@@ -257,7 +258,7 @@ Classic modal logic is linked to game semantics via model checking games~: compu
$q \in Q_S$& S &$q' \in Q_D$ such that $(q,q')\in E$\\
\midrule
$q \in Q_D$& D &$q' \in Q_S$ such that $(q,q')\in E$\\
\bottomrule\\
\bottomrule
\end{tabular}
\caption{Possible moves in a safety game}\label{tab:safety-game}
\end{table}
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@@ -295,7 +296,7 @@ Safety games are a special case of another type of games, parity games.
$q \in Q_S$& S &$q' \in Q$ such that $(q,q')\in E$&$\pr(q)$\\
\midrule
$q \in Q_D$& D &$q' \in Q$ such that $(q,q')\in E$&$\pr(q)$\\
\bottomrule\\
\bottomrule
\end{tabular}
\caption{Priorities and possible moves in a parity game}\label{tab:parity-game}
\end{table}
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@@ -1035,7 +1036,7 @@ We then get a smaller game, still characterizing codensity bisimilarities, that
$P \in\mathcal{G}$& S &\makecell{$\sigma\in\Sigma$, $k : X \rightarrow\Omega_\sigma$ such that \\$P \not\sqsubseteq(\tau_\sigma\circ Fk \circ c)^*\mathbf{\Omega}_\sigma$}\\
\midrule
$k : X \rightarrow\Omega_\sigma$& D &$P \in\mathcal{G}$ such that $P \not\sqsubseteq k^*\mathbf{\Omega}_\sigma$\\
@@ -1263,10 +1264,10 @@ Just as codensity bisimilarities (greatest fixed points of the univariate predic
\begin{tabular}{c c c c}
Position & Player & Possible moves & Priority \\
\toprule
\makecell{$i \in[1,m]$\\$u \in\mathbb{C}_X$}& S &\makecell{$i$, $u$, $\sigma\in\Sigma$, \\$f : X \rightarrow\Omega_{i,\sigma}^m$ such that \\$u \not\sqsubseteq(\tau_{i,\sigma}\circ F_i f \circ c_i)^*\mathbf{\Omega}_{i,\sigma}$}&$2i +\delta_{\eta_i =\mu}$\\
\makecell{$i \in[1,m]$,$u \in\mathbb{C}_X$}& S &\makecell{$i$, $u$, $\sigma\in\Sigma$, $f : X \rightarrow\Omega_{i,\sigma}^m$\\such that $u \not\sqsubseteq(\tau_{i,\sigma}\circ F_i f \circ c_i)^*\mathbf{\Omega}_{i,\sigma}$}&$2i +\delta_{\eta_i =\mu}$\\
\midrule
\makecell{$i \in[1,m]$\\$u \in\mathbb{C}_X$\\$\sigma\in\Sigma$\\$f : X \rightarrow\Omega_{i,\sigma}^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathbb{C}_X$\\ such that $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$}&$0$\\
\bottomrule\\
\makecell{$i \in[1,m]$,$u \in\mathbb{C}_X$\\$\sigma\in\Sigma$,$f : X \rightarrow\Omega_{i,\sigma}^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathbb{C}_X$\\ such that $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$}&$0$\\
@@ -1301,12 +1302,12 @@ Note that the actual choice of $\gamma$ is only a matter of reducing the state s
\begin{tabular}{c c c}
Position & Player & Possible moves \\
\toprule
\makecell{$i \in[1,m]$\\$u \in\mathbb{C}_X$\\$(\seq{\alpha}{k})\in\gamma^k$}& S &\makecell{$i$, $u$, $(\seq{\alpha}{k})$, $\sigma\in\Sigma$, \\$f : X \rightarrow\Omega_{i,\sigma}^m$ such that \\$u \not\sqsubseteq(\tau_{i,\sigma}\circ F_i f \circ c_i)^*\mathbf{\Omega}_{i,\sigma}$}\\
\makecell{$i \in[1,m]$,$u \in\mathbb{C}_X$\\$(\seq{\alpha}{k})\in\gamma^k$}& S &\makecell{$i$, $u$, $(\seq{\alpha}{k})$, $\sigma\in\Sigma$, $f : X \rightarrow\Omega_{i,\sigma}^m$\\ such that $u \not\sqsubseteq(\tau_{i,\sigma}\circ F_i f \circ c_i)^*\mathbf{\Omega}_{i,\sigma}$}\\
