The formula for an arbitrary social announcement, denoted by $\langle a \rangle\phi$ or $\langle a!\rangle \phi$, signifies that agent $a$ is capable of making a (sincere) social announcement such that $\phi$ holds.
$\langle a\rangle\phi$: "After $a$ makes an announcement to its followers, $\phi$ is true."
$\langle a!\rangle\phi$: "After $a$ sincerely makes an announcement to its followers, $\phi$ is true."
$\langle a!\rangle\phi$: "After $a$ sincerely makes an announcement to its followers, $\phi$ is true."
Noncompactness
The logic of $L_{SAL}$ is not compact. Here are two counterexamples.
$\{\langle a\rangle(B_{a}\bot \wedge \neg B_{b}\bot), \langle a\rangle B_{b}\bot \}\cup \{\neg B_{a}\theta ~~|~~ \theta~~is~~not~~an~~instance ~~of~~tautology~~\}$ $\{\langle a!\rangle B_{b}p \wedge \neg B_{b} p \}\cup\{\neg B_{a}\theta ~~|~~ \theta~~is~~not~~an~~instance~~of~~tautology~~\}$Framework
Language of APAL
$\mathsf{A}$ is a finite set of agents. $\mathsf{Prop}$ is the set of propositional variables. A message is defined as follows
\[ \theta := p~|~\neg \theta~|~(\theta\wedge \theta)\]
And we define language $L_{SAL}$ by specifying
\[ \phi := B_{a} \theta~|~\neg\phi~|~(\phi\wedge\phi)~|~[a: \theta]\phi~|~\langle a! \theta\rangle\phi~|~\langle a \rangle\phi~|~\langle a! \rangle\phi \]
where $p\in \mathsf{Prop}$, $a \in A$ and for each $a$ there is a modality $B_{a}$ in $L_{SAL}$
Denotation
\[V = 2^{\mathsf{Prop}}
\\
\llbracket p \rrbracket = \{x \in V ~|~ p \in x \}
\\
\llbracket \neg \theta \rrbracket = V \backslash \llbracket \theta \rrbracket
\\
\llbracket (\theta_1 \wedge \theta_2) \rrbracket = \llbracket \theta_1\rrbracket \cap \llbracket\theta_2 \rrbracket\]
Model
$~~~~~~~~(f, k)\vDash [a:\theta] \phi \Leftrightarrow [a:\theta](f, k)\vDash \phi$
in which $[a:\theta](f, k) = (f, [a:\theta]k)$
where $[a:\theta]k(b) = \begin{cases} k(b)\cap \llbracket \theta \rrbracket~~~~ {\rm if}~b\in f(a), \\ k(b)~~~~~~{\rm otherwise} \end{cases}$
$~~~~~~~~(f, k)\vDash \langle a!\theta\rangle \phi \Leftrightarrow [a:\theta](f, k)\vDash \phi~\&~k(a)\subseteq \llbracket \theta\rrbracket$
$~~~~~~~~(f, k)\vDash \langle a\rangle\phi \Leftrightarrow \exists\theta [a :\theta](f, k) \vDash \phi$
$~~~~~~~~(f, k)\vDash \langle a!\rangle\phi \Leftrightarrow \exists\theta (k(a) \subseteq \llbracket \theta\rrbracket~\&~[a :\theta](f, k) \vDash \phi)$
$~~~~~~~~(f, k)\vDash [a:\theta] \phi \Leftrightarrow [a:\theta](f, k)\vDash \phi$
in which $[a:\theta](f, k) = (f, [a:\theta]k)$
where $[a:\theta]k(b) = \begin{cases} k(b)\cap \llbracket \theta \rrbracket~~~~ {\rm if}~b\in f(a), \\ k(b)~~~~~~{\rm otherwise} \end{cases}$
$~~~~~~~~(f, k)\vDash \langle a!\theta\rangle \phi \Leftrightarrow [a:\theta](f, k)\vDash \phi~\&~k(a)\subseteq \llbracket \theta\rrbracket$
$~~~~~~~~(f, k)\vDash \langle a\rangle\phi \Leftrightarrow \exists\theta [a :\theta](f, k) \vDash \phi$
$~~~~~~~~(f, k)\vDash \langle a!\rangle\phi \Leftrightarrow \exists\theta (k(a) \subseteq \llbracket \theta\rrbracket~\&~[a :\theta](f, k) \vDash \phi)$
Kuijer's Counterexample
Arbitrary announcements, denoted by $\Diamond$, have been studied in $\mathsf{APAL}$. Currently we do not know if $\mathsf{APAL}$ has finitary axiomatization due to Kuijer's Counterexample. 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 $q\not\in Var(\psi)\cup Var(\phi)\cup Var(\chi)$.
