\def\statementEncodingTable{
\def\arraystretch{1.65}
\setlength\tabcolsep{8pt}
\begin{table}
	\caption{Translation Rules}
	\label{tab:encode}	
	\begin{center}
		\makebox[\linewidth]{
			\begin{tabular}{|ccc|}
				\hline
				KB statement&\vline&Vectorization\\
				\hline
				CX $\sqsubseteq$ CY&$\rightarrow$&$[\ 0.0,\ \frac{X}{c}, \frac{Y}{c},\ 0.0\ ]$\\
				\hline
				CX $\sqcap$ CY $\sqsubseteq$ CZ&$\rightarrow$&$[\ \frac{X}{c},\ \frac{Y}{c},\ \frac{Z}{c},\ 0.0\ ]$\\
				\hline
				CX $\sqsubseteq$ $\exists$RY.CZ&$\rightarrow$&$[0.0,\ \frac{X}{c},\ \frac{-Y}{r},\ \frac{Z}{c}]$\\
				\hline
				$\exists$RX.CY $\sqsubseteq$ CZ&$\rightarrow$&$[\frac{-X}{r},\ \frac{Y}{c},\ \frac{Z}{c},\ 0.0]$\\
				\hline
				RX $\sqsubseteq$ RY&$\rightarrow$&$[0.0, \frac{-X}{r}, \frac{-Y}{r}, 0.0]$\\
				\hline
				RX $\circ$ RY $\sqsubseteq$ RZ&$\rightarrow$&$[\frac{-X}{r}, \frac{-Y}{r}, \frac{-Z}{r}, 0.0]$\\
				\hline 
			\end{tabular}		
	}
	
	\medskip
	c = Number of Possible Concept Names \\ r = Number of Possible Role Names	
	\end{center}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{6pt}
}

\def\elpSyntax{
Expressions in \ELp\ use the following grammar:\nls
\def\arraystretch{1.35}
\begin{tabular}{lcccccc}\label{tab:syn}
\centering
$\textbf{R}$&$::=$&$N_R$\\
$\textbf{C} $&$::=$&$ N_C $&$|$&$ \textbf{C} \sqcap \textbf{C} $&$|$&$ \exists \textbf{R}.\textbf{C}$
\end{tabular}\def\arraystretch{1}
}

\def\elpSignature{
The signature $\Sigma$ for \ELp\ is defined as:\nls
$\Sigma = \left<N_I,N_C,N_R\right>\ $ with $N_I, N_C, N_R$ pairwise disjoint.\nls\noindent
$N_I$ is a set of individual names.\\
$N_C$ is a set of Concept names that includes $\top$.\\
$N_R$ is a set of Role names.\nls
}

\def\elpSemantics{
\def\arraystretch{1.15}
\setlength\tabcolsep{8pt}
\begin{table*}
	\caption{\ELp\ Semantics}
	\label{tab:sem}
	\begin{center}
		\makebox[\linewidth]{
			\begin{tabular}{ |c|c|c| } 
				%HEADER
				\hline
				Description & Expression & Semantics\\ 
				%EL+
				\hline 
				Individual & $ a $ & $ a \in \Delta^{\mathcal{I}} $\\    
				Top & $ \top $ & $ \Delta^{\mathcal{I}} $\\ 
				%Bottom & $ \bot $ & $ \emptyset $ \\
				Concept & $ C $ & $ C^{\mathcal{I}} \subseteq \Delta^{\mathcal{I}} $\\
				Role & $ R $ & $ R^{\mathcal{I}} \subseteq \Delta^{\mathcal{I}} \times \Delta^{\mathcal{I}} $\\
				Conjunction & $ C \sqcap D $ & $ C^{\mathcal{I}} \cap D^{\mathcal{I}} $\\
				Existential Restriction & $ \exists R.C $ & \{ $ a \ | \ $there is $b \in \Delta^{\mathcal{I}}$ such that $ \left( a,b \right) \in R^{\mathcal{I}}$ and $  b \in C^{\mathcal{I}} $ \}\\
				Concept Subsumption& $C \sqsubseteq D$& $ C^{\mathcal{I}} \subseteq D^{\mathcal{I}} $\\				
				Role Subsumption& $R \sqsubseteq S$& $ R^{\mathcal{I}} \subseteq S^{\mathcal{I}} $\\
				Role Chain& $R_1 \circ \cdots \circ R_n \sqsubseteq R$&$ R_1^{\mathcal{I}} \circ \cdots \circ R_n^{\mathcal{I}} \subseteq R^{\mathcal{I}} $\\
				\hline
		\end{tabular}}
		\small{with $\circ$ signifying standard binary composition}\\
	\end{center}	
\end{table*}
\def\arraystretch{1}
\setlength\tabcolsep{6pt}
}

\def\completionRules{
\setlength\tabcolsep{1pt}\def\arraystretch{1.3}
\begin{table}
\caption{\ELp\ Completion Rules}
\label{tab:comprul}	\centering
	\begin{tabular}{|lccccccl|}\hline
		(1)&$A \sqsubseteq C $& &$C \sqsubseteq D$& &&$\models$&$ A \sqsubseteq D$\\
		(2)&$A \sqsubseteq C_1$& &$A \sqsubseteq C_2$& &$C_1 \sqcap C_2 \sqsubseteq D$&$\models$&$A \sqsubseteq D$\\
		(3)&$A \sqsubseteq C$& &$C \sqsubseteq \exists R.D$& &&$\models$&$A \sqsubseteq \exists R.D$\\
		(4)&$A \sqsubseteq \exists R.B$& &$B \sqsubseteq C$& &$\exists R.C \sqsubseteq D$&$\models$&$A \sqsubseteq D$\\
		(5)&$A \sqsubseteq \exists S.D$& &$S \sqsubseteq R$& &&$\models$&$A \sqsubseteq \exists R.D$\\
		(6)&$A \sqsubseteq \exists R_1.C$& &$C \sqsubseteq \exists R_2.D$& &$R_1 \circ R_2 \sqsubseteq R$&$\models$&$A \sqsubseteq \exists R.D$\\\hline
	\end{tabular}	
\end{table}
\def\arraystretch{1}\setlength\tabcolsep{6pt}
}

