Social Announcement Logic

Dynamic Epistemic Logic in Social Networks

Rui Zhu

Supervised by Dr. Jeremy Seligman

Department of Philosophy, Humanities, UNIVERSITY OF AUCKALND

Motivations

  • Can we establish a logical framework for reasoning about following relationships through belief updates?
  • Example:
Aladdin
$\uparrow$
Bulma
$\leftarrow$ Chopper
$ \uparrow ？$
Doraemon

Framework

  • Both beliefs and announcement are expressed at the propositional level.
  • $B_{a} \theta$ means "agent $a$ believes message $\theta$."
  • $[a: \theta]\phi$ $(\langle a! \theta\rangle\phi)$ can be read as "After agent $a$ (sincerely) announces $\theta$ to its followers, $\phi$ is true."
  • The arbitrary operators $\langle a\rangle$ and $\langle a!\rangle\phi$ are used to express the capacity of agent $a$ to make a (sincerely) social announcement.
  • Agents are conservative. They never abandon their existing beliefs.

Examples

A model $m = (f, k)$ is a tuple where $f$ represents a social network and function $k$ represent an epistemic distribution .

$$ \begin{tikzpicture}[node distance=1.5cm] % Define nodes with white text \node(A) [text = white] {\textbf{model}}; \node(A1) [below of=A, text=orange] {$a$}; \node(A2) [left of=A1, text=cyan] {$b$}; \node(B1) [below of=A1, text=magenta] {$c$}; % Corrected from B1 to A1 \node(B2) [left of=B1, text=green] {$d$}; % Draw arrows \draw[->, white] (A1) .. controls +(up:19mm) and +(right:15mm) .. (A1); \draw[->, white] (A2) -- (A1); \draw[->, white] (B1) -- (A1); \draw[->, white] (B2) -- (A2); \draw[->, white] (B2) -- (B1); \end{tikzpicture} \\ \\ \begin{tikzpicture}[node distance=1.5cm ] \node(A4) [text=white] {$p,q,r$}; \node(B2) [below of= A4, text=white ] {$ p,q$}; \node(B1) [left of= B2 , text=white] {$ p,r$}; \node(B3) [right of=B2, text=white] {$ q,r $}; \node(D1) [below of=B1, text=white] {$p$}; \node(D2) [right of=D1, text=white] {$q$}; \node(D3) [right of=D2, text=white] {$r$}; \node(E) [below of=D2, text=white] {$\varnothing$}; \draw[orange, line width=1pt] (B1) circle (21pt); \draw[cyan, line width=1pt] (B1) circle (18pt); \draw[cyan, line width=1pt] (D1) circle (18pt); \draw[magenta, line width=1pt] (D1) circle (15pt); \draw[magenta, line width=1pt] (E) circle (15pt); \draw[green, line width=1pt] (B3) circle (13pt); \draw[green, line width=1pt] (D2) circle (13pt); \draw[green, line width=1pt] (D3) circle (13pt); \end{tikzpicture} $$
test

Arbitrary Announcement

Kuijer's Counterexample

Currently we do not know if $\mathsf{APAL}$ has finitary axiomatization. The following rule does not preserve validity.

From $ \langle \chi\rangle\langle q\rangle\phi\rightarrow\psi$, infer $ \langle \chi\rangle\Diamond\phi\rightarrow\psi$
where $q\not\in Var(\psi)\cup Var(\phi)\cup Var(\chi)$. Down arrow
$\vDash \langle \top\rangle\langle q\rangle(p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)\rightarrow \langle \widehat{B_{b}} p\rangle \Diamond (p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)$

$m, w_{1}\not\vDash \langle \top\rangle\Diamond(p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)\rightarrow \langle \widehat{B_{b}} p\rangle \Diamond (p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)$

Noncompactness

The logic of $L_{SAL}$ is not compact. Here are two counterexamples.

$\{\langle a\rangle(B_{a}\bot \wedge \neg B_{b}\bot), \langle a\rangle B_{b}\bot \}\cup \{\neg B_{a}\theta ~~|~~ \theta~~is~~not~~an~~instance ~~of~~tautology~~\}$

$\{\langle a!\rangle B_{b}p \wedge \neg B_{b} p \}\cup\{\neg B_{a}\theta ~~|~~ \theta~~is~~not~~an~~instance~~of~~tautology~~\}$

Main Results

By using free message model and moduolo satisfiability, we show the following rules preserve validity.