$\vDash \langle \top\rangle\langle q\rangle(p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)\rightarrow \langle \widehat{B_{b}} p\rangle \Diamond (p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)$
$m, w_{1}\not\vDash \langle \top\rangle\Diamond(p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)\rightarrow \langle \widehat{B_{b}} p\rangle \Diamond (p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)$
$m, w_{1}\not\vDash \langle \top\rangle\Diamond(p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)\rightarrow \langle \widehat{B_{b}} p\rangle \Diamond (p \wedge \neg B_{b}p \wedge \widehat{B_{a}} B_{b}p)$
Main Results
In Chapter 3, we use free message model technique to 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 $\psi$.
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 $\psi$.
$$\begin{tikzpicture}[node distance=1.2cm]
\title{Before Announcement}
\node(A1) [text=white] {$p,q,r$};
\node(A2) [right of= A1, text=white] {$p,q,s$};
\node(A4) [right of=A2, text=white] {$p, r,s $};
\node(A3) [right of=A4, text=white] {$q, r,s ...$};
\node(B2) [below of= A1, text=white] {$ p,r$};
\node(B1) [left of= B2, text=white] {$ p,q$};
\node(B3) [right of=B2, text=white] {$ p,s $};
\node(B4) [right of= B3, text=white] {$ q,r$};
\node(B5) [right of= B4, text=white] {$ q,s$};
\node(B6) [right of=B5, text=white] {$ r,s ... $};
\node(D1) [below of=B1, text=white] {$p$};
\node(D2) [right of=D1, text=white] {$q$};
\node(D3) [right of=D2, text=white] {$r$};
\node(D4) [right of=D3, text=white] {$s ...$};
\node(E) [below of=D2, text=white] {$\varnothing$};
\draw[orange, line width=1pt] (B1) circle (12pt);
\draw[orange, line width=1pt] (B2) circle (12pt);
\draw[orange, line width=1pt] (D1) circle (12pt);
\draw[orange, line width=1pt] (D3) circle (12pt);
\draw[cyan, line width=1pt, dotted] (A1) circle (10pt);
\draw[cyan, line width=1pt, dotted] (B4) circle (10pt);
\draw[cyan, line width=1pt, dotted] (D3) circle (10pt);
\draw[magenta, line width=1pt, dashed] (A1) circle (14pt);
\draw[magenta, line width=1pt, dashed] (D2) circle (14pt);
\draw[magenta, line width=1pt, dashed] (E) circle (14pt);
\end{tikzpicture}$$
$$\begin{tikzpicture}[node distance=1.2cm]
\title{After announcement}
\node(A1) [text=white] {$p,q,r$};
\node(A2) [right of= A1, text=white] {$p,q,s$};
\node(A4) [right of=A2, text=white] {$p, r,s $};
\node(A3) [right of=A4, text=white] {$q, r,s ...$};
\node(B2) [below of= A1, text=white] {$ p,r$};
\node(B1) [left of= B2, text=white] {$ p,q$};
\node(B3) [right of=B2, text=white] {$ p,s $};
\node(B4) [right of= B3, text=white] {$ q,r$};
\node(B5) [right of= B4, text=white] {$ q,s$};
\node(B6) [right of=B5, text=white] {$ r,s ... $};
\node(D1) [below of=B1, text=white] {$p$};
\node(D2) [right of=D1, text=white] {$q$};
\node(D3) [right of=D2, text=white] {$r$};
\node(D4) [right of=D3, text=white] {$s ...$};
\node(E) [below of=D2, text=white] {$\varnothing$};
\draw[orange, line width=1pt] (B4) circle (12pt);
\draw[orange, line width=1pt] (B5) circle (12pt);
\draw[orange, line width=1pt] (D2) circle (12pt);
\draw[orange, line width=1pt] (D4) circle (12pt);
\draw[cyan, line width=1pt, dotted] (A3) circle (10pt);
\draw[cyan, line width=1pt, dotted] (B6) circle (10pt);
\draw[cyan, line width=1pt, dotted] (D4) circle (10pt);
\draw[magenta, line width=1pt, dashed] (A3) circle (14pt);
\draw[magenta, line width=1pt, dashed] (D3) circle (14pt);
\draw[magenta, line width=1pt, dashed] (E) circle (14pt);
\end{tikzpicture}$$
A set of formulas is greedy if it contains a cofinite set of atomic messages.