\def\oneSequence{
\begin{figure}
\begin{mdframed}\small
\begin{multicols}{2}
\noindent seed$_0$ = C1 $\sqsubseteq$ C2\\C2 $\sqsubseteq$ C3\\C3 $\sqcap$ C4 $\sqsubseteq$ C5\\C1 $\sqsubseteq$ $\exists$R1.C1\\C1 $\sqsubseteq$ $\exists$R2.C3\\C2 $\sqsubseteq$ $\exists$R2.C3\\C2 $\sqsubseteq$ $\exists$R1.C3\\C1 $\sqsubseteq$ $\exists$R3.C4\\C2 $\sqsubseteq$ $\exists$R1.C3\\$\exists$R1.C2 $\sqsubseteq$ C4\\R1 $\sqsubseteq$ R2\\R1 $\circ$ R3 $\sqsubseteq$ R4
\end{multicols}
Which entails
\begin{multicols}{2}
\noindent Step 1:\\C1 $\sqsubseteq$ C3\\C1 $\sqsubseteq$ C4\\C1 $\sqsubseteq$ $\exists$R1.C3\\C1 $\sqsubseteq$ $\exists$R2.C1\\C1 $\sqsubseteq$ $\exists$R4.C4
\columnbreak

\noindent Step 2:\\seed$_1$ = C1 $\sqsubseteq$ C5
\end{multicols}
\end{mdframed}
\caption{First Iteration of Sequence}
\label{fig:seq}	
\end{figure}
}

\def\piecewisePic{
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{figures/visualPiecewise.png}
    \caption{Piecewise Architecture}
    \label{fig:piece}
\end{figure}
}

\def\flatPic{
\begin{figure}
    \centering
    \includegraphics[width=0.35\textwidth]{figures/visualFlat.png}
    \caption{Flat Architecture}
    \label{fig:flat}
\end{figure}
}

\def\deepPic{
\begin{figure}
    \centering
    \includegraphics[width=0.35\textwidth]{figures/visualDeep.png}
    \caption{Deep Architecture}
    \label{fig:deep}
\end{figure}
}

\def\allArcitecturePics{
\begin{figure}%
    \centering
    \subfloat[Piecewise Architecture]{{\includegraphics[width=.3\textwidth]{figures/visualPiecewise.png}}}%
    \qquad
    \subfloat[Deep Architecture]{{\includegraphics[width=0.3\textwidth]{figures/visualDeep.png}}}%
    \qquad
    \subfloat[Flat Architecture]{{\includegraphics[width=0.3\textwidth]{figures/visualFlat.png}}}%
    \caption{Architectures}%
    \label{fig:bothtrain}%
\end{figure}
}

\def\syntrainingPic{
\begin{figure}
    \centering
    \includegraphics[width=0.47\textwidth]{figures/synthtrain.png}
    \caption{Synthetic Training}
    \label{fig:syntrain}
\end{figure}
}

\def\snotrainingPic{
\begin{figure}
    \centering
    \includegraphics[width=0.47\textwidth]{figures/snotrain.png}
    \caption{SNOMED Training}
    \label{fig:snotrain}
\end{figure}
}

\def\bothTrainingPics{
\begin{figure}%
    \centering
    \subfloat[Synthetic Training]{{\includegraphics[width=0.47\textwidth]{figures/synthtrain.png}}}%
    \qquad
    \subfloat[SNOMED Training]{{\includegraphics[width=0.47\textwidth]{figures/snotrain.png}}}%
    \caption{Dataset Training Comparison}%
    \label{fig:bothtrain}%
\end{figure}
}

\def\accTable{
\def\arraystretch{1.05}
\setlength\tabcolsep{.2em}
\begin{table*}[t]
\small
\caption{Average Precision Recall and F1-score For each Distance Evaluation}
\label{tab:fscore}
    \makebox[\linewidth]{
	\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}\hline
	&\multicolumn{3}{c}{Atomic Levenshtein Distance}\vline&\multicolumn{3}{c}{Character Levenshtein Distance}\vline&\multicolumn{3}{c|}{Predicate Distance}\\\cline{2-10}
	&Precision&Recall&F1-score&Precision&Recall&F1-score&Precision&Recall&F1-score\\\hline
	&\multicolumn{9}{c|}{Synthetic Data}\\\hline
	Piecewise Prediction&0.138663&0.142208&0.140412&0.138663&0.142208&0.140412&0.138646&0.141923&0.140264\\
	Deep Prediction&\textbf{0.154398}&\textbf{0.156056}&\textbf{0.155222}&\textbf{0.154398}&\textbf{0.156056}&\textbf{0.155222}&\textbf{0.154258}&\textbf{0.155736}&\textbf{0.154993}\\
	Flat Prediction&0.140410&0.142976&0.141681&0.140410&0.142976&0.141681&0.140375&0.142687&0.141521\\
	Random Prediction&0.010951&0.0200518&0.014166&0.006833&0.012401&0.008811&0.004352&0.007908&0.007908\\
	\hline
	&\multicolumn{9}{c|}{SNOMED Data}\\
	\hline	
	Piecewise Prediction&0.010530&0.013554&0.011845&0.010530&0.013554&0.011845&0.010521&0.013554&0.011839\\
	Deep Prediction&\textbf{0.015983}&0.0172811&\textbf{0.016595}&\textbf{0.015983}&0.017281&\textbf{0.016595}&\textbf{0.015614}&0.017281&\textbf{0.016396}\\
	Flat Prediction&0.014414&\textbf{0.018300}&0.016112&0.0144140&\textbf{0.018300}&0.016112&0.013495&\textbf{0.018300}&0.015525\\
	Random Prediction&0.002807&0.006803&0.003975&0.001433&0.003444&0.002023&0.001769&0.004281&0.002504\\
	\hline
	\end{tabular}}
\end{table*}
\def\arraystretch{1}
\setlength\tabcolsep{6pt}

