total families

Total Families

To guarantee the termination and completeness of the solving of equality constraints, we need to impose rather draconian restrictions on the instances of type synonym families. More specifically, to achieve both termination and completeness, we need the Strong Termination Condition and if we settle for termination alone (accepting to be incomplete for some rather exotic programs), we need the Relaxed (Termination) Condition as defined in Type Checking with Open Type Functions. The Strong Termination Condition corresponds closely to the conditions imposed on functional dependencies. The Relaxed Condition is somewhat more liberal, but still does not permit, for example, nested applications of type families in the right-hand side of a type instance. For many practically attractive uses of type families, where the the system is actually terminating, this is still to restrictive.

Two examples

data Z; data S a;

-- meets the Relaxed Condition
type family x :+ y
type instance Z   :+ y = y
type instance S x :+ y = S (x :+ y)

-- does not even meet the Relaxed Condition
type family x :* y
type instance Z   :* y = Z
type instance S x :* y = x :* y :+ y

The family (:+) meets the Relaxed Condition, but not the Strong Termination Condition. However, (:*) meets not even the Relaxed Condition. Nevertheless, we would expect that families defined by structural recursion should not compromise termination. Another somewhat irritating aspect of this example is that we would like these definitions to be closed, but they are open in their present form.

We might want to define equality on types as

data TFalse; data TTrue;

type family TypeEq s t
type instance TypeEq s s = TTrue
type instance TypeEq s t = FFalse  -- matches only if the previous instance does not

Unfortunately, the two instances are overlapping and there is no means by which we can disambiguate the overlap by using the same textual ordering as that which we are used to from value-level functions.

Defining total families

To enable textual disambiguation of overlapping instances, we declare the equalities together (by transferring GADT syntax to type synonyms):

type TypeEq s t where
  TypeEq s s = TTrue
  TypeEq s t = TFalse

TypeEq is a standard type family, but by virtue of being total (i.e., exhaustive) it is also closed. Further equalities cannot define it any further.

Let's use the same idea for the addition/multiplication examples:

type x :+ y where
  Z   :+ y = y
  S x :+ y = S (x :+ y)

type x :* y where
  Z   :* y = Z
  S x :* y = x :* y :+ y

In contrast to TypeEq, (:+) and (:*) is not total without an extra equality. We take another idea from value-level function definitions and implicitly complete each of these definitions by a final catch all equality. So, for (:+), we assume a final

x :+ y = VOID

and for (:*) a final

x :* y = VOID

A rewrite system with total families

We denote the rewrite system for TypeEq as

Et: {TypeEq s s ~ TTrue; TypeEq s t ~ TFalse}

and that of the type-level addition and multiplication as

Et: {Z   :+ y ~ y;
     S x :+ y ~ S (x :+ y); 
     _   :+ _ ~ VOID}
    {Z   :* y ~ Z; 
     S x :* y ~ x :* y :+ y; 
     _   :* _ ~ VOID}

Matching on such rewrite rule blocks starts with the first equality. If a given family application cannot possibly match on the first equality, the second is considered, and so on. The last one is guaranteed to match, and this is what makes the definitions total. Consider the following examples:

TypeEq Int Int --> TTrue
TypeEq Int Bool --> TFalse
TypeEq a b where a and b are two rigid type variables, can't be rewritten without further information about a and b.

Critical examples concerning termination

Example 1 of the paper:

Et: {F Bool ~ F (G Int); F _ ~ VOID}
    {G _ ~ VOID}

Eg: G Int ~ Bool

Completion gives:

  Eg
=> (TOP)
  VOID ~ Bool
=> FAIL

We can be a bit more tricky and use

Et: {F [a] ~ F (G a); F _ ~ VOID}
    {G _ ~ VOID}

Eg: G x ~ [x]

Here the idea is to use a new symbol in the local given, namely a rigid type variable. Nevertheless, the only equality of G matches G x and completion of Eg leads to an inconsistency.

However, if we add another equality to G, the situation changes:

Et: {F [a] ~ F (G a); F _ ~ VOID}
    {G Int ~ Bool; G _ ~ VOID}

Eg: G x ~ [x]

Here completion can't rewrite G x, as the outcome depends on getting more information about x. However, we have the following infinite derivation:

F [x]  -->  F (G x)  -->  F [x]  -->  ...

This is although Eg is inconsistent (and hence shouldn't be used for rewriting at all).

Can we fix this???