We have a finitary complete axiomatization $\mathsf{sal}$: Any non-greedy consistent set is satisfiable.
Finitary Axiomatization
The formal system $\mathsf{sal}$ is $\mathsf{Fsal}$ plus the following:
Axioms
[SRed] $\langle a!\theta\rangle\phi\leftrightarrow (B_{a}\theta \wedge [a:\theta]\phi)$
[$K^{S}_A$] $[a!](\phi \rightarrow \psi) \rightarrow ([a!]\phi \rightarrow [a!]\psi)$
[$K^{F}_{A}$] $[a](\phi \rightarrow \psi) \rightarrow ([a]\phi \rightarrow [a]\psi)$
[$A^{S}$I] $\langle a!\theta\rangle\phi \rightarrow \langle a!\rangle\phi$
[$A^{F}$I] $[a:\theta]\phi \rightarrow \langle a\rangle\phi$
Rules
[$[a]$-Nec] From $ \phi$, infer $ [a]\phi$
[$[a!]$-Nec] From $\phi$, infer $ [a!]\phi$
[$A^{S}$E] From $\langle a!p\rangle\phi \rightarrow \psi $, infer $ \langle a!\rangle\phi \rightarrow \psi$
[$A^{F}$E] 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 $\psi$.
[$K^{F}_{A}$] $[a](\phi \rightarrow \psi) \rightarrow ([a]\phi \rightarrow [a]\psi)$
[$A^{S}$I] $\langle a!\theta\rangle\phi \rightarrow \langle a!\rangle\phi$
[$A^{F}$I] $[a:\theta]\phi \rightarrow \langle a\rangle\phi$
Rules
[$[a]$-Nec] From $ \phi$, infer $ [a]\phi$
[$[a!]$-Nec] From $\phi$, infer $ [a!]\phi$
[$A^{S}$E] From $\langle a!p\rangle\phi \rightarrow \psi $, infer $ \langle a!\rangle\phi \rightarrow \psi$
[$A^{F}$E] 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 $\psi$.
$\mathsf{sal}$ is a sound and weakly complete finitary axiomatization.
Tableau Calculus
In Chapter 4, we present a complete and non-greedy compact tableau system.
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.
Results
- Following relationship expressible: wc, cc, lc, gc.
- Definable: fc, wc.
- Failure of free announcement necessitation: Indirect axiomatizations are needed.
- Preversing validity by sincere announcement necessitation: cc, lc, gc.
Variants
In Chapter 5, we explore some potential variants of social announcement logic.
- Partial Diffusion: Only some of the speaker's followers receive and update their beliefs with the announcement.
- Cascading Broadcast: Everyone automatically broadcasts the message they receive unless it contradicts their existing beliefs.
- Unfollowing: Agents unfollow those who share messages that conflict with their current beliefs.