}

\def\distTable{
\def\arraystretch{1.05}
\setlength\tabcolsep{.2em}
\begin{table*}[t]
\small
\caption{Average Statement Edit Distances with Reasoner}
\label{tab:dists}\makebox[\linewidth]{
	\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}\hline
	&\multicolumn{3}{c}{Atomic Levenshtein Distance}\vline&\multicolumn{3}{c}{Character Levenshtein Distance}\vline&\multicolumn{3}{c|}{Predicate Distance}\\\cline{2-10}
	&From&To&Average&From&To&Average&From&To&Average\\\hline
	&\multicolumn{9}{c|}{Synthetic Data}\\\hline
	Piecewise Prediction&1.336599&1.687640&1.512119&1.533115&1.812006&1.672560&2.633427&4.587382&3.610404\\
	Deep Prediction&\textbf{1.256940}&\textbf{1.507150}&\textbf{1.382045}&\textbf{1.454787}&\textbf{1.559751}&\textbf{1.507269}&\textbf{2.504496}&\textbf{3.552074}&\textbf{3.028285}\\
	Flat Prediction&1.344946&1.584674&1.464810&1.586281&1.660409&1.623345&2.517655&3.739770&3.128713\\
	Random Prediction&1.598016&1.906369&1.752192&1.970604&1.289533&1.630068&5.467918&10.57324&8.020583\\\hline
	&\multicolumn{9}{c|}{SNOMED Data}\\\hline
	Piecewise Prediction&1.704931&\textbf{2.686562}&\textbf{2.195746}&2.016249&2.862737&2.439493&6.556592&\textbf{5.857769}&6.207181\\
	Deep Prediction&1.759633&3.052080&2.405857&2.027190&3.328850&2.678020&\textbf{4.577427}&6.179389&\textbf{5.378408}\\
	Flat Prediction&\textbf{1.691738}&2.769542&2.230640&\textbf{1.948757}&2.991328&2.470042&5.548226&6.665659&6.106942\\
	Random Prediction&1.814656&3.599629&2.707143&2.094682&\textbf{1.621700}&\textbf{1.858191}&5.169093&12.392325&8.780709\\
	\hline
	\end{tabular}}
\end{table*}
\def\arraystretch{1}
\setlength\tabcolsep{6pt}
}

\def\suppExample{
\def\arraystretch{1.25}
\begin{table*}[t]
\caption{Support Generation}
\label{tab:support}	\centering\makebox[\linewidth]{
	\begin{tabular}{|l|r|c|l|}\hline
	& New Fact&Rule&Support\\\hline
	Step 1&{\color{red}C1 $\sqsubseteq$ C3}&(1)&{\color{red}C1 $\sqsubseteq$ C2,C2 $\sqsubseteq$ C3}\\
    &{\color{blue}C1 $\sqsubseteq$ C4}&(4)&{\color{blue}C1 $\sqsubseteq$ C2,C1 $\sqsubseteq \exists$R1.C1,$\exists$R1.C2 $\sqsubseteq$ C4}\\
    &C1 $\sqsubseteq \exists$R1.C3&(3)&C1 $\sqsubseteq$ C2,C2 $\sqsubseteq \exists$R1.C3\\
    &C1 $\sqsubseteq \exists$R2.C1&(5)&C1 $\sqsubseteq \exists$R1.C1,R1 $\sqsubseteq$ R2\\
    &C1 $\sqsubseteq \exists$R4.C4&(6)&C1 $\sqsubseteq \exists$R1.C1,R1 $\circ$ R3 $\sqsubseteq$ R4,C1 $\sqsubseteq \exists$R3.C4\\\hline
    Step 2&C1 $\sqsubseteq$ C5&(2)&C3 $\sqcap$ C4 $\sqsubseteq$ C5,{\color{red}C1 $\sqsubseteq$ C2,C2 $\sqsubseteq$ C3},{\color{blue}C1 $\sqsubseteq$ C2,C1 $\sqsubseteq \exists$R1.C1,$\exists$R1.C2 $\sqsubseteq$ C4}\\\hline
	\end{tabular}}	
\end{table*}
\def\arraystretch{1}
}

