#### 15.1` `Sharing and Equality

##### 15.1.1` `Re-Examining Equality

```
data BT:
| leaf
| node(v, l :: BT, r :: BT)
end
a-tree =
node(5,
node(4, leaf, leaf),
node(4, leaf, leaf))
b-tree =
block:
four-node = node(4, leaf, leaf)
node(5,
four-node,
four-node)
end
```

`b-tree`

is morally equivalent to how we’ve written `a-tree`

, but we’ve
created a helpful binding to avoid code duplication.`a-tree`

and `b-tree`

are bound to trees with
`5`

at the root and a left and right child each containing
`4`

, we can indeed reasonably consider these trees
equivalent. Sure enough:
```
check:
a-tree is b-tree
a-tree.l is a-tree.l
a-tree.l is a-tree.r
b-tree.l is b-tree.r
end
```

However, there is another sense in which these trees are not
equivalent. concretely, `a-tree`

constructs a distinct node for
each child, while `b-tree`

uses the same node for both
children. Surely this difference should show up somehow, but we
have not yet seen a way to write a program that will tell these
apart.

`is`

operator uses the same equality test as
Pyret’s `==`

. There are, however, other equality tests in
Pyret. In particular, the way we can tell apart these data is by using
Pyret’s `identical`

function, which implements
reference equality. This checks not only whether two
values are structurally equivalent but whether they are
the result of the very same act of value construction.
With this, we can now write additional tests:
```
check:
identical(a-tree, b-tree) is false
identical(a-tree.l, a-tree.l) is true
identical(a-tree.l, a-tree.r) is false
identical(b-tree.l, b-tree.r) is true
end
```

Let’s step back for a moment and consider the behavior that gives us this
result. We can visualize the different values by putting each distinct value
in a separate location alongside the running program. We can draw the
first step as creating a `node`

with value `4`

:

`a-tree = node(5, 1001, node(4, leaf, leaf)) b-tree = block: four-node = node(4, leaf, leaf) node(5, four-node, four-node) end`

Heap

- 1001:
`node(4, leaf, leaf)`

The next step creates another node with value `4`

, distinct from the
first:

`a-tree = node(5, 1001, 1002) b-tree = block: four-node = node(4, leaf, leaf) node(5, four-node, four-node) end`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

Then the `node`

for `a-tree`

is created:

`a-tree = 1003 b-tree = block: four-node = node(4, leaf, leaf) node(5, four-node, four-node) end`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

- 1003:
`node(5, 1001, 1002)`

When evaluating the `block`

for `b-tree`

, first a single node is
created for the `four-node`

binding:

`a-tree = 1003 b-tree = block: four-node = 1004 node(5, four-node, four-node) end`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

- 1003:
`node(5, 1001, 1002)`

- 1004:
`node(4, leaf, leaf)`

These location values can be substituted just like any other, so they get
substituted for `four-node`

to continue evaluation of the
block.We skipped substituting `a-tree`

for the moment, that
will come up later.

`a-tree = 1003 b-tree = block: node(5, 1004, 1004) end`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

- 1003:
`node(5, 1001, 1002)`

- 1004:
`node(4, leaf, leaf)`

Finally, the node for `b-tree`

is created:

`a-tree = 1003 b-tree = 1005`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

- 1003:
`node(5, 1001, 1002)`

- 1004:
`node(4, leaf, leaf)`

- 1005:
`node(5, 1004, 1004)`

This visualization can help us explain the test we wrote using `identical`

.
Let’s consider the test with the appropriate location references substituted
for `a-tree`

and `b-tree`

:

`check: identical(1003, 1005) is false identical(1003.l, 1003.l) is true identical(1003.l, 1003.r) is false identical(1005.l, 1005.r) is true end`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

- 1003:
`node(5, 1001, 1002)`

- 1004:
`node(4, leaf, leaf)`

- 1005:
`node(5, 1004, 1004)`

`check: identical(1003, 1005) is false identical(1001, 1001) is true identical(1001, 1004) is false identical(1004, 1004) is true end`

Heap

- 1001:
`node(4, leaf, leaf)`

- 1002:
`node(4, leaf, leaf)`

- 1003:
`node(5, 1001, 1002)`

- 1004:
`node(4, leaf, leaf)`

- 1005:
`node(5, 1004, 1004)`

`is`

operator can also be parameterized by a different equality
predicate than the default `==`

. Thus, the above block can
equivalently be written as:We can use `is-not`

to check for expected failure of equality.
```
check:
a-tree is-not%(identical) b-tree
a-tree.l is%(identical) a-tree.l
a-tree.l is-not%(identical) a-tree.r
b-tree.l is%(identical) b-tree.r
end
```

```
check:
a-tree is b-tree
a-tree is-not%(identical) b-tree
a-tree.l is a-tree.r
a-tree.l is-not%(identical) a-tree.r
end
```

`identical`

really means
[Understanding Equality]
(Pyret has a full range of equality operations suitable for different situations).Exercise

There are many more equality tests we can and should perform even with the basic data above to make sure we really understand equality and, relatedly, storage of data in memory. What other tests should we conduct? Predict what results they should produce before running them!