Relational Semantics
$$\begin{tikzpicture}[node distance=1.1cm]
% Define nodes with white text
\node(A) [text = white] {$f$};
\node(A1) [below of=A, text=orange] {$a$};
\node(A2) [left of=A1, text=green] {$d$};
\node(B1) [below of=A1, text=cyan ] {$b$}; % Corrected from B1 to A1
\node(B2) [left of=B1, text=magenta] {$c$};
% Draw arrows
\draw[->, white] (A1) .. controls +(up:19mm) and +(right:15mm) .. (A1);
\draw[->, white] (B1) .. controls +(down:19mm) and +(right:15mm) .. (B1);
\draw[->, white] (B1) -- (A1);
\draw[->, white] (B2) -- (B1);
\end{tikzpicture}
\\
\begin{tikzpicture}[node distance=1.1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$k~~~~(w)$ };
\draw[orange, line width=1pt] (B1) circle (16pt);
\draw[cyan, line width=1pt, dotted] (B1) circle (14pt);
\draw[cyan, line width=1pt, dotted] (D1) circle (14pt);
\draw[magenta, line width=1pt, dashed] (D1) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture} $$
$$\begin{tikzpicture}[node distance=1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$[a:r]k~~~~(w_1)$ };
\draw[orange, line width=1pt] (B1) circle (16pt);
\draw[cyan, line width=1pt, dotted] (B1) circle (14pt);
\draw[magenta, line width=1pt, dashed] (D1) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}
\\
\begin{tikzpicture}[node distance=1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$[a:\neg r]k~~~~(w_2)$};
\draw[cyan, line width=1pt, dotted] (D1) circle (14pt);
\draw[magenta, line width=1pt, dashed] (D1) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}$$
$$
\begin{tikzpicture}[node distance=1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$[a:\bot]k~~~~(w_3)$ };
\draw[magenta, line width=1pt, dashed] (D1) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}
\\
\begin{tikzpicture}[node distance=1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$[b:r]k~~~~(w_4)$ };
\draw[orange, line width=1pt] (B1) circle (16pt);
\draw[cyan, line width=1pt, dotted] (B1) circle (14pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}
$$
$$
\begin{tikzpicture}[node distance=1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$[b:\neg r]k~~~~(w_5)$ };
\draw[orange, line width=1pt] (B1) circle (16pt);
\draw[cyan, line width=1pt, dotted] (D1) circle (14pt);
\draw[magenta, line width=1pt, dashed] (D1) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}
\\
\\
\begin{tikzpicture}[node distance=1cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$[b:\bot]k~~~~(w_6)$ };
\draw[orange, line width=1pt] (B1) circle (16pt);
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}$$
$$\begin{tikzpicture}[node distance=1.5cm]
\title{Arbitrary Announcement}
\node(A4) [white] {$p,q,r$};
\node(B2) [below of= A4, white] {$ p,q$};
\node(B1) [left of= B2, white] {$ p,r$};
\node(B3) [right of=B2, white] {$ q,r $};
\node(D1) [below of=B1, white] {$p$};
\node(D2) [right of=D1, white] {$q$};
\node(D3) [right of=D2, white] {$r$};
\node(E) [below of=D2, white]
{$\varnothing$};
\node(F) [below of=E, white]
{$k_{s}~~~~(w_s)$ };
\draw[green, line width=1pt, dashdotdotted] (B3) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D2) circle (12pt);
\draw[green, line width=1pt, dashdotdotted] (D3) circle (12pt);
\end{tikzpicture}
\begin{tikzpicture}[node distance=1.5cm]
\title{After announcement}
\node(A4) [white] {$w$};
\node(B1) [below left of= A4, white] {$w_{1}$};
\node(B4) [right of=B1, white] {$w_{5}$};
\node(B2) [ below of= B1, white] {$w_{2}$};
\node(B3) [right of=B2, white] {$w_{4}$};
\node(D1) [left of=B2, white] {$w_{3}$};
\node(D3) [right of=B3, white] {$w_{6}$};
\node(E) [below of=D2, white] {$w_{s}$};