\def\exampleSynPred{
\def\arraystretch{1.25}
\begin{table}
\caption{Example Synthetic Output}
\label{tab:synex}	\centering\makebox[\linewidth]{
	\begin{tabular}{|l|l|l|}\hline
	&Correct Answer&Predicted Answer\\\hline
	Step 0&C9 $\sqsubseteq$ C11&C8 $\sqsubseteq$ C9\\
	&C2 $\sqsubseteq$ C10&C1 $\sqsubseteq$ C9\\
	&C9 $\sqsubseteq$ C12&C8 $\sqsubseteq$ C9\\
	&C7 $\sqsubseteq$ C6&C8 $\sqsubseteq$ $\exists$R4.C9\\
	&C9 $\sqsubseteq$ $\exists$R4.C11&C1 $\sqsubseteq$ $\exists$R5.C9\\
	&C2 $\sqsubseteq$ $\exists$R4.C9&C8 $\sqsubseteq$ $\exists$R4.C9\\
	&C9 $\sqsubseteq$ $\exists$R5.C9&C9 $\sqsubseteq$ $\exists$R5.C9\\
	&C2 $\sqsubseteq$ $\exists$R5.C11&\\
	&C9 $\sqsubseteq$ $\exists$R7.C12&\\
	&C2 $\sqsubseteq$ $\exists$R6.C12&\\\hline
	Step 1&C9 $\sqsubseteq$ C13&C8 $\sqsubseteq$ C12\\
	&C2 $\sqsubseteq$ C11&C2 $\sqsubseteq$ C10\\
	&C2 $\sqsubseteq$ C12&C1 $\sqsubseteq$ C11\\
	&C2 $\sqsubseteq$ $\exists$R4.C11&C1 $\sqsubseteq$ $\exists$R3.C12\\
	&C2 $\sqsubseteq$ $\exists$R5.C9&C1 $\sqsubseteq$ $\exists$R4.C8\\
	&C2 $\sqsubseteq$ $\exists$R7.C12&\\\hline		
	Step 2&C2 $\sqsubseteq$ C13&C1 $\sqsubseteq$ C12\\	
	\hline
	\end{tabular}}	
\end{table}
\def\arraystretch{1}
}

\def\exampleSnoPred{
\begin{table}
\caption{Example SNOMED Outputs}
\label{tab:snoex}	\centering\makebox[\linewidth]{
	\begin{tabular}{|l|l|}\hline
Correct Answer&C6 $\sqsubseteq$ $\exists$R4.C1\\\hline
Meaning&if something is a zone of superficial fascia, then\\
&there is a subcutaneous tissue that it is PartOf\\\hline\hline
Prediction&C6 $\sqsubseteq$ $\exists$R4.C3\\\hline
Meaning&if something is a zone of superficial fascia,\\
&then there is a subcutaneous tissue of palmar\\
&area of little finger that it is PartOf\\\hline\hline
Correct Answer&C8 $\sqsubseteq$ $\exists$R3.C2\\\hline
Meaning&if something is a infrapubic region of pelvis,\\
&then there is a perineum that it is PartOf\\\hline\hline
Prediction&C9 $\sqsubseteq$ $\exists$R3.C2\\\hline
Meaning&if something is a zone of integument,\\
&then there is a perineum that it is PartOf\\
\hline
\end{tabular}}	
\end{table}
}

\def\goals{
\begin{enumerate}
    \item Can a neural network emulate reasoner behavior?
    \item Can we still achieve success without embeddings?
    \item Can learning happen with arbitrary numerical names?
    \item Will intermediate answers help our evaluation?
    \item Is F1-score sufficient to evaluate an integrated logic and neural network system? If not, what would be?
\end{enumerate}
}

\def\piecSyn{
\begin{figure*}%
    \centering
    \subfloat[Character Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/levSynP.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1levSynP.png}}}%
    \qquad
    \subfloat[Atomic Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/ISynP.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1ISynP.png}}}%
    \qquad
    \subfloat[Predicate Distances]{{\includegraphics[width=0.47\textwidth]{figures/pSynP.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1pSynP.png}}}%
    \caption{Synthetic Data Piecewise Architecture}%
    \label{fig:synp}%
\end{figure*}
}

\def\deepSyn{
\begin{figure*}%
    \centering
    \subfloat[Character Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/levSynD.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1levSynD.png}}}%
    \qquad
    \subfloat[Atomic Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/ISynD.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1ISynD.png}}}%
    \qquad
    \subfloat[Predicate Distances]{{\includegraphics[width=0.47\textwidth]{figures/pSynD.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1pSynD.png}}}%
    \caption{Synthetic Data Deep Architecture}%
    \label{fig:synd}%
\end{figure*}
}

\def\flatSyn{
\begin{figure*}%
    \centering
    \subfloat[Character Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/levSynF.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1levSynF.png}}}%
    \qquad
    \subfloat[Atomic Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/ISynF.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1ISynF.png}}}%
    \qquad
    \subfloat[Predicate Distances]{{\includegraphics[width=0.47\textwidth]{figures/pSynF.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1pSynF.png}}}%
    \caption{Synthetic Data Flat Architecture}%
    \label{fig:synf}%
\end{figure*}
}

\def\piecSno{
\begin{figure*}%
    \centering
    \subfloat[Character Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/levSnoP.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1levSnoP.png}}}%
    \qquad
    \subfloat[Atomic Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/ISnoP.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1ISnoP.png}}}%
    \qquad
    \subfloat[Predicate Distances]{{\includegraphics[width=0.47\textwidth]{figures/pSnoP.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1pSnoP.png}}}%
    \caption{SNOMED Data Piecewise Architecture}%
    \label{fig:snop}%
\end{figure*}
}

\def\deepSno{
\begin{figure*}%
    \centering
    \subfloat[Character Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/levSnoD.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1levSnoD.png}}}%
    \qquad
    \subfloat[Atomic Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/ISnoD.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1ISnoD.png}}}%
    \qquad
    \subfloat[Predicate Distances]{{\includegraphics[width=0.47\textwidth]{figures/pSnoD.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1pSnoD.png}}}%
    \caption{SNOMED Data Deep Architecture}%
    \label{fig:snod}%
\end{figure*}
}

\def\flatSno{
\begin{figure*}%
    \centering
    \subfloat[Character Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/levSnoF.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1levSnoF.png}}}%
    \qquad
    \subfloat[Atomic Levenshtein Distances]{{\includegraphics[width=0.47\textwidth]{figures/ISnoF.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1ISnoF.png}}}%
    \qquad
    \subfloat[Predicate Distances]{{\includegraphics[width=0.47\textwidth]{figures/pSnoF.png}}}%
    \qquad
    \subfloat[Precision, Recall, F1-score]{{\includegraphics[width=0.47\textwidth]{figures/f1pSnoF.png}}}%
    \caption{SNOMED Data Flat Architecture}%
    \label{fig:snof}%
\end{figure*}
}

