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Merge pull request #1689 from triska/clpb_doc

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DOC: preliminary CLP(B) documentation in DocLog format
This commit is contained in:
Mark Thom
2023-01-24 22:51:35 +01:00
committed by GitHub
parent ec55765cb2 1c08b56e05
commit df266378c4
2 changed files with 283 additions and 31 deletions
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310
src/lib/clpb.pl
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@@ -1,6 +1,6 @@
/* CLP(B): Constraint Logic Programming over Boolean Variables
Copyright (C): 2019 Markus Triska
Copyright (C): 2019-2023 Markus Triska
All rights reserved.
E-mail: triska@metalevel.at
@@ -105,60 +105,309 @@ goal_expansion(del_attr(Var, Module), (var(Var) -> put_atts(Var, -Access);true))
Access =.. [Module,_].
/**
/** Constraint Logic Programming over Boolean variables
## Introduction
This library provides CLP(B), Constraint Logic Programming over
Boolean variables. It can be used to model and solve combinatorial
problems such as verification, allocation and covering tasks.
CLP(B) is an instance of the general CLP(_X_) scheme,
extending logic programming with reasoning over specialised domains.
The implementation is based on reduced and ordered Binary Decision
Diagrams (BDDs).
Benchmarks and usage examples of this library are available from:
[*https://www.metalevel.at/clpb/*](https://www.metalevel.at/clpb/)
## Boolean expressions
A _Boolean expression_ is one of:
| `0` | false |
| `1` | true |
| _variable_ | unknown truth value |
| _atom_ | universally quantified variable |
| ~ _Expr_ | logical NOT |
| _Expr_ + _Expr_ | logical OR |
| _Expr_ * _Expr_ | logical AND |
| _Expr_ # _Expr_ | exclusive OR |
| _Var_ ^ _Expr_ | existential quantification |
| _Expr_ =:= _Expr_ | equality |
| _Expr_ =\= _Expr_ | disequality (same as #) |
| _Expr_ =< _Expr_ | less or equal (implication) |
| _Expr_ >= _Expr_ | greater or equal |
| _Expr_ < _Expr_ | less than |
| _Expr_ > _Expr_ | greater than |
| card(Is,Exprs) | cardinality constraint (_see below_) |
| `+(Exprs)` | n-fold disjunction (_see below_) |
| `*(Exprs)` | n-fold conjunction (_see below_) |
where _Expr_ again denotes a Boolean expression.
The Boolean expression `card(Is,Exprs)` is true iff the number of true
expressions in the list `Exprs` is a member of the list `Is` of
integers and integer ranges of the form `From-To`. For example, to
state that precisely two of the three variables `X`, `Y` and `Z` are
`true`, you can use `sat(card([2],[X,Y,Z]))`.
`+(Exprs)` and `*(Exprs)` denote, respectively, the disjunction and
conjunction of all elements in the list `Exprs` of Boolean
expressions.
Atoms denote parametric values that are universally quantified. All
universal quantifiers appear implicitly in front of the entire
expression. In residual goals, universally quantified variables always
appear on the right-hand side of equations. Therefore, they can be
used to express functional dependencies on input variables.
## Interface predicates
The most frequently used CLP(B) predicates are:
* `sat(+Expr)`
True iff the Boolean expression Expr is satisfiable.
* `taut(+Expr, -T)`
If Expr is a tautology with respect to the posted constraints, succeeds
with *T = 1*. If Expr cannot be satisfied, succeeds with *T = 0*.
Otherwise, it fails.
* `labeling(+Vs)`
Assigns truth values to the variables Vs such that all constraints
are satisfied.
The unification of a CLP(B) variable _X_ with a term _T_ is equivalent
to posting the constraint sat(X=:=T).
## Examples
Here is an example session with a few queries and their answers:
```
?- use_module(library(clpb)).
true.
?- sat(X*Y).
X = 1, Y = 1.
?- sat(X * ~X).
false.
?- taut(X * ~X, T).
T = 0, clpb:sat(X=:=X).
?- sat(X^Y^(X+Y)).
clpb:sat(X=:=X), clpb:sat(Y=:=Y).
?- sat(X*Y + X*Z), labeling([X,Y,Z]).
