# Higher-Order Definitions

Higher-order definitions are definitions of relations whose arguments can be relations that are not singletons.
A parameter that represents such a relation must begin with an uppercase letter.
Such a parameter is often called a *higher-order parameter*, or a *higher-order variable*.

You must annotate a higher-order definition with either `@outline`

or `@inline`

, but the former is preferable.
Note that you cannot use `@inline`

for definitions that are directly or indirectly recursive.

In the example below, `aggregate`

is higher-order, because its parameter is intended to be a relation of arbitrary arity.

```
// read query
doc"""
Sum up all values in the last column of a relation.
For compound keys, use all but the last key element as the group-by variable.
"""
@outline
def aggregate[R](prefix..., agg) {
agg = sum[key, value : R(prefix..., key, value)]
}
def data = {
("A", "a", 1); ("A", "b", 10);
("B", "a", 2); ("B", "b", 20);
}
def output:data = data
def output:aggregate1 = aggregate[data]
def output:aggregate2 = aggregate[aggregate[data]]
```

As the example shows, higher-order relations can be stacked on top of each other.
The keyword `prefix...`

is a vararg and can refer to multiple elements in a tuple.
See the Rel Primer for more information about varargs.

Higher-order relations can be even used in recursive logic.
The example below computes the transitive closure of a binary relation.
If the binary relation `R`

contains edges between the nodes of a graph, then nodes `A`

and `B`

are in the transitive closure of `R`

if and only if `B`

is reachable from `A`

:

```
// read query
// Compute the transitive closure of a binary relation.
@outline
def transitive_closure[R] = R
@outline
def transitive_closure[R] = x, y : R(x, z) and transitive_closure[R](z, y) from z
def edges = (1, 2); (2, 1); (3, 4); (4, 5)
def output = transitive_closure[edges]
```

Compare this version of `transitive_closure`

with the one shown in section `@outline`

.

There are many simple examples of higher-order definitions in the Standard Library, for instance, `union`

.

Here is a larger example, a Rel relation that emulates — though less efficiently — the Library relation `sum`

.
The relations `last`

, `enumerate`

, `count`

, and `arity`

can be found in the Standard Library:

```
// read query
@outline
def my_sum[R] = my_sum_aux[enumerate[last[R]], count[R]], arity[R] != 0
@outline
def my_sum_aux[Vector, index] = 0, index = 0
@outline
def my_sum_aux[Vector, index] = Vector[index] + my_sum_aux[Vector, index - 1], index > 0
def my_set = range[1, 10, 1]
def output:A = my_sum[my_set]
def output:B = my_sum[9]
def output:C = my_sum[{}]
def output:D = my_sum[{("a", 1); ("A", "B", 100); 50}]
def empty_arity_2(x, y) = {}
def output:E = arity[empty_arity_2]
def output:F = my_sum[empty_arity_2]
```

In many programming languages, a typical example of a higher-order function is one that maps a function over a list. Here is a relation that maps a binary relation over the elements of a unary relation:

```
// read query
// Map binary `F` over unary `R`.
@outline
def my_map[F, R] = F[x] from x in R
def double[x in Int] = x + x
def emphasis[s in String] = concat[s, "!"]
def my_arguments = range[1, 6, 2]
def output:One = my_map[double, my_arguments]
def output:Two = my_map[emphasis, {"Quietly"; "Please"}]
```

Note that in Rel a relation such as `my_map`

is unnecessary.
The following also works:

```
// read query
def double[x in Int] = x + x
def emphasis[s in String] = concat[s, "!"]
def my_arguments = range[1, 6, 2]
def output:One = double[my_arguments]
def output:Two = emphasis[{"Quietly"; "Please"}]
```

In both cases the example with `double`

is equivalent to the following:

```
// read query
def double[x in Int] = x + x
def my_arguments = range[1, 6, 2]
def output = x: my_arguments(val) and double(val, x) from val
```

A higher-order relation is not required to perform such mapping.

Notice that `my_map`

is a true higher-order relation with two higher-order parameters: `F`

and `R`

.

In particular, its second argument is not a singleton:

```
// read query
@outline
def my_map[F, R] = F[x], count[R] from x in R
def double[x in Int] = x + x
def my_arguments = range[1, 6, 2]
def output = my_map[double, my_arguments]
```

One way to think about it is that in the first case the iteration over `{1; 3; 5}`

— defined by `my_arguments`

— is carried out within the body of the definition of `my_map`

, as seen in the `from x in R`

expression.

The same logic can be expressed with first-order logic as follows:

```
// read query
def double[x in Int] = x + x, count[x]
def my_arguments = range[1, 6, 2]
def output = double[my_arguments]
```

You can write this as first-order logic thanks to scalar conversion.
Scalar conversion means that even though the argument of a relation is a relation by itself (here `range[1, 6, 2]`

), the underlying relation `double`

applies its logic to each value in `my_arguments`

.

The two examples below further illustrate the difference between a higher-order variable and a first-order variable in similar definitions:

```
// read query
@inline
def P[R] = max[R]
def output = P[{1; 2; 3}]
```

```
// read query
@inline
def P[r] = max[r]
def output = P[{1; 2; 3}]
```

## Higher-Order Keywords Shadowing Type Relations

Higher-order keywords in a declaration begin with a capital letter, similar to a type relation. This makes it possible to shadow the name of a type relation with a higher-order keyword.

Higher-order keywords in relational declarations may shadow the names of type relations.

The following example illustrates this point:

```
// read query
@outline
def all_negative(String) = forall(x in String: x < 0)
def output = all_negative({-1; -2})
```

Here, `String`

is a higher-order keyword. It shadows the type relation `String`

usually used to check whether a value is of data type String.

Next: Annotations