\section{AST patterns}

In this section we describe AST patterns through examples, while Sect.~\ref{sec:extracting-patterns} explains how patterns are extracted from student programs. Consider the following Prolog program implementing the relation \code{sister(X,Y)}\footnote{Meaning “\code{X} is a sister of \code{Y}”.}:

\begin{Verbatim}
sister(X,Y):-     % X is Y’s sister when:
  parent(P,X),
  parent(P,Y),    % X and Y share a common parent P,
  female(X),      % X is female, and
  X \textbackslash{}= Y.         % X and Y are not the same person.
\end{Verbatim}

Figure~\ref{fig:sister} shows the program’s AST with two patterns. The pattern drawn with blue dotted arrows encodes the fact that the first argument to the \code{sister} predicate also appears in the call to \code{female}. In other words, this pattern states that \code{X} must be female to be a sister. We write the pattern as the s-expression

\begin{Verbatim}[fontfamily=sf]
(clause (head (compound (functor \code{sister}) (args var)))
    (compound (functor \code{female}) (args var)))
\end{Verbatim}

\begin{figure}[htbp]
  \centering
  \begin{forest}
    for tree={
      font=\sf,
      edge=gray,
      s sep=0.05cm,
    }
[text
  [clause,name=top
    [head,name=h
      [compound,name=hc [functor,name=hf [sister,name=hfl]]
        [args,name=ha [var,name=hav [X,name=havl,draw,rectangle,thick,blue,dotted,line width=0.4mm]] [args [var [Y]]]]]]
    [and
      [compound,name=g1 [functor,name=g1f [parent,name=g1fl]]
        [args,name=g1a [var,name=g1av [P,name=g1avl,draw,rectangle,thick,red]] [args [var [X]]]]]
      [and
        [compound,name=g2 [functor,name=g2f [parent,name=g2fl]]
          [args,name=g2a [var,name=g2av [P,name=g2avl,draw,rectangle,thick,red]] [args [var [Y]]]]]
        [and
          [compound,name=g3 [functor,name=g3f [female,name=g3fl]]
            [args,name=g3a [var,name=g3av [X,name=g3avl,draw,rectangle,thick,blue,dotted,line width=0.4mm]]]]
          [binop [var [X]] [\textbackslash{}{=}] [var [Y]]]]]]]]
% first pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (top) edge[out=-10,in=-170] (g1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g1) edge[transform canvas={xshift=-1.5mm}] (g1f);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g1f) edge[transform canvas={xshift=-1.1mm}] (g1fl);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g1) edge[transform canvas={xshift=1.5mm}] (g1a);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g1a) edge[transform canvas={xshift=-1.2mm}] (g1av);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (top) edge[out=-10,in=-170,transform canvas={xshift=-2mm}] (g2);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g2) edge[transform canvas={xshift=-1.5mm}] (g2f);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g2f) edge[transform canvas={xshift=-1.1mm}] (g2fl);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g2) edge[transform canvas={xshift=1.5mm}] (g2a);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (g2a) edge[transform canvas={xshift=-1.2mm}] (g2av);
% second pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (top) edge[transform canvas={xshift=-0.8mm,yshift=0.8mm}] (h);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (h) edge[transform canvas={xshift=-1mm}] (hc);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (hc) edge[transform canvas={xshift=-1.5mm}] (hf);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (hf) edge[transform canvas={xshift=-1.1mm}] (hfl);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (hc) edge[transform canvas={xshift=1.5mm}] (ha);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (ha) edge[transform canvas={xshift=-1.2mm}] (hav);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (top) edge[out=-5,in=160] (g3);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (g3) edge[transform canvas={xshift=-1.5mm}] (g3f);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (g3f) edge[transform canvas={xshift=-1.1mm}] (g3fl);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (g3) edge[transform canvas={xshift=1.5mm}] (g3a);
\path[-{Latex[length=1.5mm,width=1mm]},thick,dotted,relative,blue] (g3a) edge[transform canvas={xshift=-1.2mm}] (g3av);
  \end{forest}
  \caption{The AST for the \code{sister} program, showing two patterns and the leaf nodes inducing them. Solid red arrows equate the first arguments in the two calls to \code{parent}. Dotted blue arrows encode the necessary condition that \code{X} must be female to be a sister.}
  \label{fig:sister}
\end{figure}

