:q}$ , and $M^{a:p_1, a:p_2}$ .

Now, $M_a \models [p_1 \cup q] \lnot [ p_2 \cup q ] \lnot (\Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top) \rightarrow \Diamond_{i_a:p_2} (i_c \land \Diamond_{i_a:p_2} \top))$ if and only if

$$M_a^{a:p_1} \models \lnot [ p_2 \cup q ] \lnot (\Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top) \rightarrow \Diamond_{i_a:p_2} (i_c \land \Diamond_{i_a:p_2} \top))$$

and

$$M_a^{a:q} \models \lnot [ p_2 \cup q ]\lnot (\Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top) \rightarrow \Diamond_{i_a:p_2} (i_c \land \Diamond_{i_a:p_2} \top)).$$

Both updated models $M_a^{i_a:p_1}$ and $M_a^{i_a:q}$ are depicted in Fig. 8. Notice that since agent b is pro topic $p_1$ , they repost it thus creating a reflexive loop labeled with $p_1$ . Agent c, who is not pro topic $p_1$ , does not have such a loop. In the case of q, none of the agents have a reflexive loop.

In model $M_a^{a:p_1}$ , propositional variable $p_1$ has been set to 1. Thus, we can choose to update the model further with a’s post on $p_2$ to satisfy the initial formula. Indeed, consider model $M^{a:p_1, a:p_2}$ . In the model, we have that $M^{a:p_1, a:p_2}_a \models \Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top) \rightarrow \Diamond_{i_a:p_2} (i_c \land \Diamond_{i_a:p_2} \top)$ since there is an $a:p_1$ -arrow to state satisfying $i_b$ from which there another $a:p_1$ -arrow. Similarly for $a:p_2$ -arrows. This corresponds to satisfying $p_1 \rightarrow p_2$ by setting both variables to 1.

In model $M_a^{a:q}$ , propositional variable $p_1$ has been set to 0. Thus, we can choose any further update of the model. Let agent a post on q once again. Since we do not discriminate between different posts on the same topic, model $M_a^{a:q, a:q}$ looks exactly like $M_a^{a:q}$ . Moreover, it is clear that $M_a^{a:q, a:q} \models \Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top) \rightarrow \Diamond_{i_a:p_2} (i_c \land \Diamond_{i_a:p_2} \top)$ since $M_a^{a:q, a:q} \not \models \Diamond_{i_a:p_1} (i_b \land \Diamond_{i_a:p_1} \top)$ . This corresponds to satisfying $p_1 \rightarrow p_2$ by setting both variables to 0.