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Merge pull request #1724 from triska/clpz_corrections

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CLP(ℤ) corrections
This commit is contained in:
Mark Thom
2023-01-28 20:11:48 +01:00
committed by GitHub
parent 2bdba261ea 814b631543
commit ac0cea8a73
1 changed files with 79 additions and 62 deletions
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src/lib/clpz.pl
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@@ -282,39 +282,39 @@ exclude_([L|Ls0], Goal, Ls) :-
:- op(700, xfx, cis_lt).
:- op(1200, xfx, ++>).
/** <module> Constraint Logic Programming over Integers
/** Constraint Logic Programming over Integers
## Introduction
This library provides CLP(ℤ): Constraint Logic Programming over
Integers.
CLP(ℤ) is an instance of the general CLP(.) scheme, extending logic
CLP(ℤ) is an instance of the general CLP(_X_) scheme, extending logic
programming with reasoning over specialised domains. CLP(ℤ) lets us
reason about *integers* in a way that honors the relational nature
of Prolog.
There are two major use cases of CLP(ℤ) constraints:
1. [*declarative integer arithmetic*](<#clpz-integer-arith>)
1. [*declarative integer arithmetic*](#clpz-integer-arith)
2. solving *combinatorial problems* such as planning, scheduling
and allocation tasks.
The predicates of this library can be classified as:
* _arithmetic_ constraints like `#=/2`, `#>/2` and `#\=/2`
* _arithmetic_ constraints like `(#=)/2`, `(#>)/2` and `(#\=)/2`
* the _membership_ constraints `in/2` and `ins/2`
* the _enumeration_ predicates `indomain/1`, `label/1` and `labeling/2`
* _combinatorial_ constraints like `all_distinct/1` and `global_cardinality/2`
* _reification_ predicates such as `#<==>/2`
* _reflection_ predicates such as `fd_dom/2`
In most cases, [_arithmetic constraints_](<#clpz-arith-constraints>)
In most cases, [_arithmetic constraints_](#clpz-arith-constraints)
are the only predicates you will ever need from this library. When
reasoning over integers, simply replace low-level arithmetic
predicates like `(is)/2` and `(>)/2` by the corresponding CLP(ℤ)
constraints like #=/2 and #>/2 to honor and preserve declarative
constraints like `(#=)/2` and `(#>)/2` to honor and preserve declarative
properties of your programs. For satisfactory performance, arithmetic
constraints are implicitly rewritten at compilation time so that
low-level fallback predicates are automatically used whenever
@@ -323,7 +323,7 @@ possible.
Almost all Prolog programs also reason about integers. Therefore, it
is highly advisable that you make CLP(ℤ) constraints available in all
your programs. One way to do this is to put the following directive in
your =|~/.scryerrc|= initialisation file:
your `~/.scryerrc` initialisation file:
```
:- use_module(library(clpz)).
@@ -362,7 +362,8 @@ constraints is to use the dedicated `clpz` tag on
foremost CLP(ℤ) experts regularly participate in these discussions
and will help you for free on this platform.
## Arithmetic constraints {#clpz-arith-constraints}
{#clpz-arith-constraints}
## Arithmetic constraints
In modern Prolog systems, *arithmetic constraints* subsume and
supersede low-level predicates over integers. The main advantage of
@@ -408,14 +409,15 @@ The bitwise operations `(\)/1`, `(/\)/2`, `(\/)/2`, `(>>)/2`,
`(<<)/2`, `lsb/1`, `msb/1`, `popcount/1` and `(xor)/2` are also
supported.
## Declarative integer arithmetic {#clpz-integer-arith}
{#clpz-integer-arith}
## Declarative integer arithmetic
The [_arithmetic constraints_](<#clpz-arith-constraints>) #=/2, #>/2
etc. are meant to be used _instead_ of the primitives `(is)/2`,
`(=:=)/2`, `(>)/2` etc. over integers. Almost all Prolog programs also
reason about integers. Therefore, it is recommended that you put the
following directive in your =|~/.scryerrc|= initialisation file to make
CLP(ℤ) constraints available in all your programs:
The [_arithmetic constraints_](#clpz-arith-constraints) `(#=)/2`,
`(#>)/2` etc. are meant to be used _instead_ of the primitives
`(is)/2`, `(=:=)/2`, `(>)/2` etc. over integers. Almost all Prolog
programs also reason about integers. Therefore, it is recommended that
you put the following directive in your =|~/.scryerrc|= initialisation
file to make CLP(ℤ) constraints available in all your programs:
```
:- use_module(library(clpz)).
