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- library(aggregate): Aggregation operators on backtrackable predicates
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## A.1 library(aggregate): Aggregation operators on backtrackable predicates

- Compatibility
- Quintus, SICStus 4. The forall/2
is a SWI-Prolog built-in and
term_variables/3
is a SWI-Prolog built-in 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 non-backtrackable assignment on a term.

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 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:

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*

- [nondet]
**aggregate**(`+Template, :Goal, -Result`) - Aggregate bindings in
`Goal`according to`Template`. The aggregate/3 version performs bagof/3 on`Goal`. - [nondet]
**aggregate**(`+Template, +Discriminator, :Goal, -Result`) - Aggregate bindings in
`Goal`according to`Template`. The aggregate/4 version performs setof/3 on`Goal`. - [semidet]
**aggregate_all**(`+Template, :Goal, -Result`) - 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 minumum and maximum of an empty set is undefined. - [semidet]
**aggregate_all**(`+Template, +Discriminator, :Goal, -Result`) - 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. **foreach**(`:Generator, :Goal`)- True if conjunction of results is true. Unlike forall/2,
which runs a failure-driven loop that proves
`Goal`for each solution of`Generator`, foreach/2 creates a conjunction. Each member of the conjunction is a copy of`Goal`, where the variables it shares with`Generator`are filled with the values from the corresponding solution.The implementation executes forall/2 if

`Goal`does not contain any variables that are not shared with`Generator`.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.

- bug
`Goal`is copied repeatedly, which may cause problems if attributed variables are involved.

- [det]
**free_variables**(`: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
`Generator`. All variables as yet unbound are universally quantified, unless`free_variables(Generator, Template, OldList, NewList)`

finds this set using OldList as an accumulator.- author
- - Richard O'Keefe

- Jan Wielemaker (made some SWI-Prolog enhancements) - license
- Public domain (from DEC10 library).
- To be done
- - Distinguish between control-structures and data terms.

- Exploit our built-in term_variables/2 at some places?

- [semidet,multifile]sandbox:
**safe_meta**(`+Goal, -Called`) - Declare the aggregate meta-calls safe. This cannot be proven due to the
manipulations of the argument
`Goal`.