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\section{AST patterns}
We encode structural patterns in ASTs using tree regular expressions (TREs). An ordinary regular expression describes the set of strings matching certain constraints; similarly, a TRE describes the set of trees containing certain nodes and relations. Since TREs describe structure, they are themselves represented as trees. More specifically, both ASTs and TREs are ordered rooted trees.
In this work we used TREs to encode (only) child and sibling relations in ASTs. We write them as S-expressions, such as \pattern{(a (b \textbf{\textasciicircum}~d~\textbf{.}~e~\$) c)}. This expression matches any tree satisfying the following constraints (see Fig.~\ref{fig:tre-example} for an example):
\begin{itemize}
\item the root \pattern{a} has at least two children, \pattern{b} and \pattern{c}, adjacent and in that order; and
\item the node \pattern{b} has three children: \pattern{d}, followed by any node, followed by \pattern{e}.
\end{itemize}
\noindent
Analogous to ordinary regular expressions, caret (\texttt{\textbf{\textasciicircum}}) and dollar sign (\texttt{\$}) anchor a node to be respectively the first or last child of its parent. A period (\texttt{\textbf{.}}) is a wildcard that matches any node.
\begin{figure}[htbp]
\centering
\begin{forest}
for tree={
font=\sf,
edge=darkgray,
l sep=0,
l=0.9cm,
}
[a,name=a
[f]
[b,name=b
[d,name=d] [g,name=g] [e,name=e]]
[c,name=c [j] [k]]
[h]
]
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (a) edge[transform canvas={xshift=0.8mm,yshift=-0.2mm}] (b);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (a) edge[transform canvas={xshift=-0.9mm,yshift=-0.3mm}] (c);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (b) edge[transform canvas={xshift=-1.1mm,yshift=0.2mm}] (d);
\draw[opacity=0] (b) -- node[anchor=east,thick,blue,opacity=1,font=\large] {\texttt{\textasciicircum}} (d);
\path[draw,thick,relative,blue,transform canvas={xshift=-0.7mm}] (b) -- ($ (b)!0.6!(g) $);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (b) edge[transform canvas={xshift=1.1mm,yshift=0.2mm}] (e);
\draw[opacity=0] (b) -- node[anchor=west,thick,blue,opacity=1,font=\scriptsize,transform canvas={xshift=0mm,yshift=1mm}] {\texttt{\$}} (e);
\end{forest}
\caption{A tree matching a pattern (blue arrows besides the edges). In the pattern, each arrow $x→y$ means that node $x$ has a child $y$. A shorter line without an arrowhead (e.g. \pattern{b}$\boldsymbol{-}$ \pattern{g}) indicates a wildcard, where the child can be any node. Anchors \texttt{\textbf{\textasciicircum}} and \texttt{\$} mean that the pattern will match only the first or last child.}
\label{fig:tre-example}
\end{figure}
With TREs we encode interesting patterns in a program while disregarding irrelevant parts. Take for example the following, nearly correct Python function that prints the divisors of its argument $n$:
\begin{Verbatim}
def divisors(n):
for d in range(1, n):
if n % d == 0:
print(d)
\end{Verbatim}
Figure~\ref{fig:patterns-example} shows the simplified AST for this program, with two patterns overlaid. These patterns are represented by the S-expressions
\begin{figure}[htb]
\centering
\begin{forest}
for tree={
font=\sf,
edge=darkgray,
l sep=0,
l=0.9cm,
}
[Function, name=def, draw,rectangle,red,dashed
[name, name=name1 [divisors, name=divisors, font=\bfseries\sffamily]]
[args, name=args1
[Var, name=args1-1, draw,rectangle,blue [n, font=\bfseries\sffamily, l=0.7cm]]]
[body, name=body1, before computing xy={s=2cm}
[For, name=for, l=1.2cm, draw,rectangle,red,dashed
[target
[Var, l=0.9cm [d, font=\bfseries\sffamily, l=0.7cm]]]
[iter, name=iter
[Call, name=call, l=0.9cm
[func, name=func, l=0.8cm [range, name=range, font=\bfseries\sffamily, l=0.7cm]]
[args, name=args2, l=0.8cm
[Num, name=args2-1, l=1cm [1, font=\bfseries\sffamily, l=0.7cm]]
[Var, name=args2-2, l=1cm,draw,rectangle,blue [n, font=\bfseries\sffamily, l=0.7cm]]]]]
[body, name=body2
[If, name=if, draw,rectangle,red,dashed
[test, l=0.7cm [Compare, l=0.8cm
[BinOp
[Var [n, font=\bfseries\sffamily, l=0.6cm]]
[\%, font=\bfseries\sffamily]
[Var [d, font=\bfseries\sffamily, l=0.6cm]]]
[{==}, font=\bfseries\sffamily, l=0.7cm]
[Num [0, font=\bfseries\sffamily, l=0.7cm]]]]
[body, l=0.7cm [Call, l=0.8cm
[func [print, font=\bfseries\sffamily, l=0.7cm]]
[args [Var, l=0.7cm [d, font=\bfseries\sffamily, l=0.6cm]]]]]]]]]]
% first pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red,dashed] (def) edge[transform canvas={xshift=1.1mm,yshift=0.2mm}] (body1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red,dashed] (body1) edge[transform canvas={xshift=0.8mm}] (for);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red,dashed] (for) edge[transform canvas={xshift=1.1mm,yshift=0.5mm}] (body2);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red,dashed] (body2) edge[transform canvas={xshift=0.8mm}] (if);
% second pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (def) edge[transform canvas={xshift=-1.1mm,yshift=0.2mm}] (name1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (def) edge[transform canvas={xshift=-0.8mm}] (args1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (args1) edge[transform canvas={xshift=-0.8mm}] (args1-1);