@@ -460,7 +462,7 @@ and should therefore be deferred to more advanced lectures.
For supported expressions, CLP(ℤ) constraints are drop-in
replacements of these low-level arithmetic predicates, often yielding
more general programs. See [`n_factorial/2`](<#clpz-factorial>) for an
more general programs. See [`n_factorial/2`](#clpz-factorial) for an
example.
This library uses goal_expansion/2 to automatically rewrite
@@ -497,9 +499,10 @@ primitives by providing declarative alternatives that are meant to be
used instead.
## Example: Factorial relation {#clpz-factorial}
{#clpz-factorial}
## Example: Factorial relation
We illustrate the benefit of using #=/2 for more generality with a
We illustrate the benefit of using `(#=)/2` for more generality with a
simple example.
Consider first a rather conventional definition of `n_factorial/2`,
@@ -560,7 +563,7 @@ us from _all_ procedural phenomena. For example, the two programs do
not even have the same _termination properties_ in all cases.
Instead, the primary benefit of CLP(ℤ) constraints is that they allow
you to try different execution orders and apply [*declarative
debugging*](https://www.metalevel.at/prolog/debugging.html)
debugging*](https://www.metalevel.at/prolog/debugging)
techniques _at all_! Reordering goals (and clauses) can significantly
impact the performance of Prolog programs, and you are free to try
different variants if you use declarative approaches. Moreover, since
@@ -568,27 +571,29 @@ all CLP(ℤ) constraints _always terminate_, placing them earlier can
at most _improve_, never worsen, the termination properties of your
programs. An additional benefit of CLP(ℤ) constraints is that they
eliminate the complexity of introducing `(is)/2` and `(=:=)/2` to
beginners, since _both_ predicates are subsumed by #=/2 when reasoning
over integers.
beginners, since _both_ predicates are subsumed by `(#=)/2` when
reasoning over integers.
## Combinatorial constraints {#clpz-combinatorial}
{#clpz-combinatorial}
## Combinatorial constraints
In addition to subsuming and replacing low-level arithmetic
predicates, CLP(ℤ) constraints are often used to solve combinatorial
problems such as planning, scheduling and allocation tasks. Among the
most frequently used *combinatorial constraints* are all_distinct/1,
global_cardinality/2 and cumulative/2. This library also provides
several other constraints like disjoint2/1 and automaton/8, which are
most frequently used *combinatorial constraints* are `all_distinct/1`,
`global_cardinality/2` and `cumulative/2`. This library also provides
several other constraints like `disjoint2/1` and `automaton/8`, which are
useful in more specialized applications.
## Domains {#clpz-domains}
{#clpz-domains}
## Domains
Each CLP(ℤ) variable has an associated set of admissible integers,
which we call the variable's *domain*. Initially, the domain of each
CLP(ℤ) variable is the set of _all_ integers. CLP(ℤ) constraints
like #=/2, #>/2 and #\=/2 can at most reduce, and never extend, the
domains of their arguments. The constraints in/2 and ins/2 let us
explicitly state domains of CLP(ℤ) variables. The process of
CLP(ℤ) variable is the set of _all_ integers. CLP(ℤ) constraints like
`(#=)/2`, `(#>)/2` and `(#\=)/2` can at most reduce, and never extend,
the domains of their arguments. The constraints `(in)/2` and `(ins)/2`
let us explicitly state domains of CLP(ℤ) variables. The process of
determining and adjusting domains of variables is called constraint
*propagation*, and it is performed automatically by this library. When
the domain of a variable contains only one element, then the variable
@@ -597,7 +602,8 @@ is automatically unified to that element.
Domains are taken into account when further constraints are stated,
and by enumeration predicates like labeling/2.
## Example: Sudoku {#clpz-sudoku}
{#clpz-sudoku}
## Example: Sudoku
As another example, consider _Sudoku_: It is a popular puzzle
over integers that can be easily solved with CLP(ℤ) constraints.
