This library provides aggregating operators over the solutions of a
predicate. The operations are a generalisation of the bagof/3, setof/3
and findall/3 builtin predicates. Aggregations that can be computed
incrementally avoid findall/3 and run in constant memory. 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:
smallest_country(Name, Area) :
aggregate(min(A, N), country(N, A), min(Area, Name)).
There are four aggregation predicates (aggregate/3, aggregate/4, aggregate_all/3 and aggregate/4), distinguished on two properties.
 aggregate vs. aggregate_all

The aggregate predicates use setof/3 (aggregate/4) or bagof/3
(aggregate/3), dealing with existential qualified variables
(
Var^Goal
) and providing multiple solutions for the remaining free
variables in Goal. The aggregate_all/3 predicate uses findall/3,
implicitly qualifying all free variables and providing exactly one
solution, while aggregate_all/4 uses sort/2 over solutions that
Discriminator (see below) generated using findall/3.
 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
country(belgium, 11000000)
may
succeed twice, we can use the following to avoid counting the
population of Belgium twice:
aggregate(sum(P), Name, country(Name, P), Total)
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 r(min(X), max(X))
computes both the minimum and maximum binding
for X.
 count
 Count number of solutions. Same as
sum(1)
.
 sum(Expr)
 Sum of Expr for all solutions.
 min(Expr)
 Minimum of Expr for all solutions.
 min(Expr, Witness)
 A term
min(Min, Witness)
, where Min is the minimal version
of Expr over all solutions, and Witness is any other template
applied to solutions that produced Min. If multiple
solutions provide the same minimum, Witness corresponds to
the first solution.
 max(Expr)
 Maximum of Expr for all solutions.
 max(Expr, Witness)
 As
min(Expr, Witness)
, but producing the maximum result.
 set(X)
 An ordered set with all solutions for X.
 bag(X)
 A list of all solutions for X.
Acknowledgements
The development of this library was sponsored by SecuritEase,
http://www.securitease.com
 Compatibility
  Quintus, SICStus 4. The forall/2 is a SWIProlog builtin and
term_variables/3 is a SWIProlog builtin with
different semantics.
 To be done
  Analysing the aggregation template and compiling a predicate
for the list aggregation can be done at compile time.
  aggregate_all/3 can be rewritten to run in constant space using
nonbacktrackable assignment on a term.
 aggregate(+Template, :Goal, Result) is nondet
 Aggregate bindings in Goal according to Template. The aggregate/3
version performs bagof/3 on Goal.
 aggregate(+Template, +Discriminator, :Goal, Result) is nondet
 Aggregate bindings in Goal according to Template. The aggregate/4
version performs setof/3 on Goal.
 aggregate_all(+Template, :Goal, Result) is semidet
 Aggregate bindings in Goal according to Template. The
aggregate_all/3 version performs findall/3 on Goal. Note that this
predicate fails if Template contains one or more of
min(X)
, max(X)
,
min(X,Witness)
or max(X,Witness)
and Goal has no solutions, i.e.,
the minimum and maximum of an empty set is undefined.
The Template values count
, sum(X)
, max(X)
, min(X)
, max(X,W)
and
min(X,W)
are processed incrementally rather than using findall/3 and
run in constant memory.
 aggregate_all(+Template, +Discriminator, :Goal, Result) is semidet
 Aggregate bindings in Goal according to Template. The
aggregate_all/4 version performs findall/3 followed by sort/2 on
Goal. See aggregate_all/3 to understand why this predicate can
fail.
 clean_body(+Goal0, Goal) is det[private]
 Remove redundant
true
from Goal0.
 template_to_pattern(+Template, Pattern, Post, Vars, Aggregate)[private]
 Determine which parts of the goal we must remember in the
findall/3 pattern.
 Arguments:

Post   is a bodyterm that evaluates expressions to reduce
storage requirements. 
Vars   is a list of intermediate variables that must be
added to the existential variables for bagof/3. 
Aggregate   defines the aggregation operation to execute. 
 needs_one(+Ops, OneOrZero)[private]
 If one of the operations in Ops needs at least one answer,
unify OneOrZero to 1. Else 0.
 aggregate_list(+Op, +List, Answer) is semidet[private]
 Aggregate the answer from the list produced by findall/3,
bagof/3 or setof/3. The latter two cases deal with compound
answers.
 To be done
  Compile code for incremental state update, which we will use
for aggregate_all/3 as well. We should be using goal_expansion
to generate these clauses.
 min_pair(+Pairs, Key, Value) is det[private]
 max_pair(+Pairs, Key, Value) is det[private]
 True if KeyValue has the smallest/largest key in Pairs. If
multiple pairs share the smallest/largest key, the first pair is
returned.
 step(+AggregateAction, +New, +State0, State1)[private]
 state0(+Op, State, Finish)[private]
 state1(+Op, +First, State, Finish)[private]
 foreach(:Generator, :Goal)
 True when the conjunction of instances of Goal created from
solutions for Generator is true. Except for term copying, this could
be implemented as below.
foreach(Generator, Goal) :
findall(Goal, Generator, Goals),
maplist(call, Goals).
The actual implementation uses findall/3 on a template created from
the variables shared between Generator and Goal. Subsequently, it
uses every instance of this template to instantiate Goal, call Goal
and undo only the instantiation of the template and not other
instantiations created by running Goal. Here is an example:
? foreach(between(1,4,X), dif(X,Y)), Y = 5.
Y = 5.
? foreach(between(1,4,X), dif(X,Y)), Y = 3.
false.
The predicate foreach/2 is mostly used if Goal performs
backtrackable destructive assignment on terms. Attributed variables
(underlying constraints) are an example. Another example of a
backtrackable data structure is in library(hashtable). If we care
only about the side effects (I/O, dynamic database, etc.) or the
truth value of Goal, forall/2 is a faster and simpler alternative.
If Goal instantiates its arguments it is will often fail as the
argument cannot be instantiated to multiple values. It is possible
to incrementally grow an argument:
? foreach(between(1,4,X), member(X, L)).
L = [1,2,3,4_].
Note that SWIProlog up to version 8.3.4 created copies of Goal
using copy_term/2 for each iteration.
 free_variables(:Generator, +Template, +VarList0, VarList) is det
 Find free variables in bagof/setof template. In order to handle
variables properly, we have to find all the universally
quantified variables in the Generator. All variables as yet
unbound are universally quantified, unless
 they occur in the template
 they are bound by X^P, setof/3, or bagof/3
free_variables(Generator, Template, OldList, NewList)
finds this
set using OldList as an accumulator.
 author
  Richard O'Keefe
  Jan Wielemaker (made some SWIProlog enhancements)
 license
  Public domain (from DEC10 library).
 To be done
  Distinguish between controlstructures and data terms.
  Exploit our builtin term_variables/2 at some places?
 term_is_free_of(+Term, +Var) is semidet[private]
 True if Var does not appear in Term. This has been rewritten
from the DEC10 library source to exploit our nondeterministic
arg/3.
 list_is_free_of(+List, +Var) is semidet[private]
 True if Var is not in List.
 sandbox:safe_meta(+Goal, Called) is semidet[multifile]
 Declare the aggregate metacalls safe. This cannot be proven due
to the manipulations of the argument Goal.