\section{Notation}
\label{sec:notation}
In Table \ref{tab:notation}, we present an overview of the notation used in the paper.
% $[N]$ denotes the set $\{1, 2, \ldots, N\}$.
% \todo{uncomment out table}
%% TABLE (commented out to avoid long render times)
%\begin{landscape}
\begin{table}[!htb]
{\renewcommand{\arraystretch}{2}%
	\caption{Notation used in our paper, with notes on interpretation.} \label{tab:notation}
	\resizebox{\textwidth}{!}{\begin{tabular}{|lll|}
	\hline 
        % Queries, copilot responses, and associated attributes
        \multicolumn{3}{|l|}{\textbf{Queries, copilot responses, and associated attributes} explanation here} \\ \hline
        \multicolumn{1}{|l|}{Fully-specified query} 
            & \multicolumn{1}{l|}{
            $\begin{aligned}
            q_f \in Q
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Partially-specified/masked query}
            & \multicolumn{1}{l|}{
            $\begin{aligned}
            q_m \in Q
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Copilot response strategy}             
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            \pi: Q \rightarrow Y
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Copilot response (given $q, \pi$)}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            y \in Y \ | \ q \in Q, \pi \in \Pi
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Minimum viable copilot response}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        % Synthetic query generation
        \multicolumn{3}{|l|}{\textbf{Synthetic query generation} explanation here} \\ \hline
        \multicolumn{1}{|l|}{Intent space}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            \mathcal{I}~\coloneq~\{\text{exercise, gift, movie, plant, travel}\}
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Intent-specific attribute space}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            \Theta_i \coloneq \{\theta_1, \dots, \theta_{|\Theta_i|}\}
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Attribute-specific options}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            X_{\theta \in \Theta_i} \coloneq \{x \ | \ x \text{ is an option for } \theta\}
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Intent-specific query template}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Number of attributes to omit}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            n \sim U(\{1, \dots, |\Theta_i|\})
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Masked attributes}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            \Theta^m_i  \sim U(S_n(\Theta_i))
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Revealed attributes}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            \Theta^r_i \coloneq \Theta_i \setminus \Theta^m_i
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{``Depends on'' attributes}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        % Single-step response evaluation functions
        \multicolumn{3}{|l|}{\textbf{Single-turn response evaluation functions} explanation} \\ \hline
        \multicolumn{1}{|l|}{Utility function}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            \textbf{utility}_{ST}: (q_m, y_\pi) \in Q \times Y_\pi \mapsto \frac{|\phi(y_\pi)|^\gamma}{|q_{\Theta^m_i}|} \in [0,1]
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Cost function}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Cost-aware utility function}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Single-turn oracle policy}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline

        % Multi-step response evaluation functions
        \multicolumn{3}{|l|}{\textbf{Multi-turn response evaluation functions} explanation here} \\ \hline
        \multicolumn{1}{|l|}{Utility function}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Cost function}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Cost-aware utility function}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline
        \multicolumn{1}{|l|}{Multi-turn oracle policy}
        & \multicolumn{1}{l|}{
            $\begin{aligned}
            
