Fix vertical placing of Xop

3 files changed, 11 insertions, 11 deletions

diff --git a/Assignment1/assignment1.tex b/Assignment1/assignment1.tex
index c1e6a0d..d96839e 100644
--- a/Assignment1/assignment1.tex
+++ b/Assignment1/assignment1.tex
@@ -90,11 +90,11 @@
 \DeclareMathOperator{\defeq}{\overset{\text{def}}{=}}
 \DeclareMathOperator{\Uop}{\mathbf{U}}
 \DeclareMathOperator{\Wop}{\mathbf{W}}
-\DeclareMathOperator{\Xop}{\bigcirc}
+\DeclareMathOperator{\Xop}{\raisebox{1pt}{$\bigcirc$}}
 \DeclareMathOperator{\Fop}{\lozenge}
 \DeclareMathOperator{\Gop}{\square}
 \DeclareMathOperator{\Sop}{\mathbf{S}}
-\DeclareMathOperator{\Pop}{\bigcirc^{--1}}
+\DeclareMathOperator{\Pop}{\raisebox{1pt}{$\bigcirc$}^{--1}}
 
 \newcommand{\wff}{\emph{wff}} % Well-formed formula
 \newcommand{\wffn}{\ifmmode\expandafter\text\fi{\wff\textsuperscript{0}}} % Pure present wff
diff --git a/Assignment1/semantics.tex b/Assignment1/semantics.tex
index dcc8e6b..1bca0b2 100644
--- a/Assignment1/semantics.tex
+++ b/Assignment1/semantics.tex
@@ -24,12 +24,12 @@ We use $\sigma \vDash \phi$ as a shorthand for $\sigma \vDash_0 \phi$.
 \end{definition}
 \camil
 Note that we must redefine the satisfaction relation for the LTL operators, because $\vDash$ is now a ternary relation.
-The difference with LTL formulae is well illustrated by the $\bigcirc$ operator.
-In the LTL case, $\sigma=A_0A_1A_2\dots\vDash\bigcirc\phi$ iff $\sigma[1\dots]=A_1A_2\dots\vDash\phi$.
-Thus, whether $\bigcirc\phi$ holds for $\sigma$ cannot depend on $A_0$.
-In the PLTL case, $\sigma\vDash_i\bigcirc\phi$ iff $\sigma\vDash_{i+1}\phi$.
-The information about $A_0$ is not lost, so whether $\bigcirc\phi$ holds for $\sigma$ may depend on $A_0$,
-	as is trivially the case with $\bigcirc\bigcirc^{-1}a$.
+The difference with LTL formulae is well illustrated by the $\Xop$ operator.
+In the LTL case, $\sigma=A_0A_1A_2\dots\vDash\Xop\phi$ iff $\sigma[1\dots]=A_1A_2\dots\vDash\phi$.
+Thus, whether $\Xop\phi$ holds for $\sigma$ cannot depend on $A_0$.
+In the PLTL case, $\sigma\vDash_i\Xop\phi$ iff $\sigma\vDash_{i+1}\phi$.
+The information about $A_0$ is not lost, so whether $\Xop\phi$ holds for $\sigma$ may depend on $A_0$,
+	as is trivially the case with $\Xop\Xop^{-1}a$.
 
 \begin{figure}
 	\centering
@@ -40,7 +40,7 @@ The information about $A_0$ is not lost, so whether $\bigcirc\phi$ holds for $\s
 			\sigma &\vDash_i & a                 &\text{iff} & a\in A_i\\
 			\sigma &\vDash_i & \phi_1\land\phi_2 &\text{iff} & \text{$\sigma\vDash_i\phi_1$ and $\sigma\vDash_i\phi_2$}\\
 			\sigma &\vDash_i & \lnot\phi         &\text{iff} & \sigma \nvDash_i \phi\\
-			\sigma &\vDash_i & \bigcirc\phi      &\text{iff} & \sigma \vDash_{i+1} \phi\\
+			\sigma &\vDash_i & \Xop\phi          &\text{iff} & \sigma \vDash_{i+1} \phi\\
 			\sigma &\vDash_i & \phi_1\Uop\phi_2  &\text{iff} & \exists_{j \ge i}.\text{$\sigma \vDash_j \phi_2$ and $\sigma \vDash_k \phi_1$ for all $i \le k < j$}\\
 			\sigma &\vDash_i & \phi_1\Sop\phi_2  &\text{iff} & \exists_{0 \le j \le i}.\text{$\sigma \vDash_j \phi_2$ and $\sigma \vDash_k \phi_1$ for all $j < k \le i$}\\
 			\sigma &\vDash_i & \Pop\phi          &\text{iff} & \text{$i \geq 1$ and $\sigma \vDash_{i-1} \phi$}
diff --git a/Assignment1/syntax.tex b/Assignment1/syntax.tex
index 5d2830a..6d86281 100644
--- a/Assignment1/syntax.tex
+++ b/Assignment1/syntax.tex
@@ -22,7 +22,7 @@ The $\Sop$-modality is comparable to $\Uop$: $\phi_1\Sop\phi_2$ holds if $\phi_2
 			\enspace\middle|\enspace a
 			\enspace\middle|\enspace \phi_1 \land \phi_2
 			\enspace\middle|\enspace \lnot \phi
-			\enspace\middle|\enspace \bigcirc \phi
+			\enspace\middle|\enspace \Xop \phi
 			\enspace\middle|\enspace \phi_1 \Uop \phi_2
 			\enspace\middle|\enspace \Pop \phi
 			\enspace\middle|\enspace \phi_1 \Sop \phi_2
@@ -208,6 +208,6 @@ Before turning to the formal semantics in the next subsection, we provide some e
 	These include
 		$\mathbf X^{-1}, \mathbf G^{-1}, \mathbf F^{-1}$~\citep{Markey2003},
 		but also \raisebox{-1pt}{\tikz\draw[black,fill=black](0,0)circle(.4em);}$,\blacksquare,\blacklozenge$~\citep{Gabbay1989}
-		and $\stackinset{c}{}{c}{}{$\cdot$}{$\bigcirc$},\boxdot,\stackinset{c}{}{c}{}{$\cdot$}{$\lozenge$}$~\citep{Havelund2002}.
+		and $\stackinset{c}{}{c}{}{$\cdot$}{$\Xop$},\boxdot,\stackinset{c}{}{c}{}{$\cdot$}{$\lozenge$}$~\citep{Havelund2002}.
 \end{remark}
 \cbend