Constraints Processing System
This section briefly overviews how constraints processing works. The system described here follows loosely the CHR specification, and has been in particular heavily influenced by JCHR^{1} implementation. The distinctive features are builtin support for logical variables, terms, and pattern matching. Alternative body branches are a deviation from the standard CHR.
Terms and unification
Constraints processing relies heavily on the use of terms as data type. Abstractly speaking, terms are functions of zero or more arguments. Any opaque value captured by a term must be a POJO.
f(g(), h(k()))
p(val("foo"), q())
node(name("List"), arg(node(name("Int"), arg())))
(examples of terms)
The unique feature of terms is that they support unification. Unification of two terms is possible with or without variables being used as subterms of either participant, but in the latter case the two terms must be equal.
A term variable ranges over terms. A substitution is a mapping of variables to terms. Unification searches for a substitution $\sigma$, such that for two terms being unified, $f$ and $g$, the following holds: $\sigma f = \sigma g$. For example:
[X > g()] unifies f(X, h(X)) and f(g(), h(g()))
[X > Z, Y > g(X)] unifies f(X, g(X)) and f(Z, Y)
The latter unifier is the most general one, since other substitutions that also unify the two terms are all subsumed by it:
[X > g(X), Y > g(g(X)), Z > g(X)]
[X > g(g(X)), Y > g(g(g(X))), Z > g(g(X))]
Unification may fail because the arity or the symbols of the two terms mismatch. Also, since infinite terms are not supported, unification may fail because of occurs check, which means a variable can’t be unified with a term, that itself or through its descendants has this variable as one of its arguments.
f(X, h(X)) and f(g(), h(k())) can't be unified because g != k
f(X, g(X)) and f(g(X), g(h()) can't be unified because of
failed occurs check
Pattern matching is possible when variables are only used by one of the terms, which then serves as a pattern. To test if a pattern matches a given term can be implemented by a linear time algorithm, whereas full unification is slightly more complicated.
Logical variables
Logical variables serve to identify an object that is to be determined in the future. They are monotonic, in the sense that once a variable is assigned a particular value, it stays assigned to that value. In addition, they implement a unionfind data structure^{2} a.k.a. “disjoint set”. Any free logical variable can be assigned a class by setting its “parent” field to point to the class’s representative. All logical variables belonging to the same class are treated as exactly one variable. Logical variables notify observers when they become ground and when their parent (class representative) changes.
val X = Logical("X")
val Y = Logical("Y")
assertTrue(X.isFree() && y.isFree())
X.set("foo")
assertTrue(X.value() == "foo"))
Y.union(X)
assertTrue(Y.find() == X)
assertTrue(Y.find().value() == "foo")
(example of using logical variables)
A term variable can also be a logical variable, so that when two terms are unified, the substitution already has the calculated value for that variable.
val X = Logical("X")
val t1 = term("f", var(X), term("g", var(X)))
val t2 = term("f", term("h"), term("g", term("h")))
assertTrue(X.isFree())
val substitution = t1.unify(t2)
assertTrue(substitution.isValid())
assertTrue(X.find().value() == term("h"))
Constraints and predicates
Constraints are, simply put, tuples with fixed arity and a symbol attached. In some respects constraints correspond to rows in a database table. Logically they can be understood as facts, relations, or propositions. An argument to a constraint can be a term, a logical variable, or any POJO, except another constraint.
Constraints are activated from a constraint rule’s body and stay active until there are no possible matches with any of constraint rule’s heads. If a constraint is not discarded by a constraint rule, it is stored for future use. Stored constraints represent the program’s state.
The following figure shows the lifecycle of a constraint. The big rounded square in the middle contains the states, in which a constraint is considered “alive”: it can be either “active” or “stored”, but available for filling up vacant positions in a constraint rule’s head. A stored constraint can be reactivated if one of its arguments changes, such as when a logical variable becomes ground.
A successfully fired constraint rule, which declares one or more of constraints in its head to be “replaced”, causes these to be terminated.
(lifecycle of a constraint)
Predicates
Whereas a constraint serves to embody a fact or a relation among objects simply by being a witness of such a fact or a relation, a predicate^{3} helps to establish a fact or a relation, or check if one exists, by means of executing a procedure. Same is true for facts and propositions.
Predicates must implement ask/tell protocol. If a predicate is invoked from constraint rule’s guard clause, it represents a query (ask), and if it is invoked from the body, it is an assertion (tell).
Example of a predicate
Example of ask/tell
Example of failed predicate
Constraint rules
Constraints program is built from constraint rules. Each rule has three parts: the part that is responsible for triggering the constraint rule, called “head”, the part that checks for preconditions, called “guard”, and the part that is evaluated when constraint rule is fired, which is called “body”.
Head
Head is a set of constraints which are all required to be alive in order for a constraint rule to fire. This set is divided into “kept” part and “replaced” part, the latter containing constraints that are to be discarded as soon as the constraint rule fires.
A constraint rule is triggered when there are constraint occurrences matching all constraints specified in constraint rule’s head. These occurrences include the active constraint, plus any additional matching constraints that are currently alive, filling the other vacant slots.
There is some terminology inherited from CHR that can be useful when discussing the kinds of constraint rules. In the following table E
and E'
are the set of constraints in constraint rule’s head, C
is a conjunction of predicates serving as guard, and G
is a conjunction of predicates and constraints in constraint rule’s body.
“kept” set E 
“replaced” set E’ 
Notation  Designation 