From $\langle a!p\rangle\phi \rightarrow \psi $, infer $ \langle a!\rangle\phi \rightarrow \psi$

From $[a:p]\phi \rightarrow \psi$, infer $\langle a\rangle\phi\rightarrow \psi$

where $p$ is a propositional variable that does not occur $\phi$ or $\psi$.

Thus, by using non-greedy set we have a finitary complete axiomatization $\mathsf{sal}$.

Tableau Calculus

  • In Chapter four, we present a complete and non-greedy compact tableau system.
  • This system allows us to easily construct proofs for theorems in $\mathsf{sal}$, such as the distribution of $\langle a!\rangle[a!]$ over conjunction.
  • Also, it can generate models to satisfy any consistent non-greedy set of formulas.

we test the following set of formulas: \[ \{ \langle a!\rangle B_{b}q \wedge \neg B_{b}q, ~~~~ [a:p]\neg B_{b}p\} \\ \Rightarrow unsatisfiable. \\ \{ \langle a!\rangle (B_{b}q \wedge \neg B_{c}r), ~~~~ \langle b!p\rangle (B_{c}p \wedge \neg B_{a}p), ~~~[a:r](\neg B_{b}r \wedge B_{c}r) \} \\ \Rightarrow \{cFa, \neg aFb, \neg bFa, B_{b}p \} \]

Epistemic Distribution

Coherences

We define five coherences for our original models.

  • Global coherence: All agents' beliefs are collectively consistent.
  • Local coherence: No individual agent's beliefs become inconsistent following sincere announcements.
  • Weak coherence: Each agent's beliefs are internally consistent.
  • Cohesive coherence: Agents with inconsistent beliefs are separated.
  • Flowing coherence: No contradictory flow can be made.
$$\begin{tikzpicture} \definecolor{bubbles}{rgb}{0.91, 1.0, 1.0} \definecolor{capri}{rgb}{0.0, 0.75, 1.0} \definecolor{celadon}{rgb}{0.67, 0.88, 0.69} \definecolor{cottoncandy}{rgb}{1.0, 0.74, 0.85} \definecolor{deepcerise}{rgb}{0.85, 0.2, 0.53} \definecolor{chartreuse}{rgb}{0.87, 1.0, 0.0} \definecolor{britishracinggreen}{rgb}{0.0, 0.26, 0.15} \begin{scope} [fill opacity = .5] \draw[fill= bubbles] (-5,5) rectangle (8,-3); \draw[fill=celadon, draw = black] (-1.5,1) circle (3); \draw[fill=deepcerise, draw = black] (1.55,1) circle (2.87); \draw[fill=cottoncandy, draw = black] (2.65,1) ellipse (5cm and 3.2cm); % Adjusted this line \draw[fill=capri, draw = black] (0,1) circle (1); \node at (-4,4) {\color{chartreuse}\LARGE\textbf{SAL}}; \node at (-3,2.5) {\color{britishracinggreen}\LARGE\textbf{wc}}; \node at (5,2.5) {\color{britishracinggreen}\LARGE\textbf{fc}}; \node at (2.5, 2.5) {\color{britishracinggreen}\LARGE\textbf{cc}}; \node at (0,2.5) {\color{britishracinggreen}\LARGE\textbf{lc}}; \node at (0, 1) {\color{britishracinggreen}\LARGE\textbf{gc}}; \end{scope} \end{tikzpicture} $$

Implications

By implementing coherent models, the expressiveness of $L_{SAL}$ is enhanced. These models assist in reasoning about the structure of social networks. Some allow us to represent "following relationships", but some logical properties might be lost and may not be characterizable.

Main Results

  • Following relationship expressible: wc, cc, lc, gc.
  • Characterizable: fc, wc.
  • Failure of free announcement necessitation: Indirect axiomatizations are needed. An axiomatization without the necessitation rule is available with respect to wc.
  • Preversing validity by sincere announcement necessitation: cc, lc, gc.

Application

  • By applying a cascading announcement from social announcement logic, we can explore how sociological phenomena such as peer pressure, the bandwagon effect, and the rise of polarization within echo chambers come to exist.

Given 20 agents with existing beliefs, we allow them to randomly make cascading announcements 200 times.

When an agent receives a belief that contradicts their current one, they will unfollow the speaker.

Additionally, agents can form following relationships 40 times, with agents in the same echo chamber being 3 times more likely to establish such relationships.

Achievements

  • Establishment of the Framework of Social Announcement Logic
  • Proof of the Properties of Social Announcement Logic
  • Model Transformation Technique Dealing with Arbitrary Operators
  • The Development of A Tableau System
  • Practical Applications for Interdisciplinary Studies

Thank you