\def\predAcc{
\begin{figure*}%
    \centering
    \subfloat[Synthetic Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1pSynP.png}}}%
    \subfloat[Synthetic Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1pSynD.png}}}%
    \qquad
    \subfloat[Synthetic Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1pSynF.png}}}%
    \subfloat[SNOMED Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1pSnoP.png}}}%
    \qquad
    \subfloat[SNOMED Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1pSnoD.png}}}%
    \subfloat[SNOMED Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1pSnoF.png}}}%
    \caption{Predicate Distance Precision, Recall, and F1-score}%
    \label{fig:fpred}%
\end{figure*}
}

\def\levAcc{
\begin{figure*}%
    \centering
    \subfloat[Synthetic Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1levSynP.png}}}%
    \subfloat[Synthetic Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1levSynD.png}}}%
    \qquad
    \subfloat[Synthetic Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1levSynF.png}}}%
    \subfloat[SNOMED Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1levSnoP.png}}}%
    \qquad
    \subfloat[SNOMED Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1levSnoD.png}}}%
    \subfloat[SNOMED Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1levSnoF.png}}}%
    \caption{Character Levenshtein Distance Precision, Recall, and F1-score}%
    \label{fig:flev}%
\end{figure*}
}

\def\atomAcc{
\begin{figure*}%
    \centering
    \subfloat[Synthetic Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1ISynP.png}}}%
    \subfloat[Synthetic Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1ISynD.png}}}%
    \qquad
    \subfloat[Synthetic Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1ISynF.png}}}%
    \subfloat[SNOMED Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1ISnoP.png}}}%
    \qquad
    \subfloat[SNOMED Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1ISnoD.png}}}%
    \subfloat[SNOMED Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/f1ISnoF.png}}}%
    \caption{Atomic Levenshtein Distance Precision, Recall, and F1-score}%
    \label{fig:fatom}%
\end{figure*}
}

\def\predDist{
\begin{figure*}%
    \centering
    \subfloat[Synthetic Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/pSynP.png}}}%
    \subfloat[Synthetic Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/pSynD.png}}}%
    \qquad
    \subfloat[Synthetic Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/pSynF.png}}}%
    \subfloat[SNOMED Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/pSnoP.png}}}%
    \qquad
    \subfloat[SNOMED Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/pSnoD.png}}}%
    \subfloat[SNOMED Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/pSnoF.png}}}%
    \caption{Predicate Distance}%
    \label{fig:distpred}%
\end{figure*}
}

\def\levDist{
\begin{figure*}%
    \centering
    \subfloat[Synthetic Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/levSynP.png}}}%
    \subfloat[Synthetic Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/levSynD.png}}}%
    \qquad
    \subfloat[Synthetic Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/levSynF.png}}}%    
    \subfloat[SNOMED Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/levSnoP.png}}}
    \qquad    
    \subfloat[SNOMED Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/levSnoD.png}}}%
    \subfloat[SNOMED Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/levSnoF.png}}}%
    \caption{Character Levenshtein Distance}%
    \label{fig:distlev}%
\end{figure*}
}

\def\atomDist{
\begin{figure*}%
    \centering
    \subfloat[Synthetic Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/ISynP.png}}}%
    \subfloat[Synthetic Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/ISynD.png}}}%
    \qquad
    \subfloat[Synthetic Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/ISynF.png}}}%
    \subfloat[SNOMED Data Piecewise Architecture]{{\includegraphics[width=0.49\textwidth]{figures/ISnoP.png}}}%
    \qquad
    \subfloat[SNOMED Data Deep Architecture]{{\includegraphics[width=0.49\textwidth]{figures/ISnoD.png}}}%
    \subfloat[SNOMED Data Flat Architecture]{{\includegraphics[width=0.49\textwidth]{figures/ISnoF.png}}}%
    \caption{Atomic Levenshtein Distance}%
    \label{fig:distatom}%
\end{figure*}
}

\def\avgAll{
\begin{table}[ht]
\begin{center}
\def\arraystretch{1.25}
\setlength\tabcolsep{8pt}
\begin{tabular}{lrrr}
 & \multicolumn{1}{l}{Read File} & \multicolumn{1}{l}{Write File} & \multicolumn{1}{l}{Get NNF} \\
 \hline
OWL API & 167.1696 & 91.61577 & 23.737692\\
\fowl & 1275.384 & 68.87327 & 13.186410
\end{tabular}%
\caption{Average of times in milliseconds for tested ontologies}\label{tab:raw}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
\end{center}
\end{table}
}

\def\avgAx{
\begin{table}[ht]
\begin{center}
\def\arraystretch{1.25}
\setlength\tabcolsep{8pt}
\begin{tabular}{lrrr}
 & \multicolumn{1}{l}{Read File} & \multicolumn{1}{l}{Write File} & \multicolumn{1}{l}{Get NNF} \\
 \hline
OWL API & 85.367181 & 8.82415379 & 1.8501445 \\
\fowl & 60.724610 & 6.01522460 & 0.6555225
\end{tabular}%
\caption{Average times in nanoseconds scaled by number of axioms}\label{tab:scal}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
\end{center}
\end{table}
}

\def\goAll{
\begin{table}
\begin{center}
\def\arraystretch{1.5}
\setlength\tabcolsep{4pt}
%\resizebox{\textwidth}{!}{%
\begin{tabular}{lrrr}
 & \multicolumn{1}{l}{Read File} & \multicolumn{1}{l}{Write File} & \multicolumn{1}{l}{Get NNF} \\
 \hline
OWL API & 7248.2998 & 4453.8336 & 1390.7188 \\
\fowl & 55845.850 & 2918.0029 & 872.97672
\end{tabular}
\caption{Times in milliseconds for the Gene Ontology}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
\end{center}
\end{table}
}