empty  nonempty  E’ <=> C  G 
Simplification 
nonempty  empty  E => C  G 
Propagation 
nonempty  nonempty  E \ E’ <=> C  G 
Simpagation 
Essentially, a constraint rule with only “kept” constraints in its head is a “propagation”, the one with only “replaced” constraints is a “simplification”, and the one that has both “kept” and “replaced” constraints is a combination of the two.
In addition, we define a fourth kind of constraint rule, an “auto” constraint rule with an empty head. As the name implies, such constraint rule is triggered automatically on start of constraints program execution.
Guard
Guard is a conjunction of predicates, which are checked before a constraint rule is fired. Predicates in a guard are queried.
Body
Body is a conjunction of predicates and constraint activations. When triggered, each body clause is evaluated in order, with predicates serving as assertions and constraint activations producing new constraints. Each newly activated constraint is checked against any constraint rules that can be fired, and so on.
Triggering a constraint rule
The procedure processActive()
accepts a constraint occurrence that has been activated and proceeds as follows until this constraint is either discarded or suspended (moved to store). This procedure calls itself recursively on every constraint activation, so there might be many active constraints at any given moment, organised into a stack, which matches exactly the call stack of processActive()
.
First, the procedure searches for “relevant” constraint rules that declare a constraint in their heads matching the active constraint occurrence currently being processed. These constraint rules form a FIFO queue, corresponding to the order of constraint rules within a rule table.
For each “relevant” constraint rule, which at least partially matches the currently active constraint occurrence, the vacant slots in constraint rule’s head are filled with matching constraints from the store, which were previously suspended. Once a full match with constraint rule’s head is established, the guard is checked.
A constraint rule together with a set of constraints matching the ones declared by its head, one of which must be the active constraint, is considered to have been triggered after the guard check returns true. Otherwise this constraint rule is skipped.
After a constraint rule is triggered, first the constraints marked by constraint rule’s head as replaced are deactivated, and the logical variables defined by patterns are assigned appropriate values by asserting equality through equals predicate. Then the constraints and predicates defined in constraint rule’s body are activated onebyone. Every activation of a constraint is a recursive call to processActive()
.
If after having processed all items in constraint rule’s body the constraint occurrence currently being processed is still “alive”, meaning it has not being discarded during one of the recursive calls to this procedure, the process of matching and triggering constraint rules continues with the next constraint rule from the queue.
When there are no more constraint rules that can be matched, or the active constraint has been discarded, the procedure processActive()
finishes. If, when the procedure is finished the active constraint is still “alive”, it is suspended and moved to the store.
Alternative body
Example of executing a program
Consider the following example of the Euclid algorithm taken from the CHR book^{4}:
gcd(0) <=> true
gcd(N) \ gcd(M) <=> 0 < N, N =< M  gcd(M  N)
To illustrate the idea of using stored constraints to fill vacant slots when matching a constraint rule, let’s see what happens when this program is evaluated with 4 and 6 as arguments to gcd/1
constraint.
Constraints store  Active constraint  Match  Result 