Patterns describe relations between nodes in a program’s AST. Specifically, the pattern ($a$ $b$ $c$) means that the nodes $b$ and $c$ are descended from $a$, and that $b$ precedes $c$ in a depth-first tree walk. In general, an AST matches the pattern (\textsf{name} $p_1$ … $p_k$) if it contains a node $n$ labeled \textsf{name}; the subtree rooted at $n$ must also contain, in depth-first order, distinct nodes $n_1$ to $n_k$ matching subpatterns $p_1$ to $p_k$.

We handle syntactic variations in programs by omitting certain nodes from patterns. For example, by not including \code{and} nodes, the above pattern can match predicates with goals in any order (any arrangement of \code{and} nodes in the AST). This way, the same pattern will match any clause containing the “\textsf{sister}-\textsf{female}” relation, regardless of other goals.

The second pattern in Fig.~\ref{fig:sister}, drawn with solid red arrows, encodes the fact that the two calls to \code{parent} share the first argument. In other words, \code{X}~and~\code{Y} must have the same parent~\code{P}.

\begin{Verbatim}[fontfamily=sf]
(clause (compound (functor \code{parent}) (args var))
    (compound (functor \code{parent}) (args var)))
\end{Verbatim}

All patterns used in this paper have the same basic structure: a union of paths from some \textsf{clause} to a pair of leaf nodes with variables (or values). Some local context is included with each leaf, such as the functor of a compound term. Each pattern only refers to a single variable, so we do not include its name with the \textsf{var} nodes. We regard such patterns as the basic units of meaning in Prolog programs: each pattern encodes an interaction between two parts of the program accessing the same object (variable).

A relation between any two objects in a program is insufficient to reason about the program’s behavior on the whole. In the tutoring context, however, there are patterns that strongly indicate the presence of certain bugs. Take for example the following incorrect program to sum a list:

\begin{Verbatim}
sum([],0).          % \textit{base case:} the empty list sums to zero
sum([H|T],Sum):-    % \textit{recursive case:}
  sum(T,Sum),       %  sum the tail and
  Sum is Sum + H.   %  add first element (\textit{bug:} reused variable)
\end{Verbatim}

This error is fairly common with Prolog novices: the variable \code{Sum} is used to represent both the sum of the whole list, and the sum of only the tail elements. The last line fails since Prolog cannot unify \code{Sum} with a (usually) different value of \code{Sum\,+\,H}. The program’s AST is displayed in Fig.~\ref{fig:sum}.