\def\goAx{
\begin{table}
\begin{center}
\def\arraystretch{1.5}
\setlength\tabcolsep{4pt}
\begin{tabular}{lrrr}
 & \multicolumn{1}{l}{Read File} & \multicolumn{1}{l}{Write File} & \multicolumn{1}{l}{Get NNF} \\
 \hline
OWL API & 8.895726006 & 5.466120964 & 1.7068076 \\
\fowl & 68.53874628 & 3.581219801 & 1.0713907
\end{tabular}
\caption{Times in nanoseconds for GO, scaled by the number of axioms}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
\end{center}
\end{table}
}

\def\axiompatternenglish{
\begin{figure}[t]
\begin{compactenum}
    \item Every $A$ is also a $B$.
    \item $A$ and $B$ are disjoint.
    \item The domain of $R$ is $A$.
    \item For every $B$ which has an inverse $R$-filler, this inverse $R$-filler is in $A$. In other words, the domain of $R$ scoped with $B$ is $A$.
    \item The range of $R$ is $B$.
    \item For every $A$ which has an $R$-filler, this $R$-filler is in $B$. In other words, the range of $R$ scoped with $A$ is $B$.
    \item For every $A$ there has to be an $R$-filler in $B$.
    \item For every $B$ there has to be an inverse $R$-filler in $A$.
    \item $R$ is functional.
    \item $R$ has at most one filler in $B$.
    \item For every $A$ there is at most one $R$-filler.
    \item For every $A$ there is at most one $R$-filler in $B$.
    \item $R$ is inverse functional.
    \item $R$ has at most one inverse filler in $A$.
    \item For every $B$ there is at most one inverse $R$-filler.
    \item For every $B$ there is at most one inverse $R$-filler in $A$.
    \item $A$ may have a filler for $R$ in $B$.
\end{compactenum}
\caption{Axioms}
\label{fig:axioms}
\end{figure}
}

\def\sourceaxiomcounts{
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{axiomcounts.pdf}
\caption{Axiom Expressibility Counts, Log Scale}
\label{fig:axiomcounts}
\end{figure}
}

\def\sourceaxiompercent{
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{axiompercents.pdf}%*8/10]{axiompercents.pdf}
\caption{$\overline{ae}_{\mathcal{P}}(\mathcal{A})$, By Source}
\label{fig:axiompercents}
\end{figure}
}

\def\overallcomposition{
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth*16/20]{overallcomposition.pdf}
\caption{Overall Distribution of Axioms}
\label{fig:overallcomp}
\end{figure}
}

\def\owlaxaxioms{
\begin{table}[t]
\centering
	\begin{tabular}{llp{0.1cm}ll}
    Subclass & \subClass &               &    \begin{tabular}{c}Functional\end{tabular}& \functional \\
    Disjoint Classes & \disjointClasses &    &   \begin{tabular}{c}Qualified Functional\end{tabular}& \qualifiedFunctional \\
    Domain & \domain &            &            \begin{tabular}{c}Scoped Functional\end{tabular}& \scopedFunctional \\ 
    Scoped Domain & \scopedDomain &    &   \begin{tabular}{c}Qualified Scoped Functional\end{tabular}& \qualifiedScopedFunctional \\
    Range & \range &             &            Inverse Functional& \inverseFunctional \\
    Scoped Range & \scopedRange &       &            \begin{tabular}{c}Inverse Qualified Functional\end{tabular} & \inverseQualifiedFunctional \\
    Existential & \existential &     &      \begin{tabular}{c}Inverse Scoped Functional\end{tabular}& \inverseScopedFunctional \\
    Inverse Existential & \inverseExistential &   &  \begin{tabular}{c}Inverse Qualified Scoped Functional\end{tabular}& \inverseQualifiedScopedFunctional \\
     &  &                 &                        Structural Tautology & \structuralTautology 
    \end{tabular}
    \small{A,B,R are variable terms that contain at most one class or role name, or a data range}\par\null\par
\caption{OWLAx Axiom Patterns}	
\label{tab:owlaxax}	
\end{table}
}

\def\ontologies{
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
\begin{table}[t]
\centering
\begin{tabular}{lrrrrrrrr}
&LOV&Hydrography&Anatomy&Conference&ODP&Ontobee&Misc\\ \hline
classes &  20075 & 341 & 6048 & 498 & 577 & 509925 & 80796 \\
roles & 10106 & 121 & 5 & 226 & 600 & 10453 & 535 \\
data properties & 8340 & 77 & 0 & 85 & 71 & 480 & 66 \\
axioms & 274991 & 8873 & 41407 & 3037 & 7378 & 5071892 & 816572 \\
logical axioms & 82450 & 5463 & 16383 & 2153 & 3223 & 963880 & 272334 \\ 
normalized axioms & 94937&1748&9951&2774&3917&1208950&391015 \\
ontologies & 250 & 4 & 2 & 7 & 80 & 171 & 4 \\\hline
\end{tabular}%
\caption[Ontology Statistics]{Ontology Statistics \footnotemark}
\label{tab:ontstats}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{1pt}
\footnotetext{Logical axioms are axioms that are neither declarations nor annotations}
}

\def\overalltable{
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
\begin{table}[t]
\centering
\begin{tabular}{lccc}
&$\overline{ae}_{\mathcal{P}}(\mathcal{A})$&$\overline{oe}_{\mathcal{P}}(\mathcal{O})$&StdDev $\overline{oe}_{\mathcal{P}}(\mathcal{O})$\\ \hline
 All Axioms &  82.2\% &  82.9\% &  0.206 \\
 Class Axioms &  83.8\% &  92.9\% &  0.165 \\
 Simple Class Axioms&  99.8\% &  96.5\% &  0.153 \\ \hline
\end{tabular}
\caption{Overall Average Expressibility}
\label{tab:covoverall}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{1pt}
}