\draw[opacity=0] (args1) -- node[anchor=east,thick,blue,opacity=1,font=\large] {\texttt{\textasciicircum}} (args1-1);
\draw[opacity=0] (args1) -- node[anchor=west,thick,blue,opacity=1,font=\scriptsize,transform canvas={xshift=-0.5mm,yshift=0.6mm}] {\texttt{\$}} (args1-1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (def) edge[transform canvas={xshift=-1mm,yshift=-0.3mm}] (body1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (body1) edge[transform canvas={xshift=-0.8mm}] (for);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (for) edge[transform canvas={xshift=1mm,yshift=-0.5mm}] (iter);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (iter) edge[transform canvas={xshift=0.8mm}] (call);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (call) edge[transform canvas={xshift=-1.2mm}] (func);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (call) edge[transform canvas={xshift=1.2mm}] (args2);
\path[draw,thick,relative,blue,transform canvas={xshift=-0.7mm}] (args2) -- ($ (args2)!0.55!(args2-1) $);
\draw[opacity=0] (args2) -- node[anchor=east,thick,blue,opacity=1,font=\large,transform canvas={xshift=0.6mm,yshift=0.4mm}] {\texttt{\textasciicircum}} (args2-1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,blue] (args2) edge[transform canvas={xshift=1mm}] (args2-2);
\draw[opacity=0] (args2) -- node[anchor=west,thick,blue,opacity=1,font=\scriptsize,transform canvas={xshift=0mm,yshift=1mm}] {\texttt{\$}} (args2-2);
\end{forest}
\caption{The AST for the \code{divisors} program with two patterns. Leaf nodes (in bold) correspond to terminals in the program, i.e. names and values. Dashed red arrows represent the pattern describing the control structure of the program. Solid blue arrows encode the incorrect second argument to the \code{range} function.}
\label{fig:patterns-example}
\end{figure}
\begin{enumerate}
\item \pattern{(Function (body (For (body If))))} and
\item \pattern{(Function (name divisors) (args \textbf{\textasciicircum}~Var \$)}
\item[] ~~~~\pattern{(body (For (iter (Call (func range) (args \textbf{\textasciicircum}~\textbf{.} Var \$))))))}.
\end{enumerate}
The first TRE encodes a single path in the AST and describes the program’s block structure: \textsf{Function}$\boldsymbol{-}$\textsf{For}$\boldsymbol{-}$\textsf{If}. The second TRE relates the argument in the definition of \code{divisors} with the last argument to \code{range} that provides the iterator in the for loop. Since S-expressions are not easy to read, we will instead represent TREs by highlighting relevant text in examples of matching programs:
\begin{Verbatim}
\red{\dashuline{\textbf{def}}} divisors(\blue{\underline{\textbf{n}}}):
\red{\dashuline{\textbf{for}}} d in range(1, \blue{\underline{\textbf{n}}}):
\red{\dashuline{\textbf{if}}} n % d == 0:
print(d)
\end{Verbatim}
The second pattern describes a common mistake for this problem: \code{range(1,n)} will only generate values up to \code{n-1}, so \code{n} will not be printed as its own divisor. A correct pattern would include the binary operator \code{+} on the path to \code{n}, indicating a call to \code{range(1,n+1)}.
\subsection{Constructing patterns}
Patterns are extracted automatically from student programs. We first canonicalize each program~\cite{rivers2015data-driven} using code from ITAP\footnote{Available at \url{https://github.com/krivers/ITAP-django}.}. To construct TREs describing individual patterns, we select a subset of nodes in the AST, and walk the tree from each selected node to the root, including all nodes along those paths.
Depending on node type we also include some nodes adjacent to such paths. For each comparison and unary/binary expression on the path we include the corresponding operator. For function definitions and calls we include the function name. Finally, in all argument lists we include the anchors (\textbf{\texttt{\textasciicircum}}~and~\code{\$}) and a wildcard (\textbf{\texttt{.}}) for each argument not on the path. This allows our TREs to discriminate between e.g. the first and second argument to a function.
While pattern extraction is completely automated, we have manually defined the kinds of node subsets that are selected. After analyzing solutions to several programming problems, we decided to use the following kinds of patterns. Figure~\ref{fig:patterns-example} shows two examples of the first two kinds of patterns.
\begin{enumerate}
\item
We select each pair of leaf nodes referring to the same variable.
\item
For each control-flow node $n$ we construct a pattern from the set $\{n\}$; we do the same for each \textsf{Call} node representing a function call.
\item
For each expression (such as \code{(F-32)*5/9}) we select the different combinations of literal and variable nodes in the expression. In these patterns we include at most one node referring to a variable.
\end{enumerate}
Note that in every constructed pattern, all \textsf{Var} nodes refer to the same variable. We found that patterns constructed from such nodesets are useful for discriminating between programs. As we show in Sect.~\ref{sec:interpreting-rules}, they are also easily interpreted in terms of bugs and strategies for a given problem.
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