\node(F) [below of=E, white] {$m'$};
\draw [->][cyan, line width=1pt, dotted] (B1) -- (D3);
\draw [->][cyan, line width=1pt, dotted] (A4) -- (B3);
\draw [->][cyan, line width=1pt, dotted] (A4) -- (B4);
\draw [->, cyan, line width=1pt, bend left =30, dotted] (A4) to (D3);
\draw [->][cyan, line width=1pt, dotted] (B3) -- (D3);
\draw [->][cyan, line width=1pt, dotted] (B4) -- (D3);
\draw [->][cyan, line width=1pt, dotted] (B2) -- (E);
\draw [->][cyan, line width=1pt, dotted] (D1) -- (E);
\draw [->][cyan, line width=1pt, dotted] (B1) -- (B3);
\draw [->][orange, line width=1pt] (B4) -- (D1);
\draw [->][orange, line width=1pt] (A4) -- (B1);
\draw [->][orange, line width=1pt] (A4) -- (B2);
\draw [->, orange, line width=1pt, bend right =30] (A4) to (D1);
\draw [->][orange, line width=1pt] (B1) -- (D1);
\draw [->][orange, line width=1pt] (B2) -- (D1);
\draw [->][orange, line width=1pt] (B3) -- (E);
\draw [->][orange, line width=1pt] (D3) -- (E);
\draw [->][orange, line width=1pt] (B4) -- (B2);
\end{tikzpicture}$$
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
- Variants Exploration for Interdisciplinary Studies
Future Directions
- Axiomatization of Sincere Arbitrary Social Announcement Logic
- Extension and Variants Investigation
- Finite Model Property and Decidability of Arbitrary Social Announcement Logic
- Relation to Public Announcement Logic
- Reduction of Group Announcement
Progress in Public Announcement Logic
The free message model technique can also be used in Public Announcement Logic (PAL).
Boolean Arbitrary Public Announcement Logic (BAPAL) is non-compact.
$\{\Diamond B_{a}(\neg B_{a} p \wedge \widehat{B_{b}}p)\wedge \Diamond\widehat{B_{a}} B_{b}\neg p\}\cup \bigcup_{i>0} \{B_{a}\neg p_{i}\}$
Arbitrary Public Announcement Logic (APAL) has a finitary axiomatization.
From $ \langle \chi\rangle\langle \widehat{B_{c}} p \wedge \neg p\rangle\phi\rightarrow\psi$, infer $ \langle \chi\rangle\Diamond\phi\rightarrow\psi$ where both $p$ and $c$ are free.
Complete Sequence
$\langle G! \rangle \phi \leftrightarrow \bigvee_{\vec{a!} \in S(G)} \langle \vec{a!} \rangle \phi$ where $S(G)$ represents all permutations of $G$.
To perfectly reduce a Group Announcement, the goal is to find the shortest sequence that includes all permutations.
$ \langle a! \rangle \langle b! \rangle \langle a! \rangle \phi \leftrightarrow \langle a! \rangle \langle b! \rangle \phi \vee \langle b! \rangle \langle a! \rangle \phi $
For $|G| = 4$, the shortest sequence length is $12$: $\langle a! \rangle \langle b! \rangle \langle c! \rangle \langle d! \rangle \langle a! \rangle \langle b! \rangle \langle c! \rangle \langle a! \rangle \langle d! \rangle \langle b! \rangle \langle a! \rangle \langle c! \rangle$
This remains an unsolved problem and is often associated with the well-known "Haruhi Problem".
| Size | Length |
|---|---|
| n = 3 | 7 |
| n = 4 | 12 |
| n = 5 | 19 |
| n = 6 | 28 |
| n = 7 | 39 |
Reference
- Plaza, Jan (1989). “Logics of public announcements”
- Gerbrandy, Jelle and Willem Groeneveld (1997). “Reasoning about Information Change”
- van Ditmarsch, Hans, Wiebe van Der Hoek, and Barteld Kooi (2007). "Dynamic epis- temic logic"
- Seligman, Jeremy, Fenrong Liu, and Patrick Girard (2013). “Facebook and the epistemic logic of friendship”
- Kuijer, Bouke (2015). "Unsoundness of R$_{\Box}$"
- Christoff, Zoé (2016). “Dynamic logics of networks: information flow and the spread of opin- ion”
- Xiong, Zuojun (2017). “On the logic of multicast messaging and balance in social networks”
- Xiong, Zuojun et al. (2017). “Towards a logic of tweeting”