\def\sourceontcov{
\begin{table}[t]
\centering
\begin{tabular}{lccccccc}
 & LOV & Hydrography & Anatomy & Conference & ODP & Ontobee & Misc \\ \hline
All Axioms & 81.3\% & 77.5\% & 99.9\% & 87.3\% & 76.5\% & 88.3\% & 67.2\% \\
Class Axioms & 92.7\% & 89.7\% & 100\% & 92.8\% & 96.4\% & 91.8\% & 99.4\% \\
Simple Class Axioms & 95.2\% & 97.6\% & 100\% & 95.2\% & 97.7\% & 97.9\% & 99.4\% \\ \hline
\end{tabular}%
\caption{$\overline{oe}_{\mathcal{P}}(\mathcal{O})$ By Source}
\label{tab:ontcovsource}
\end{table}
}


\def\sourceontstd{
\begin{table}[t]
\centering
\begin{tabular}{lccccccc}
 & LOV & Hydrography & Anatomy & Conference & ODP & Ontobee & Misc \\ \hline
All Axioms & 0.227 & 0.132 & 0.001 & 0.063 & 0.184 & 0.176 & 0.131 \\
Class Axioms & 0.190 & 0.106 & 0.000 & 0.030 & 0.075 & 0.159 & 0.139 \\
Simple Class Axioms & 0.187 & 0.036 & 0.000 & 0.013 & 0.065 & 0.131 & 0.007 \\ \hline
\end{tabular}
\caption{$\overline{oe}_{\mathcal{P}}(\mathcal{O})$ Standard Deviation By Source}
\label{tab:ontstdsource}
\end{table}
}

\def\sourceaxcov{
\begin{table}[t]
\centering
\begin{tabular}{lccccccc}
 & LOV & Hydrography & Anatomy & Conference & ODP & Ontobee & Misc \\ \hline
All Axioms & 78.7\% & 77.1\% & 99.9\% & 89.3\% & 85.7\% & 88.7\% & 62.4\% \\
Class Axioms & 92.6\% & 84.5\% & 100\% & 96.2\% & 97.3\% & 89.9\% & 62.5\% \\
Simple Class Axioms & 99.2\% & 96.5\% & 100\% & 99.2\% & 99.1\% & 99.8\% & 99.9\% \\ \hline
\end{tabular}%
\caption{$\overline{ae}_{\mathcal{P}}(\mathcal{A})$ By Source}
\label{tab:axcovsource}
\end{table}
}

\def\sourcepersimp{
\begin{table}[t]
\centering
\begin{tabular}{lccccccc}
 & LOV & Hydrography & Anatomy & Conference & ODP & Ontobee & Misc \\ \hline
Subclass & 42.4\% & 46.2\% & 66.8\% & 57.6\% & 56.4\% & 60.2\% & 41.0\% \\
Existential & 05.9\% & 07.4\% & 33.1\% & 06.7\% & 05.7\% & 26.5\% & 20.7\% \\
Subclass + Existential & 48.3\% & 53.3\% & 99.9\% & 64.3\% & 62.1\% & 86.7\% & 61.7\% \\ \hline
\end{tabular}%
\caption{Percent Subclass and Existential By Source}
\label{tab:sourcepersimp}
\end{table}
}

\def\profontcov{
\def\arraystretch{1}
\setlength\tabcolsep{4pt}
\begin{table}[t]
\centering
\begin{tabular}{lccccc}
 & EL & QL & RL & DL & Full \\ \hline
All Axioms & 94.6\% & 87.7\% & 81.7\% & 82.8\% & 82.9\% \\
Class Axioms & 97.5\% & 97.1\% & 95.2\% & 94.4\% & 92.9\% \\
Simple Class Axioms & 98.1\% & 97.1\% & 95.2\% & 97.2\% & 96.5\% \\ \hline
\end{tabular}
\caption{$\overline{oe}_{\mathcal{P}}(\mathcal{O})$ By Profile}
\label{tab:profperont}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
}

\def\profontstd{
\def\arraystretch{1}
\setlength\tabcolsep{4pt}
\begin{table}[t]
\centering
\begin{tabular}{lccccc}
 & EL & QL & RL & DL & Full \\ \hline
All Axioms & 0.179 & 0.233 & 0.302 & 0.198 & 0.206 \\
Class Axioms & 0.139 & 0.169 & 0.218 & 0.124 & 0.165 \\
Simple Class Axioms & 0.179 & 0.233 & 0.302 & 0.198 & 0.153 \\ \hline
\end{tabular}
\caption{$\overline{oe}_{\mathcal{P}}(\mathcal{O})$ Standard Deviation By Profile}
\label{tab:profstddev}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
}

\def\profaxcov{
\def\arraystretch{1}
\setlength\tabcolsep{4pt}
\begin{table}[t]
\centering
\begin{tabular}{lccccc}
 & EL & QL & RL & DL & Full \\ \hline
All Axioms & 98.7\% & 99.7\% & 97.8\% & 91.4\% & 82.2\% \\
Class Axioms & 98.8\% & 100\% & 100\% & 92.7\% & 83.8\% \\
Simple Class Axioms & 100\% & 100\% & 100\% & 99.9\% & 99.8\% \\ \hline
\end{tabular}
\caption{$\overline{ae}_{\mathcal{P}}(\mathcal{A})$ By Profile}
\label{tab:profperax}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
}

