/* CLP(B): Constraint Logic Programming over Boolean Variables
- Copyright (C): 2019 Markus Triska
+ Copyright (C): 2019-2023 Markus Triska
All rights reserved.
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
-
- ```
- node(ID, Var, Low, High, Aux)
- ```
+ *) The integers 0 or 1, denoting false and true, respectively, or
+ *) A node of the form
- 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.
%
% ```
% ?- 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.
% ```
%
% ```
% ?- 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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