18 Solver types

Solver types are an experimental addition to the language supporting the implementation of constraint solvers. A program may place constraints on and between variables of a solver type, limiting the values those variables may take on before they are actually bound. For example, if X and Y are variables belonging to a constrained integer solver type, we might place constraints upon them such that X > 3 + Y and Y =< 7. A later attempt to unify Y with 10 will fail (it would violate the second constraint); similarly an attempt to unify X with 5 and Y with 4 would fail (it would violate the first constraint).

18.1 The ‘any’ inst

Variables with solver types can have one of three possible insts: free, ground or any. A variable with a solver type with inst any may not (yet) be semantically ground, in the following sense: if a variable is semantically ground, then the set of values it unifies with form an equivalence class; if a variable is non-ground, then the set of values it unifies with do not form an equivalence class.

More formally, X is ground if for values Y and Z that unify with X, it is the case that Y and Z also unify with each other. X is non-ground if there are values Y and Z that unify with X, but which do not unify with each other.

A non-solver type value will have inst any if it is constructed using one or more inst any values.

The builtin modes ia and oa are equivalent to in(any) and out(any) respectively.

18.2 Abstract solver type declarations

The type declarations

:- solver type t1.
:- solver type t2(T1, T2).

declare types t1/0 and t2/2 to be abstract solver types. Abstract solver type declarations are identical to ordinary abstract type declarations except for the solver keyword.

18.3 Solver type definitions

A solver type definition takes the following form:

:- solver type solver_type
    where   representation   is representation_type,
            ground           is ground_inst,
            any              is any_inst,
            constraint_store is mutable_decls,
            equality         is equality_pred,
            comparison       is comparison_pred.

The representation attribute is mandatory. The ground_inst and any_inst attributes are optional and default to ground. The constraint_store attribute is mandatory: mutable_decls must be either a single mutable declaration (see Module-local mutable variables), or a comma separated list of mutable declarations in brackets. The equality and comparison attributes are optional, although a solver type without equality would not be very useful. The attributes that are not omitted must appear in the order shown above.

The representation_type is the type used to implement the solver_type. A two-tier scheme of this kind is necessary for a number of reasons, including

• a semantic gap is necessary to accommodate the fact that non-ground solver_type values may be represented by ground representation_type values (in the context of the corresponding constraint solver state);
• this approach greatly simplifies the definition of equality and comparison predicates for the solver_type.

The ground_inst is the inst associated with representation_type values denoting ground solver_type values.

The any_inst is the inst associated with representation_type values denoting any solver_type values.

The compiler constructs four impure functions for converting between solver_type values and representation_type values (name is the function symbol used to name solver_type and arity is the number of type parameters it takes):

:- impure func 'representation of ground name/arity'(solver_type) =
                        representation_type.
:-        mode 'representation of ground name/arity'(in) =
                        out(ground_inst) is det.

:- impure func 'representation of any name/arity'(solver_type) =
                        representation_type.
:-        mode 'representation of any name/arity'(in(any)) =
                        out(any_inst) is det.

:- impure func 'representation to ground name/arity'(representation_type) =
                        solver_type.
:-        mode 'representation to ground name/arity'(in(ground_inst)) =
                        out is det.

:- impure func 'representation to any name/arity'(representation_type) =
                        solver_type.
:-        mode 'representation to any name/arity'(in(any_inst)) =
                        out(any) is det.

These functions are impure because of the semantic gap issue mentioned above.

Solver types may be exported from their defining module, but only in an abstract form. This requires the full definition to appear in the implementation section of the module, and an abstract declaration like the following in its interface:

:- solver type solver_type.

If a solver type is exported, then its representation type, and, if specified, its equality and/or comparison predicates must also exported from the same module.

If a solver type has no equality predicate specified, then the compiler will generate an equality predicate that throws an exception of type ‘exception.software_error/0’ when called.

Likewise, if a solver type has no equality comparison specified, then the compiler will generate a comparison predicate that throws an exception of type ‘exception.software_error/0’ when called.

If provided, any mutable declarations given for the constraint_store attribute are equivalent to separate mutable declarations; their association with the solver type is for the purposes of documentation. That is,

:- solver type t
    where …,
          constraint_store is [ mutable(a, int, 42, ground, []),
                                mutable(b, string, "Hi", ground, [])
                               ],
          …

is equivalent to

:- solver type t
    where …
:- mutable(a, int, 42, ground, []).
:- mutable(b, string, "Hi", ground, []).

18.4 Implementing solver types

A solver type is an abstraction, implemented using a combination of a private representation type and a constraint store.

The constraint store is an (impure) piece of state used to keep track of the extant constraints on variables of the solver type. This will typically be implemented using foreign code.

It is important that changes to the constraint store are properly trailed (see Trailing) so that changes can be undone on backtracking.

The solver type implementation should provide functions and predicates

• to construct and deconstruct solver type values,
• to place constraints on solver type variables,
• to convert any solver type variables to ground if possible (this is obviously an impure operation — see Impurity),
• to convert solver type values to non-solver type values (again, this is impure and requires the argument solver type values be sufficiently ground),
• to ask questions about the extant constraints on solver type variables without constraining them further (this too is impure because the set of constraints on a variable may change during execution of the program).

18.5 Solver types and negated contexts

Mercury’s negation and if-then-else goals (and hence also inequalities and universal quantifications) are implemented using negation as failure, meaning that the failure to find a proof of one statement is regarded as a proof of its negation. Negation as failure is sound provided that no non-local variable becomes further bound during the execution of a goal which may be negated. This includes negated goals themselves, as well as the conditions of if-then-elses, which are negated iff they fail without producing any solution, and the bodies of pred or func expressions, which may be called or applied in one of the other contexts, or indeed in another pred or func expression.

Mercury checks that any solver variables that are used in the above contexts are used in such a way that negation as failure remains sound. In the case of negation and if-then-else goals, if any non-local solver type variable or higher-order variable with inst any is used in a negated context, the goal must be placed inside a promise_pure, promise_semipure, or promise_impure scope. The first two promises assert that (among other things) no solver variable becomes further bound in the negated context. The third promise makes the weaker assertion that the goal satisfies the requirements of all impure goals (namely, that it does not interfere with the semantics of other pure goals).

In the case of pred and func expressions, Mercury allows three possibilities. The higher-order value may be considered ground, which means that all non-local variables used in the body of the expression (including those with other higher-order values) must themselves be ground. Higher-order values that are ground can be safely called or applied in any context, including negated contexts, since none of their (ground) non-local variables can become further bound by doing so. Alternatively, the higher-order value may be considered to have inst any, which allows non-local variables used in the body of the expression to have inst any. Calling or applying these values may further bind non-local variables, so if this occurs in a negated context then, as in the case of solver variables, a promise will be required around the negation or if-then-else.

Pred and func expressions with inst any are written using any_pred and any_func in place of pred and func, respectively.

The third possibility is that the higher-order value can be given an impure type (see Higher-order impurity).