            \end{aligned}$}
            &  test\\ \hline


        
        % % MDP Variables
        % \multicolumn{3}{|l|}{\textbf{MDP Variables} Here, timestep \(t\in [T] = \{1, 2, \dots T\}\) (subscript) and arm index \(i\in [N] = \{1, 2, \dots N\}\) (superscript) are implied.} \\ \hline
        % \multicolumn{1}{|l|}{State space} & \multicolumn{1}{l|}{\(s\in\mathcal{S}=\{0,1\}\)} & \(s=\begin{cases} 1 & \textrm{arm is in the `good' state.}\\ 0  &  \text{else%, patient did not take their medication
        % }\end{cases}\) \\ \hline
        % \multicolumn{1}{|l|}{Belief space} & \multicolumn{1}{l|}{\(\begin{aligned}
        % 	b \in\mathcal{B} &= [0,1]\\
        % 	b_{t+1} &= \begin{cases}
        % 	s_{t+1} & \textrm{if known} \\
        % 	b_{t}P_{1,1}^{0} + (1-b_{t})P_{0,1}^{0} & \textrm{else}
        % 	\end{cases}
        % 	\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}If an arm's true state is unknown, \\ the recursively computed belief state approximates it. \end{tabular}} \\ \hline
        % \multicolumn{1}{|l|}{Action space} & \multicolumn{1}{l|}{\(a\in\mathcal{A} = \{0,1\}\)} & \(a=\begin{cases} 1 & \textrm{pull arm (i.e., provide intervention)}\\ 0  &  \text{else, don't pull}\end{cases}\) \\ \hline
        % % MDP Functions
        % \textbf{MDP Functions} &  &  \\ \hline
        % \multicolumn{1}{|l|}{Transition function} & \multicolumn{1}{l|}{\(\begin{aligned}
        % 	P \colon \mathcal{S}\times\mathcal{A}\times\mathcal{S} &\to [0,1]\\
        % 	s_t, a_t, s_{t+1} &\mapsto \Pr(s_{t+1} \mid s_t, a_t)
        % 	\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}The probability of an arm \\ going from state \(s_t\) to \(s_{t+1}\), given action \(a_t\). \\ Equivalent (matrix) notation: \(P^{a_t}_{s_t,s_{t+1}}\). \end{tabular}} \\ \hline % P_a(s,s^\prime)
        % \multicolumn{1}{|l|}{Reward function} & \multicolumn{1}{l|}{\(\begin{aligned}
        % 	r \colon \mathcal{S}\text{ or }\mathcal{B} &\to \mathbb{R} % \\
        % 	% r(s) &\mapsto s
        % 	\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}\(r(b)\) is used in computing the Whittle index. \end{tabular}} \\ \hline
        % \multicolumn{1}{|l|}{Policy function} & \multicolumn{1}{l|}{\(\begin{aligned}
        % 	\pi \colon \mathcal{S} &\to \mathcal{A}% \\
        % 	%\pi(s) &\mapsto a
        % 	\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}A policy for actions.\\ The set of optimal policies is \(\pi^* \in \Pi^*\). \end{tabular}} \\ \hline
        % % RMAB Variables
        % \textbf{RMAB Variables} &  &  \\ \hline
        % \multicolumn{1}{|l|}{Timestep} & \multicolumn{1}{l|}{\(\{t\in \mathbb{N} \mid t\leq T\}\)} & This timestep is implicit in the MDP. \\ \hline
        % \multicolumn{1}{|l|}{Arm index} & \multicolumn{1}{l|}{\(i\in\{1,2,\dots,N\}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}Each arm can represent a patient.\\ \(k\) arms can be pulled at any timestep \(t\).\end{tabular}} \\ \hline
        % % RMAB objective functions
        % \multicolumn{3}{|l|}{\textbf{Objective Functions} The objective is to find a policy \(\pi^* = \max_\pi \mathbb{E}_\pi [R(\cdot)]\).} \\ \hline
        % \multicolumn{1}{|l|}{Discounted reward function} & \multicolumn{1}{l|}{\(\begin{aligned}
        % 	R_\beta^\pi \colon \mathcal{S}^N &\to \mathbb{R}\\
        % 	s_0^1,s_0^2,\dots,s_0^N &\mapsto  \sum_{i \in [N]} \sum_{t\in[T]} \beta^{t-1} r(s^i_t)
        % 	\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}\(\beta\in [0,1]\) is some \textit{discount parameter}.\end{tabular}} \\ \hline
        %    \multicolumn{3}{|l|}{\textbf{Fairness-motivated Constraint Functions} 
        %    %that motivate distributive fairness.
        %    } \\ \hline
        % \multicolumn{1}{|l|}{Integer periodicity } & \multicolumn{1}{l|}{\(\begin{aligned}
        %           \bigwedge_{j=0}^{\lceil\frac{T}{\nu} \rceil} \sum_{t=j\nu+1}^{(j+1)\nu} a_t^i \geq 1
        %           \end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}} A form of time-indexed fairness. \\ Guarantees arm \(i\) is pulled at least once \\ within each period of \(\nu\) timesteps.\end{tabular}} \\ \hline
        % \multicolumn{1}{|l|}{Minimum selection fraction } & \multicolumn{1}{l|}{\(\begin{aligned}
        %   		\bigwedge_{i\in[N]} \frac{1}{T}\sum_{t=1}^{T} a_t^i \geq \psi 
        %   		\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}} A form of time-indexed fairness. \\ Arm \(i\) should be pulled at least some \\ minimum fraction \(\psi \in (0,1)\) of timesteps.\end{tabular}} \\ \hline
        % \multicolumn{1}{|l|}{Probabilistic} & \multicolumn{1}{l|}{\(\begin{aligned}
        %   		\bigwedge_{i\in[N]} \bigwedge_{t\in[T]} \Pr(a_t^i =1 \mid i,t) \in [\ell, u]
        %   		\end{aligned}\)} & {\renewcommand{\arraystretch}{1.25}%
        % 	\begin{tabular}[c]{@{}l@{}}Pull each arm with probability \(p_i \in [\ell, u]\), \\ where \(\ell \in \left(0, \frac{k}{N}\right]\) and \(u \in \left[\frac{k}{N}, 1\right]\).\end{tabular}} \\ \hline
	\end{tabular}
 } 
}
\end{table}