@@ -650,7 +656,8 @@ In this concrete case, the constraint solver is strong enough to find
the unique solution without any search.
## Residual goals {#clpz-residual-goals}
{#clpz-residual-goals}
## Residual goals
Here is an example session with a few queries and their answers:
@@ -683,10 +690,10 @@ goals, it is clear that the constraint solver has deduced additional
domain restrictions in many cases.
To inspect residual goals, it is best to let the toplevel display them
for us. Wrap the call of your predicate into call_residue_vars/2 to
for us. Wrap the call of your predicate into `call_residue_vars/2` to
make sure that all constrained variables are displayed. To make the
constraints a variable is involved in available as a Prolog term for
further reasoning within your program, use copy_term/3. For example:
further reasoning within your program, use `copy_term/3`. For example:
```
?- X #= Y + Z, X in 0..5, copy_term([X,Y,Z], [X,Y,Z], Gs).
@@ -695,12 +702,13 @@ X in 0..5,
Y+Z#=X.
```
This library also provides _reflection_ predicates (like fd_dom/2,
fd_size/2 etc.) with which we can inspect a variable's current
This library also provides _reflection_ predicates (like `fd_dom/2`,
`fd_size/2` etc.) with which we can inspect a variable's current
domain. These predicates can be useful if you want to implement your
own labeling strategies.
## Core relations and search {#clpz-search}
{#clpz-search}
## Core relations and search
Using CLP(ℤ) constraints to solve combinatorial tasks typically
consists of two phases:
@@ -732,7 +740,7 @@ puzzle([S,E,N,D] + [M,O,R,E] = [M,O,N,E,Y]) :-
M #\= 0, S #\= 0.
```
Notice that we are _not_ using labeling/2 in this predicate, so that
Notice that we are _not_ using `labeling/2` in this predicate, so that
we can first execute and observe the modeling part in isolation.
Sample query and its result (actual variables replaced for
readability):
@@ -772,7 +780,8 @@ to reduce the domains of remaining variables to singleton sets. In
general though, it is necessary to label all variables to obtain
ground solutions.
## Example: Eight queens puzzle {#clpz-n-queens}
{#clpz-n-queens}
## Example: Eight queens puzzle
We illustrate the concepts of the preceding sections by means of the
so-called _eight queens puzzle_. The task is to place 8 queens on an
@@ -851,9 +860,10 @@ separated the core relation from the actual search.
## Optimisation {#clpz-optimisation}
{#clpz-optimisation}
## Optimisation
We can use labeling/2 to minimize or maximize the value of a CLP(ℤ)
We can use `labeling/2` to minimize or maximize the value of a CLP(ℤ)
expression, and generate solutions in increasing or decreasing order
of the value. See the labeling options `min(Expr)` and `max(Expr)`,
respectively.
@@ -868,7 +878,7 @@ If necessary, we can use `once/1` to commit to the first optimal
solution. However, it is often very valuable to see alternative
solutions that are _also_ optimal, so that we can choose among optimal
solutions by other criteria. For the sake of
[*purity*](https://www.metalevel.at/prolog/purity.html) and
[*purity*](https://www.metalevel.at/prolog/purity) and
completeness, we recommend to avoid `once/1` and other constructs that
lead to impurities in CLP(ℤ) programs.
@@ -876,12 +886,14 @@ Related to optimisation with CLP(ℤ) constraints are `library(simplex)`
and CLP(Q) which reason about _linear_ constraints over rational
numbers.
## Reification {#clpz-reification}
{#clpz-reification}
## Reification
The constraints in/2, #=/2, #\=/2, #</2, #>/2, #=</2, and #>=/2 can be
_reified_, which means reflecting their truth values into Boolean
values represented by the integers 0 and 1. Let P and Q denote
reifiable constraints or Boolean variables, then:
The constraints `(in)/2`, `(#=)/2`, `(#\=)/2`, `(#<)/2`, `(#>)/2`,
`(#=<)/2`, and `(#>=)/2` can be _reified_, which means reflecting
their truth values into Boolean values represented by the integers 0
and 1. Let P and Q denote reifiable constraints or Boolean variables,
then:
| #\ Q | True iff Q is false |
| P #\/ Q | True iff either P or Q |
@@ -896,7 +908,8 @@ The constraints of this table are reifiable as well.