{} 
gcd(4) 
no constraint rules matched  suspend gcd(4) 
On activating gcd(4)
there are no matching constraint rules, so the constraint is suspended.
Constraints store  Active constraint  Match  Result 

{gcd(4)} 
gcd(6) 
gcd(6) \ gcd(4) 
guard condition fails 
{gcd(4)} 
gcd(6) 
gcd(4) \ gcd(6) 
discard gcd(6) , activate gcd(2) 
Notice how both combinations of gcd(4)
and gcd(6)
constraints are tested. In general, if there are $n$ usages of constraint foo
in a given constraint rule’s head, there will be $n!$ potential matches to be evaluated, so that all permutations are covered.
The last activation above is a recursive call to processActive()
, so the cycle repeats. The active constraint has been deactivated, but contents of the store is unchanged.
Constraints store  Active constraint  Match  Result 

{gcd(4)} 
gcd(2) 
gcd(2) \ gcd(4) 
discard gcd(4) , activate gcd(0) 
{} 
gcd(0) 
gcd(0) 
discard gcd(0) 
There are no more constraint rules to match for gcd(2)
as the active constraint, so it is suspended. But the constraint gcd(6)
has been already discarded, so there are no more active constraints and the process stops. Here is the full trace of this program execution.
Constraints store  Active constraint  Match  Result 

{} 
gcd(4) 
no constraint rules matched  suspend gcd(4) 
{gcd(4)} 
gcd(6) 
gcd(6) \ gcd(4) 
guard condition fails 
{gcd(4)} 
gcd(6) 
gcd(4) \ gcd(6) 
discard gcd(6) , activate gcd(2) 
{gcd(4)} 
gcd(2) 
gcd(2) \ gcd(4) 
discard gcd(4) , activate gcd(0) 
{} 
gcd(0) 
gcd(0) 
discard gcd(0) 
{} 
gcd(2) 
no constraint rules matched  suspend gcd(2) 
After the program has finished, the constraint store consists of only one constraint gcd(2)
, which is the GCD of 4 and 6.
Semantics of a constraint rules program
The paper by Betz and Frühwirth^{5} gives an excellent treatise on semantics of CHR using linear logic. Since this constraints processing system is based on CHR, this semantics is valid for it also.
Linear logic abandons boolean values and replaces them with consumable resources. Consequently, it introduces its own set of connectives, from which we are interested only in the following: $!,\otimes,\multimap$. The multiplicative conjunction $\otimes$ combines two or more resources that are all available at the same time. The symbol $!$ (of course) marks a resource that can’t be exhausted. And instead of implication $\rightarrow$ one uses $\multimap$ symbol, the meaning of which is a transformation of one resource to another. The formula $A \multimap B$ is read: “consuming $A$, produce $B$ ”, that is given a resource $A$, we can assume $B$, but $A$ can no longer be used.
Simplification is represented by a proposition stating that from valid guard condition $C$, which can be reused, we can imply the following: given the set of constraints $E$ matching the constraint rule’s lefthand side, we can find such substitution $\sigma$, such that we can now assume the set of constraints produced from constraint rule’s body: $\sigma G$. All free variables in this proposition are universally quantified and the proposition itself can be reused infinitely.
\[ (E \Leftrightarrow C  G)^L := !\forall((!C^L) \multimap (E^L \multimap \exists\bar{y}G^L)) \]
Propagation is different from simplification in that the set of constraints $E$, which triggered the constraint rule, is made available together with the constraints produced by the body, so the constraints in $E$ are not “consumed”.
\[ (E \Rightarrow C  G)^L := !\forall((!C^L) \multimap (E^L \multimap E^L\otimes\exists\bar{y}G^L)) \]

K.U.Leuven JCHR System https://dtai.cs.kuleuven.be/CHR/JCHR/ ↩

a.k.a. “builtin constraints” in CHR literature ↩

Constraint Handling Rules http://www.informatik.uniulm.de/pm/fileadmin/pm/home/fruehwirth/constrainthandlingrulesbook.html ↩

Betz, H. and Frühwirth, T., 2005, October. A linearlogic semantics for constraint handling rules. In International Conference on Principles and Practice of Constraint Programming (pp. 137151). Springer, Berlin, Heidelberg. ↩