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+thick, text centered, draw=black, fill=red!30] \tikzstyle{io} = [trapezium, trapezium stretches=true, % A later addition +thick, trapezium left angle=70, trapezium right angle=110, minimum width=3cm, @@ -29,18 +31,20 @@ \tikzstyle{process} = [rectangle, minimum width=3.5cm, minimum height=1cm, +thick, text centered, -text width=3.5cm, +text width=4cm, draw=black, fill=orange!30] \tikzstyle{decision} = [diamond, minimum width=3cm, minimum height=1cm, +thick, text centered, draw=black, fill=green!30] -\tikzstyle{arrow} = [thick,->,>=stealth] +\tikzstyle{arrow} = [->,thick] \begin{document} % Document @@ -48,61 +52,67 @@ \section{Method} -The syntax of most programming languages is context-free. Our proposed method is simple. We construct a context-free grammar representing the intersection between the langauge syntax and an automaton recognizing the Levenshtein edit ball up to a fixed distance. Since CFLs are closed under intersection with regular languages, this is admissible. Three outcomes are possible: +The syntax of most programming languages is context-free. Our proposed method is simple. We construct a context-free grammar representing the intersection between the langauge syntax and an automaton recognizing the Levenshtein ball of a given radius. Since CFLs are closed under intersection with regular languages, this is admissible. Three outcomes are possible: \begin{enumerate} \item $\mathcal{G}_\cap$ is empty, in which case there is no repair within the given radius. In this case, we simply increase the radius and try again. - \item $\mathcal{G}_\cap$ is small, in which case we simply enumerate all possible repairs. This is the case for ~80\% of the Python dataset. - \item $\mathcal{G}_\cap$ is too large to exhaustively sample, in which case we use a PCFG to sample $\mathcal{L}(\mathcal{G})$ without replacement, then reject duplicates. This is the case for ~20\% of the Python dataset. + \item $\mathcal{L}(\mathcal{G}_\cap)$ is small, in which case we simply enumerate all possible repairs. Enumeration is tractable for $\sim 80\%$ of the Python dataset in $\leq 90$s. + \item $\mathcal{L}(\mathcal{G}_\cap)$ is too large to enumerate, so we sample from the intersection grammar $\mathcal{G}_\cap$. Sampling is necessary for $\sim20\%$ of the Python dataset. \end{enumerate} -After sampling, we use an n-gram model to rank the samples, then return the top-k results by likelihood. We depict the flowchart below: +When ambiguous, we use an n-gram model to rank and return the top-k results by likelihood. This procedure is depicted in the flowchart below: \begin{figure}[h!] \begin{center} \resizebox{0.6\textwidth}{!}{ \begin{tikzpicture}[node distance=2cm] -% \node (start) [startstop] {Start}; - \node (in1) [io] {Input}; - \node (rin1) [io, right of=in1, xshift=3cm] {L-NFA}; - \node (lin1) [io, left of=in1, xshift=-3cm] {[P]CFG}; + \node (start) [startstop] {Start}; + \node (pro1) [process, below of=start, yshift=-0.5cm] {$\mathcal{G}_\cap \leftarrow \mathcal{G}\cap\Delta(\err\sigma, d)$}; + \node (pcfg) [io, left of=pro1, xshift=-3cm] {[P]CFG}; + \node (lnfa) [io, right of=pro1, xshift=3cm] {L-WFA}; - \node (pro1) [process, below of=in1] {$\mathcal{G}_\cap \leftarrow \mathcal{G}\cap\Delta(\err\sigma, d)$}; + \node (code) [io, right of=start,xshift=3cm] {Code}; + \node (synt) [io, left of=start,xshift=-3cm] {Syntax}; \node (dec1) [decision, below of=pro1, yshift=-0.5cm] {$[\mathcal{G}_{\cap} = \varnothing]$}; - \node (pro2b) [process, right of=dec1, xshift=3cm] {Increase radius}; + \node (pro2b) [process, right of=dec1, xshift=3cm] {Increase radius, $d$}; - \node (dec2) [decision, below of=dec1, yshift=-0.8cm] {$|\mathcal{L}(\mathcal{G}_\cap)|$}; + \node (dec2) [decision, below of=dec1, yshift=-1cm] {$|\mathcal{L}(\mathcal{G}_\cap)|$}; - \node (samp1) [process, left of=dec2, xshift=-3cm] {Denumerate $\mathcal{L}(\mathcal{G}_\cap)$}; - \node (samp2) [process, right of=dec2, xshift=3cm] {Sample [P]CFG}; + \node (samp1) [process, left of=dec2, xshift=-3cm] {Enumerate $\sigma' \in \mathcal{L}(\mathcal{G}_\cap)$}; + \node (samp2) [process, right of=dec2, xshift=3cm] {Sample $\sigma' \sim P(\mathcal{G}_\cap)$}; - \node (uin1) [io, below of=dec2, yshift=0.1cm] {VOMC}; - \node (rank1) [process, below of=uin1, yshift=0.1cm] {Rerank solutions}; + \node (rank) [process, below of=dec2, yshift=-2.5cm] {Rerank by $L_\theta(\sigma')$}; + \node (vlmc) [io, below of=rank, yshift=-0.1cm] {VLMC}; % \node (out1) [io, below of=pro2a] {Output}; -% \node (stop) [startstop, below of=out1] {Stop}; + \node (stop) [startstop, above of=rank] {Done}; -% \draw [arrow] (start) -- (in1); +% \draw [arrow] (dec0) -- node[anchor=east] {no} (pro1); - \draw [arrow] (rin1) -- (in1); - \draw [arrow] (lin1) -- (in1); + \draw [arrow] (start) -- (code); + \draw [arrow] (start) -- (synt); + \draw [arrow] (code) -- (lnfa); + \draw [arrow] (synt) -- (pcfg); + \draw [arrow] (lnfa) -- (pro1); + \draw [arrow] (pcfg) -- (pro1); - \draw [arrow] (in1) -- (pro1); +% \draw [arrow] (in1) -- (pro1); \draw [arrow] (pro1) -- (dec1); \draw [arrow] (dec1) -- node[anchor=south] {yes} (pro2b); \draw [arrow] (dec1) -- node[anchor=east] {no} (dec2); - \draw [arrow] (pro2b) -- (rin1); + \draw [arrow] (pro2b) -- (lnfa); \draw [arrow] (dec2) -- node[anchor=south] {small} (samp1); \draw [arrow] (dec2) -- node[anchor=south] {large} (samp2); - \draw [arrow] (uin1) -- (rank1); - \draw [arrow] (samp1) |- (rank1); - \draw [arrow] (samp2) |- (rank1); + \draw [arrow] (vlmc) -- (rank); + \draw [arrow] (samp1) |- (rank); + \draw [arrow] (samp2) |- (rank); % \draw [arrow] (pro2a) -- (out1); -% \draw [arrow] (out1) -- (stop); + \draw [arrow] (rank) -- (stop); + \draw [arrow] (dec2) -- node[anchor=east] {1} (stop); \end{tikzpicture} }