When reasoning over Boolean variables, also consider using CLP(B)
constraints as provided by `library(clpb)`.
## Enabling monotonic CLP(ℤ) {#clpz-monotonicity}
{#clpz-monotonicity}
## Enabling monotonic CLP(ℤ)
In the default execution mode, CLP(ℤ) constraints still exhibit some
non-relational properties. For example, _adding_ constraints can yield
@@ -932,7 +945,8 @@ expressions with the functor `(?)/1` or `(#)/1`. For example:
The wrapper can be omitted for variables that are already constrained
to integers.
## Custom constraints {#clpz-custom-constraints}
{#clpz-custom-constraints}
## Custom constraints
We can define custom constraints. The mechanism to do this is not yet
finalised, and we welcome suggestions and descriptions of use cases
@@ -958,19 +972,19 @@ clpz:run_propagator(oneground(X, Y, Z), MState) :-
).
```
First, clpz:make_propagator/2 is used to transform a user-defined
First, `clpz:make_propagator/2` is used to transform a user-defined
representation of the new constraint to an internal form. With
clpz:init_propagator/2, this internal form is then attached to X and
`clpz:init_propagator/2`, this internal form is then attached to X and
Y. From now on, the propagator will be invoked whenever the domains of
X or Y are changed. Then, clpz:trigger_once/1 is used to give the
X or Y are changed. Then, `clpz:trigger_once/1` is used to give the
propagator its first chance for propagation even though the variables'
domains have not yet changed. Finally, clpz:run_propagator/2 is
domains have not yet changed. Finally, `clpz:run_propagator/2` is
extended to define the actual propagator. As explained, this predicate
is automatically called by the constraint solver. The first argument
is the user-defined representation of the constraint as used in
clpz:make_propagator/2, and the second argument is a mutable state
`clpz:make_propagator/2`, and the second argument is a mutable state
that can be used to prevent further invocations of the propagator when
the constraint has become entailed, by using clpz:kill/1. An example
the constraint has become entailed, by using `clpz:kill/1`. An example
of using the new constraint:
```
@@ -2922,7 +2936,7 @@ X #=< Y :- Y #>= X.
%% #=(?X, ?Y)
%
% The arithmetic expression X equals Y. When reasoning over integers,
% replace is/2 by #=/2 to obtain more general relations.
% replace `(is)/2` by `(#=)/2` to obtain more general relations.
X #= Y :- clpz_equal(X, Y).
@@ -2962,8 +2976,8 @@ expr_conds(A0>>B0, A>>B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0/\B0, A/\B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0\/B0, A\/B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(xor(A0,B0), xor(A,B)) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(lsb(A0), lsb(A)) --> expr_conds(A0, A).
expr_conds(msb(A0), msb(A)) --> expr_conds(A0, A).
% expr_conds(lsb(A0), lsb(A)) --> expr_conds(A0, A).
% expr_conds(msb(A0), msb(A)) --> expr_conds(A0, A).
expr_conds(popcount(A0), Count) -->
expr_conds(A0, A),
[I is A, arithmetic:popcount(I, Count)].
@@ -3539,8 +3553,8 @@ parse_reified(E, R, D,
m(A^B) => [d(D), p(pexp(A,B,R)), a(A,B,R)],
% bitwise operations
m(\A) => [function(D,\,A,R)],
m(msb(A)) => [function(D,msb,A,R)],
m(lsb(A)) => [function(D,lsb,A,R)],
m(msb(A)) => [g(#A#>0) ,function(D,msb,A,R)],
m(lsb(A)) => [g(#A#>0), function(D,lsb,A,R)],
m(popcount(A)) => [function(D,popcount,A,R)],
m(sign(A)) => [function(D,sign,A,R)],
m(A<<B) => [function(D,<<,A,B,R)],
@@ -5683,8 +5697,11 @@ run_propagator(pfunction(Op,A,B,R), MState) -->
run_propagator(pfunction(Op,A,R), MState) -->
( integer(A) ->
kill(MState),
Expr =.. [Op,A],
R is Expr
( Op == msb -> { msb(A, R) }
; Op == lsb -> { lsb(A, R) }
; Expr =.. [Op,A],
R is Expr
)
; []
).