\def\profpersimp{
\def\arraystretch{1}
\setlength\tabcolsep{4pt}
\begin{table}[t]
\centering
\begin{tabular}{lccccc}
                       & EL     & QL     & RL     & DL     & Full   \\ \hline
Subclass               & 52.5\% & 78.2\% & 95.1\% & 60.9\% & 54.8\% \\
Existential            & 46.1\% & 21.1\% & 0\% & 29.2\% & 23.9\% \\
Subclass + Existential & 98.6\% & 99.3\% & 95.1\% & 90.2\% & 78.8\% \\ \hline
\end{tabular}
\caption{Percent Subclass and Existential By Profile}
\label{tab:profpersimp}
\end{table}
\def\arraystretch{1}
\setlength\tabcolsep{2pt}
}

\def\transforms{
\begin{table}[t]
\centering
\begin{tabular}{lr}
\multicolumn{1}{c}{Ontology Axiom} & \multicolumn{1}{c}{Substituted Axiom}\\
\hline
ReflexiveObjectProperty(R) & $\top \sqsubseteq \exists \text{R}.\textbf{Self}$\\
IrreflexiveObjectProperty(R) & $ \exists \text{R}.\textbf{Self} \sqsubseteq \bot$\\
FunctionalObjectProperty(R) & $\top \sqsubseteq \ \leqslant\text{1R.}\top$\\
FunctionalDataProperty(S) & $\top \sqsubseteq \ \leqslant\text{1S}.\top$\\
InverseFunctionalObjectProperty(R) & $\top \sqsubseteq \ \leqslant\text{1R}^-.\top $\\
ObjectPropertyRange(R C) & $\top \sqsubseteq \forall\text{R.C}$\\
DataPropertyRange(S D) & $\top \sqsubseteq \forall \text{S.D}$\\
ObjectPropertyDomain(R C) & $\exists \text{R}.\top \sqsubseteq \text{C}$\\
DataPropertyDomain(S C) & $\exists\text{S}.\top \sqsubseteq \text{C}$
\end{tabular}
\\{\small R is an ObjectProperty, S is a DataProperty, C is a Class, and D is a DataRange}
\caption{Axiom Transformations}
\label{tab:axtrans}
\end{table}
}

\def\axex{
\begin{table}[t]
\centering
	\begin{tabular}{c|cccc}
	\multicolumn{1}{c}{}&\multicolumn{1}{c}{EL} & \multicolumn{1}{c}{QL} & \multicolumn{1}{c}{RL} & \multicolumn{1}{c}{DL}\\\hline
    %& EL & QL & RL & DL\\ \hline
    \subClass & X & X & X & X \\%\hline
    \disjointClasses & X & & X & X\\%\hline
    \domain & X & X & X & X \\\hline
    \scopedDomain & X & & X & X \\%\hline
    \range & & & & X \\%\hline
    \scopedRange & & & X & X\\\hline
    \existential & X & X & & X \\%\hline
    \inverseExistential & & X & & X \\%\hline
    \functional & & & & X \\\hline
    \qualifiedFunctional & & & & X \\%\hline
    \scopedFunctional & & & X & X \\%\hline
    \qualifiedScopedFunctional & & & X & X \\\hline
    \inverseFunctional & & & & X \\%\hline
    \inverseQualifiedFunctional & & & & X \\%\hline
    \inverseScopedFunctional & & & X & X\\\hline
    \inverseQualifiedScopedFunctional & & & X & X \\%\hline
    \structuralTautology & & &  & X \\\hline
    \end{tabular}
\caption{Profile Axiom Pattern Expressibility}	
\label{tab:axex}	
\end{table}
}

\def\axextwo{
\begin{table}[t]
\centering
	\begin{tabular}{c|cccc|c|cccc}
	\multicolumn{1}{c}{}&\multicolumn{1}{c}{EL} & \multicolumn{1}{c}{QL} & \multicolumn{1}{c}{RL} & \multicolumn{1}{c}{DL}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{EL} & \multicolumn{1}{c}{QL} & \multicolumn{1}{c}{RL} & \multicolumn{1}{c}{DL}\\\hline
    %& EL & QL & RL & DL\\ \hline
    \subClass & X & X & X & X & \functional & & & & X \\%\hline
    \disjointClasses & X & & X & X & \qualifiedFunctional & & & & X\\%\hline
    \domain & X & X & X & X  & \scopedFunctional & & & X & X\\
    \scopedDomain & X & & X & X & \qualifiedScopedFunctional & & & X & X\\%\hline
    \range & & & & X & \inverseFunctional & & & & X\\%\hline
    \scopedRange & & & X & X & \inverseQualifiedFunctional & & & & X\\
    \existential & X & X & & X & \inverseScopedFunctional & & & X & X\\
    \inverseExistential & & X & & X & \inverseQualifiedScopedFunctional & & & X & X\\
    &&&&&\structuralTautology & & &  & X \\\hline
    \end{tabular}
\caption{Profile Axiom Pattern Expressibility}	
\label{tab:axex}	
\end{table}
}

\def\elcov{
\begin{table}[t]
\centering
	\begin{tabular}{lp{1cm}l}
	\multicolumn{1}{c}{Axiom Pattern} &&\multicolumn{1}{c}{$\mathcal{EL}^{++}$ Normal Class Axiom}\\\hline
    \subClass && \subClass \\ 
    \disjointClasses && A $\sqcap$ B $\sqsubseteq$ C, when C = $\bot$\\
    \domain && \scopedDomain, when A = $\top$ \\
    \scopedDomain && \scopedDomain \\
    \existential &&  \existential\\\hline
    \end{tabular}
\caption{$\mathcal{EL}^{++}$ Axiom Pattern Expressibility}	
\label{tab:elcov}	
\end{table}
}