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\section{library(aggregate): Aggregation operators on backtrackable predicates}

\label{sec:aggregate}

\begin{tags}
    \tag{Compatibility}
Quintus, SICStus 4. The \predref{forall}{2} is a SWI-Prolog built-in and
\predref{term_variables}{3} is a SWI-Prolog built-in with
\textbf{different semantics}.\mtag{To be done}- Analysing the aggregation template and compiling a predicate
for the list aggregation can be done at compile time. \\- \predref{aggregate_all}{3} can be rewritten to run in constant space using
non-backtrackable assignment on a term.
\end{tags}

This library provides aggregating operators over the solutions of a
predicate. The operations are a generalisation of the \predref{bagof}{3}, \predref{setof}{3}
and \predref{findall}{3} built-in predicates. The defined aggregation operations
are counting, computing the sum, minimum, maximum, a bag of solutions
and a set of solutions. We first give a simple example, computing the
country with the smallest area:

\begin{code}
smallest_country(Name, Area) :-
        aggregate(min(A, N), country(N, A), min(Area, Name)).
\end{code}

There are four aggregation predicates (\predref{aggregate}{3}, \predref{aggregate}{4}, \predref{aggregate_all}{3} and \predref{aggregate}{4}), distinguished on two properties.

\begin{description}
    \item[aggregate vs. aggregate_all] 
The aggregate predicates use \predref{setof}{3} (\predref{aggregate}{4}) or \predref{bagof}{3}
(\predref{aggregate}{3}), dealing with existential qualified variables
(Var\Shat{}Goal) and providing multiple solutions for the remaining free
variables in Goal. The \predref{aggregate_all}{3} predicate uses \predref{findall}{3},
implicitly qualifying all free variables and providing exactly one
solution, while \predref{aggregate_all}{4} uses \predref{sort}{2} over solutions that
Discriminator (see below) generated using \predref{findall}{3}.
    \item[The Discriminator argument] 
The versions with 4 arguments deduplicate redundant solutions of
Goal. Solutions for which both the template variables and
Discriminator are identical will be treated as one solution. For
example, if we wish to compute the total population of all
countries, and for some reason \verb$country(belgium, 11000000)$ may
succeed twice, we can use the following to avoid counting the
population of Belgium twice:

\begin{code}
    aggregate(sum(P), Name, country(Name, P), Total)
\end{code}

\end{description}

All aggregation predicates support the following operators below in
Template. In addition, they allow for an arbitrary named compound term,
where each of the arguments is a term from the list below. For example,
the term \verb$r(min(X), max(X))$ computes both the minimum and maximum binding
for X.

\begin{description}
    \termitem{count}{}
Count number of solutions. Same as \verb$sum(1)$.
    \termitem{sum}{Expr}
Sum of \arg{Expr} for all solutions.
    \termitem{min}{Expr}
Minimum of \arg{Expr} for all solutions.
    \termitem{min}{Expr, Witness}
A term \verb$min(Min, Witness)$, where Min is the minimal version
of \arg{Expr} over all solutions, and \arg{Witness} is any other template
applied to solutions that produced Min. If multiple
solutions provide the same minimum, \arg{Witness} corresponds to
the first solution.
    \termitem{max}{Expr}
Maximum of \arg{Expr} for all solutions.
    \termitem{max}{Expr, Witness}
As \verb$min(Expr, Witness)$, but producing the maximum result.
    \termitem{set}{X}
An ordered set with all solutions for \arg{X}.
    \termitem{bag}{X}
A list of all solutions for \arg{X}.
\end{description}

\textbf{Acknowledgements}

\textit{The development of this library was sponsored by SecuritEase,
\url{http://www.securitease.com}
}\vspace{0.7cm}

\begin{description}
    \predicate[nondet]{aggregate}{3}{+Template, :Goal, -Result}
Aggregate bindings in \arg{Goal} according to \arg{Template}. The \predref{aggregate}{3}
version performs \predref{bagof}{3} on \arg{Goal}.

    \predicate[nondet]{aggregate}{4}{+Template, +Discriminator, :Goal, -Result}
Aggregate bindings in \arg{Goal} according to \arg{Template}. The \predref{aggregate}{4}
version performs \predref{setof}{3} on \arg{Goal}.

    \predicate[semidet]{aggregate_all}{3}{+Template, :Goal, -Result}
Aggregate bindings in \arg{Goal} according to \arg{Template}. The
\predref{aggregate_all}{3} version performs \predref{findall}{3} on \arg{Goal}. Note that
this predicate fails if \arg{Template} contains one or more of \verb$min(X)$,
\verb$max(X)$, \verb$min(X,Witness)$ or \verb$max(X,Witness)$ and \arg{Goal} has no
solutions, i.e., the minumum and maximum of an empty set is
undefined.

    \predicate[semidet]{aggregate_all}{4}{+Template, +Discriminator, :Goal, -Result}
Aggregate bindings in \arg{Goal} according to \arg{Template}. The
\predref{aggregate_all}{4} version performs \predref{findall}{3} followed by \predref{sort}{2} on
\arg{Goal}. See \predref{aggregate_all}{3} to understand why this predicate can
fail.

    \predicate{foreach}{2}{:Generator, :Goal}
True if conjunction of results is true. Unlike \predref{forall}{2}, which
runs a failure-driven loop that proves \arg{Goal} for each solution of
\arg{Generator}, \predref{foreach}{2} creates a conjunction. Each member of the
conjunction is a copy of \arg{Goal}, where the variables it shares
with \arg{Generator} are filled with the values from the corresponding
solution.

The implementation executes \predref{forall}{2} if \arg{Goal} does not contain
any variables that are not shared with \arg{Generator}.

Here is an example:

\begin{code}
?- foreach(between(1,4,X), dif(X,Y)), Y = 5.
Y = 5.
?- foreach(between(1,4,X), dif(X,Y)), Y = 3.
false.
\end{code}

\begin{tags}
    \tag{bug}
\arg{Goal} is copied repeatedly, which may cause problems if
attributed variables are involved.
\end{tags}

    \predicate[det]{free_variables}{4}{:Generator, +Template, +VarList0, -VarList}
Find free variables in bagof/setof template. In order to handle
variables properly, we have to find all the universally
quantified variables in the \arg{Generator}. All variables as yet
unbound are universally quantified, unless

\begin{enumerate}
    \item they occur in the template
    \item they are bound by X\Shat{}P, \predref{setof}{3}, or \predref{bagof}{3}
\end{enumerate}

\verb$free_variables(Generator, Template, OldList, NewList)$ finds this
set using OldList as an accumulator.

\begin{tags}
\mtag{author}- Richard O'Keefe \\- Jan Wielemaker (made some SWI-Prolog enhancements)
    \tag{license}
Public domain (from DEC10 library).\mtag{To be done}- Distinguish between control-structures and data terms. \\- Exploit our built-in \predref{term_variables}{2} at some places?
\end{tags}

\qpredicate[semidet,multifile]{sandbox}{safe_meta}{2}{+Goal, -Called}Declare the aggregate meta-calls safe. This cannot be proven due
to the manipulations of the argument \arg{Goal}.
\end{description}