X = 1, Y = 0, Z = 1
; X = 1, Y = 1, Z = 0
; X = 1, Y = 1, Z = 1.
?- sat(X =< Y), sat(Y =< Z), taut(X =< Z, T).
T = 1, clpb:sat(X=:=X*Y), clpb:sat(Y=:=Y*Z).
?- sat(1#X#a#b).
sat(X=:=a#b).
```
The pending residual goals constrain remaining variables to Boolean
expressions and are declaratively equivalent to the original query.
The last example illustrates that when applicable, remaining variables
are expressed as functions of universally quantified variables.
## Obtaining BDDs
By default, CLP(B) residual goals appear in (approximately) algebraic
normal form (ANF). This projection is often computationally expensive.
We can assert `clpb:clpb_residuals(bdd)` to see the BDD representation
of all constraints. This results in faster projection to residual
goals, and is also useful for learning more about BDDs. For example:
```
?- asserta(clpb:clpb_residuals(bdd)).
true.
?- sat(X#Y).
node(3)- (v(X, 0)->node(2);node(1)),
node(1)- (v(Y, 1)->true;false),
node(2)- (v(Y, 1)->false;true).
```
Note that this representation cannot be pasted back on the toplevel,
and its details are subject to change. Use copy_term/3 to obtain
such answers as Prolog terms.
The variable order of the BDD is determined by the order in which the
variables first appear in constraints. To obtain different orders,
we can for example use:
```
?- sat(+[1,Y,X]), sat(X#Y).
node(3)- (v(Y, 0)->node(2);node(1)),
node(1)- (v(X, 1)->true;false),
node(2)- (v(X, 1)->false;true).
```
## Enabling monotonic CLP(B)
In the default execution mode, CLP(B) constraints are _not_ monotonic.
This means that _adding_ constraints can yield new solutions. For
example:
```
?- sat(X=:=1), X = 1+0.
false.
?- X = 1+0, sat(X=:=1), X = 1+0.
X = 1+0.
```
This behaviour is highly problematic from a logical point of view, and
it may render [*declarative
debugging*](https://www.metalevel.at/prolog/debugging)
techniques inapplicable.
Assert `clpb:monotonic` to make CLP(B) *monotonic*. If this mode is
enabled, then you must wrap CLP(B) variables with the functor
`v/1`. For example:
```
?- asserta(clpb:monotonic).
true.
?- sat(v(X)=:=1#1).
X = 0.
```
## Example: Pigeons
In this example, we are attempting to place _I_ pigeons into _J_ holes
in such a way that each hole contains at most one pigeon. One
interesting property of this task is that it can be formulated using
only _cardinality constraints_ (`card/2`). Another interesting aspect
is that this task has no short resolution refutations in general.
In the following, we use [*Prolog DCG
notation*](https://www.metalevel.at/prolog/dcg) to describe a
list `Cs` of CLP(B) constraints that must all be satisfied.
```
:- use_module(library(clpb)).
:- use_module(library(clpz)).
:- use_module(library(lists)).
:- use_module(library(dcgs)).
pigeon(I, J, Rows, Cs) :-
length(Rows, I), length(Row, J),
maplist(same_length(Row), Rows),
transpose(Rows, TRows),
phrase((all_cards(Rows,[1]),all_cards(TRows,[0,1])), Cs).
all_cards([], _) --> [].
all_cards([Ls|Lss], Cs) --> [card(Cs,Ls)], all_cards(Lss, Cs).
```
Example queries:
```
?- pigeon(9, 8, Rows, Cs), sat(*(Cs)).
false.
?- pigeon(2, 3, Rows, Cs), sat(*(Cs)),
append(Rows, Vs), labeling(Vs),
maplist(portray_clause, Rows).
[0,0,1].
[0,1,0].
etc.
```
## Example: Boolean circuit
Consider a Boolean circuit that express the Boolean function =|XOR|=
with 4 =|NAND|= gates. We can model such a circuit with CLP(B)
constraints as follows:
```
:- use_module(library(clpb)).
nand_gate(X, Y, Z) :- sat(Z =:= ~(X*Y)).
xor(X, Y, Z) :-
nand_gate(X, Y, T1),
nand_gate(X, T1, T2),
nand_gate(Y, T1, T3),
nand_gate(T2, T3, Z).