\begin{figure}[htbp]
  \centering
  \begin{forest}
for tree={
    font=\sf,
    edge=gray,
    s sep=0.05cm,
}
[text
  [clause,name=c1
    [head,name=c1h
      [compound,name=c1hc [functor,name=c1hf [sum,name=c1hfv]]
        [args,name=c1ha1 [\code{[\,]},name=c1ha1v,draw,rectangle,thick,dotted,line width=0.4mm,blue]
          [args,name=c1ha2 [\code{0},name=c1ha2v,draw,rectangle,thick,dotted,line width=0.4mm,blue]]]]]]
  [clause,name=c2
    [head,name=c2h
      [compound,name=c2hc [functor,name=c2hf [sum,name=c2hfv]]
        [args,name=c2ha1 [list [var [H]] [var [T]]] [args,name=c2ha2 [var,name=c2ha2v [Sum,draw,rectangle,thick,red]]]]]]
    [and
      [compound,name=c2c [functor,name=c2cf [sum,name=c2cfv]]
        [args,name=c2ca1 [var [T]] [args,name=c2ca2 [var,name=c2ca2v [Sum,draw,rectangle,thick,red]]]]]
      [binop,name=c2b [var,name=c2bv [Sum,draw,rectangle,thick,dashed,orange]] [is,name=c2bo]
        [binop,name=c2bb [var,name=c2bbv [Sum,draw,rectangle,thick,dashed,orange]] [+,name=c2bbo] [var [H]]]]]]]
% first pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1) edge[transform canvas={xshift=-1mm}] (c1h);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1h) edge[transform canvas={xshift=-1mm}] (c1hc);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1hc) edge[transform canvas={xshift=-1.2mm}] (c1hf);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1hf) edge[transform canvas={xshift=-1mm}] (c1hfv);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1hc) edge[transform canvas={xshift=1.2mm}] (c1ha1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1ha1) edge[transform canvas={xshift=-1.2mm}] (c1ha1v);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1ha1) edge[transform canvas={xshift=1.2mm}] (c1ha2);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dotted,line width=0.4mm,blue] (c1ha2) edge[transform canvas={xshift=1mm}] (c1ha2v);
% second pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2) edge[transform canvas={xshift=-0.8mm,yshift=0.8mm}] (c2h);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2h) edge[transform canvas={xshift=-1mm}] (c2hc);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2hc) edge[transform canvas={xshift=-1.2mm}] (c2hf);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2hf) edge[transform canvas={xshift=-0.8mm,yshift=0.8mm}] (c2hfv);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2hc) edge[transform canvas={xshift=1.2mm}] (c2ha1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2ha1) edge[transform canvas={xshift=1.2mm}] (c2ha2);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2ha2) edge[transform canvas={xshift=1mm}] (c2ha2v);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2) edge[out=-15,in=-170] (c2c);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2c) edge[transform canvas={xshift=-1.2mm}] (c2cf);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2cf) edge[transform canvas={xshift=-1mm}] (c2cfv);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2c) edge[transform canvas={xshift=1.2mm}] (c2ca1);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2ca1) edge[transform canvas={xshift=1.2mm}] (c2ca2);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,red] (c2ca2) edge[transform canvas={xshift=1mm}] (c2ca2v);
% third pattern
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dashed,orange] (c2) edge[out=20,in=150] (c2b);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dashed,orange] (c2b) edge[transform canvas={xshift=-1.1mm}] (c2bv);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dashed,orange] (c2b) edge[transform canvas={xshift=0.8mm}] (c2bo);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dashed,orange] (c2b) edge[transform canvas={xshift=1mm}] (c2bb);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dashed,orange] (c2bb) edge[transform canvas={xshift=-1mm}] (c2bbv);
\path[-{Latex[length=1.5mm,width=1mm]},thick,relative,dashed,orange] (c2bb) edge[transform canvas={xshift=0.9mm}] (c2bbo);
  \end{forest}
  \caption{AST for the buggy \texttt{sum} program. Dotted arrows relate the correct values in the base case. Solid and dashed arrows denote two patterns describing incorrect reuse of the \code{Sum} variable.}
  \label{fig:sum}
\end{figure}

Several patterns capture this mistake. Solid red arrows in Fig.~\ref{fig:sum} show one example -- \code{Sum} returned by the predicate should not be the same as the \code{Sum} from the recursive call:

\begin{Verbatim}[fontfamily=sf]
(clause (head (compound (functor \code{sum}) (args (args var))))
    (compound (functor \code{sum}) (args (args (var)))))
\end{Verbatim}

\noindent
The second pattern, drawn with dashed orange arrows in the figure, indicates the likely error in the arithmetic expression:

\begin{Verbatim}[fontfamily=sf]
(clause (binop (var \code{Sum}) \code{is} (binop (var \code{Sum}) \code{+})))
\end{Verbatim}

The leftmost pattern in Fig.~\ref{fig:sum}, drawn with dotted blue arrows, describes the correct relation between the two constants in the base-case rule:

\begin{Verbatim}[fontfamily=sf]
(clause (head (compound (functor \code{sum}) (args \code{[]} (args \code{0})))))
\end{Verbatim}

\noindent
We include such patterns in our feature set to cover the base-case clauses in recursive programs, which often include no variables.

While the patterns used in this paper appear to be useful for analyzing Prolog programs, it is likely that other kinds of patterns will be needed for other programming languages. In Python, for example, variables can take on different values and be accessed from many places. This will likely require patterns relating more than two instances of a variable, or multiple variables.

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