\midrule
\makecell{$i \in[1,m]\mid\eta_i =\mu$\\$u \in\mathbb{C}_X$\\$(\seq{\alpha}{k})\in\gamma^k$\\$\sigma\in\Sigma$\\$f : X \rightarrow\Omega_{i,\sigma}^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathbb{C}_X$, \\$(\seq{\delta}{k})\in\gamma^k$ such that \\$(\seq{\delta}{k})\preceq_j (\seq{\beta}{k})$, \\$(\seq{\beta}{k})\prec_i (\seq{\alpha}{k})$\\ for some $(\seq{\beta}{k})$\\ and $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$}\\
\makecell{$i \in[1,m]\mid\eta_i =\mu$\\$u \in\mathbb{C}_X$\\$(\seq{\alpha}{k})\in\gamma^k$\\$\sigma\in\Sigma$,$f : X \rightarrow\Omega_{i,\sigma}^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathbb{C}_X$, $(\seq{\delta}{k})\in\gamma^k$\\such that $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$ and, for some $(\seq{\beta}{k})$, \\$(\seq{\delta}{k})\preceq_j (\seq{\beta}{k})\prec_i (\seq{\alpha}{k})$}\\
\midrule
\makecell{$i \in[1,m]\mid\eta_i =\nu$\\$u \in\mathbb{C}_X$\\$(\seq{\alpha}{k})\in\gamma^k$\\$\sigma\in\Sigma$\\$f : X \rightarrow\Omega_{i,\sigma}^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathbb{C}_X$, \\$(\seq{\delta}{k})\in\gamma^k$ such that \\$(\seq{\delta}{k})\preceq_j (\seq{\beta}{k})$, \\$(\seq{\beta}{k})\preceq_i(\seq{\alpha}{k})$\\ for some $(\seq{\beta}{k})$\\ and $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$}\\
\bottomrule\\
\makecell{$i \in[1,m]\mid\eta_i =\nu$\\$u \in\mathbb{C}_X$\\$(\seq{\alpha}{k})\in\gamma^k$\\$\sigma\in\Sigma$,$f : X \rightarrow\Omega_{i,\sigma}^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathbb{C}_X$, $(\seq{\delta}{k})\in\gamma^k$\\such that $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$ and, for some $(\seq{\beta}{k})$, \\$(\seq{\delta}{k})\preceq_j(\seq{\beta}{k})\preceq_i (\seq{\alpha}{k})$}\\
\bottomrule
\end{tabular}
\caption{Codensity parity game with ordinals}\label{tab:codensity-parity-game-ordinals}
\end{table}
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@@ -1516,10 +1517,10 @@ Once again, the underlying graphs of codensity parity games are quite large, but
\begin{tabular}{c c c c}
Position & Player & Possible moves & Priority \\
\toprule
\makecell{$i \in[1,m]$\\$u \in\mathcal{G}$}& S &\makecell{$i$, $u$, $\sigma\in\Sigma$, \\$f : X \rightarrow\Omega^m$ such that \\$u \not\sqsubseteq(\tau_{i,\sigma}\circ F_i f \circ c_i)^*\mathbf{\Omega}_{i,\sigma}$}&$2i +\delta_{\eta_i =\mu}$\\
\makecell{$i \in[1,m]$,$u \in\mathcal{G}$}& S &\makecell{$i$, $u$, $\sigma\in\Sigma$, $f : X \rightarrow\Omega^m$\\such that $u \not\sqsubseteq(\tau_{i,\sigma}\circ F_i f \circ c_i)^*\mathbf{\Omega}_{i,\sigma}$}&$2i +\delta_{\eta_i =\mu}$\\
\midrule
\makecell{$i \in[1,m]$\\$u \in\mathcal{G}$\\$\sigma\in\Sigma$\\$f : X \rightarrow\Omega^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathcal{G}$\\ such that $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$}&$0$\\
\bottomrule\\
\makecell{$i \in[1,m]$,$u \in\mathcal{G}$\\$\sigma\in\Sigma$,$f : X \rightarrow\Omega^m$}& D &\makecell{$j \in[1,m]$, $v \in\mathcal{G}$\\ such that $v \not\sqsubseteq f_j^*\mathbf{\Omega}_{i,\sigma}$}&$0$\\