```
Using universally quantified variables, we can show that the circuit
does compute =|XOR|= as intended:
```
?- xor(x, y, Z).
sat(Z=:=x#y).
```
## Acknowledgments
The interface predicates of this library follow the example of
[*SICStus Prolog*](https://sicstus.sics.se).
Use SICStus Prolog for higher performance in many cases.
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Each CLP(B) variable belongs to exactly one BDD. Each CLP(B)
variable gets an attribute (in module "clpb") of the form:
```
index_root(Index,Root)
```
index_root(Index,Root)
where Index is the variable's unique integer index, and Root is the
root of the BDD that the variable belongs to.
Each CLP(B) variable also gets an attribute in module `clpb_hash`: an
Each CLP(B) variable also gets an attribute in module clpb_hash: an
association table node(LID,HID) -> Node, to keep the BDD reduced.
The association table of each variable must be rebuilt on occasion
to remove nodes that are no longer reachable. We rebuild the
association tables of involved variables after BDDs are merged to
build a new root. This only serves to reclaim memory: Keeping a
node in a local table even when it no longer occurs in any BDD does
not affect the solver's correctness. However, `apply_shortcut/4`
not affect the solver's correctness. However, apply_shortcut/4
relies on the invariant that every node that occurs in the relevant
BDDs is also registered in the table of its branching variable.
A root is a logical variable with a single attribute ("clpb\_bdd")
A root is a logical variable with a single attribute ("clpb_bdd")
of the form:
```
Sat-BDD
```
Sat-BDD
where Sat is the SAT formula (in original form) that corresponds to
BDD. Sat is necessary to rebuild the BDD after variable aliasing,
and to project all remaining constraints to a list of `sat/1` goals.
and to project all remaining constraints to a list of sat/1 goals.
Finally, a BDD is either:
* The integers 0 or 1, denoting false and true, respectively, or
* A node of the form
*) The integers 0 or 1, denoting false and true, respectively, or
*) A node of the form
```
node(ID, Var, Low, High, Aux)
```
Where ID is the node's unique integer ID, Var is the
node's branching variable, and Low and High are the
node's low (Var = 0) and high (Var = 1) children. Aux
is a free variable, one for each node, that can be used
to attach attributes and store intermediate results.
node(ID, Var, Low, High, Aux)
Where ID is the node's unique integer ID, Var is the
node's branching variable, and Low and High are the
node's low (Var = 0) and high (Var = 1) children. Aux
is a free variable, one for each node, that can be used
to attach attributes and store intermediate results.
Variable aliasing is treated as a conjunction of corresponding SAT
formulae.
You should think of CLP(B) as a potentially vast collection of BDDs
that can range from small to gigantic in size, and which can merge.
*/
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Type checking.
@@ -1117,16 +1366,14 @@ indomain(1).
%
% ```
% ?- sat(A =< B), Vs = [A,B], sat_count(+[1|Vs], Count).
% Vs = [A, B],
% Count = 3,
% sat(A=:=A*B).
% Vs = [A,B], Count = 3, clpb:sat(A=:=A*B).
%
% ?- length(Vs, 120),
% sat_count(+Vs, CountOr),
% sat_count(*(Vs), CountAnd).
% Vs = [...],
% CountOr = 1329227995784915872903807060280344575,
% CountAnd = 1.
% Vs = [...],
% CountOr = 1329227995784915872903807060280344575,
% CountAnd = 1.
% ```
@@ -1266,7 +1513,8 @@ random_bindings(VNum, Node) -->
%
% ```
% ?- sat(A#B), weighted_maximum([1,2,1], [A,B,C], Maximum).
% A = 0, B = 1, C = 1, Maximum = 3.
% A = 0, B = 1, C = 1, Maximum = 3
% ; false.
% ```
weighted_maximum(Ws, Vars, Max) :-

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src/lib/clpz.pl
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@@ -5842,6 +5842,10 @@ run_propagator(preified_slash(X, Y, D, R), MState) -->
( Y == 0 ->
kill(MState),
D = 0
; Y == 1 ->
kill(MState),
D = 1,
R = X
; nonvar(X),
nonvar(Y) ->
kill(MState),