##### 15.1.2` `The Cost of Evaluating References

From a complexity viewpoint, it’s important for us to understand how
these references work. As we have hinted, `four-node`

is computed
only once, and each use of it refers to the same value: if, instead,
it was evaluated each time we referred to `four-node`

, there
would be no real difference between `a-tree`

and `b-tree`

,
and the above tests would not distinguish between them.

`L = range(0, 100)`

```
L1 = link(1, L)
L2 = link(-1, L)
```

`L`

to be
considerably more than that for a single `link`

operation. Therefore, the question is how long it takes to compute
`L1`

and `L2`

after `L`

has been computed: constant
time, or time proportional to the length of `L`

?`L`

is computed once and bound to `L`

; subsequent
expressions refer to this value (hence “reference”)
rather than reconstructing it, as reference equality shows:
```
check:
L1.rest is%(identical) L
L2.rest is%(identical) L
L1.rest is%(identical) L2.rest
end
```

`L`

to it, and see
whether the resulting argument is `identical`

to the original:
```
fun check-for-no-copy(another-l):
identical(another-l, L)
end
check:
check-for-no-copy(L) is true
end
```

```
check:
L satisfies check-for-no-copy
end
```

`.rest`

) nor
user-defined ones (like `check-for-no-copy`

) make copies of their
arguments.Strictly speaking, of course, we cannot
conclude that no copy was made. Pyret could have made a copy,
discarded it, and still passed a reference to the original. Given how
perverse this would be, we can assume—##### 15.1.3` `Notations for Equality

Until now we have used `==`

for equality. Now we have learned that it’s
only one of multiple equality operators, and that there is another one called
`identical`

. However, these two have somewhat subtly different syntactic
properties. `identical`

is a name for a function, which can
therefore be used to refer to it like any other function (e.g., when we need to
mention it in a `is-not`

clause). In contrast, `==`

is a binary
operator, which can only be used in the middle of expressions.

`identical`

and a function name equivalent of
`==`

. They do, in fact, exist! The operation performed by `==`

is
called `equal-always`

. Therefore, we can write the first block of tests
equivalently, but more explicitly, as
```
check:
a-tree is%(equal-always) b-tree
a-tree.l is%(equal-always) a-tree.l
a-tree.l is%(equal-always) a-tree.r
b-tree.l is%(equal-always) b-tree.r
end
```

`identical`

is `<=>`

.
Thus, we can equivalently write `check-for-no-copy`

as
```
fun check-for-no-copy(another-l):
another-l <=> L
end
```

##### 15.1.4` `On the Internet, Nobody Knows You’re a DAG

Despite the name we’ve given it, `b-tree`

is not actually a
tree. In a tree, by definition, there are no shared nodes,
whereas in `b-tree`

the node named by `four-node`

is shared
by two parts of the tree. Despite this, traversing `b-tree`

will
still terminate, because there are no cyclic references in it:
if you start from any node and visit its “children”, you cannot end
up back at that node. There is a special name for a value with such a
shape: directed acyclic graph (DAG).

Many important data structures are actually a DAG underneath. For instance, consider Web sites. It is common to think of a site as a tree of pages: the top-level refers to several sections, each of which refers to sub-sections, and so on. However, sometimes an entry needs to be cataloged under multiple sections. For instance, an academic department might organize pages by people, teaching, and research. In the first of these pages it lists the people who work there; in the second, the list of courses; and in the third, the list of research groups. In turn, the courses might have references to the people teaching them, and the research groups are populated by these same people. Since we want only one page per person (for both maintenance and search indexing purposes), all these personnel links refer back to the same page for people.

```
data Content:
| page(s :: String)
| section(title :: String, sub :: List<Content>)
end
```

```
people-pages :: Content =
section("People",
[list: page("Church"),
page("Dijkstra"),
page("Hopper") ])
```

`fun get-person(n): get(people-pages.sub, n) end`

```
theory-pages :: Content =
section("Theory",
[list: get-person(0), get-person(1)])
systems-pages :: Content =
section("Systems",
[list: get-person(1), get-person(2)])
```

```
site :: Content =
section("Computing Sciences",
[list: theory-pages, systems-pages])
```

```
check:
theory = get(site.sub, 0)
systems = get(site.sub, 1)
theory-dijkstra = get(theory.sub, 1)
systems-dijkstra = get(systems.sub, 0)
theory-dijkstra is systems-dijkstra
theory-dijkstra is%(identical) systems-dijkstra
end
```

##### 15.1.5` `It’s Always Been a DAG

What we may not realize is that we’ve actually been creating a DAG for longer
than we think. To see this, consider `a-tree`

, which very clearly seems to
be a tree. But look more closely not at the `node`

s but rather at the
`leaf`

(s). How many actual `leaf`

s do we create?

`leaf`

:
the data definition does not list any fields, and when constructing a
`BT`

value, we simply write `leaf`

, not (say)
`leaf()`

. Still, it would be nice to know what is happening behind the
scenes. To check, we can simply ask Pyret:
```
check:
leaf is%(identical) leaf
end
```

`leaf <=> leaf`

here, because
that is just an expression whose result is ignored. We have to write `is`

to register this as a test whose result is checked and reported.
and this check passes. That is, when we write a variant without any
fields, Pyret automatically creates a singleton: it makes just one
instance and uses that instance everywhere. This leads to a more efficient
memory representation, because there is no reason to have lots of distinct
`leaf`

s each taking up their own memory. However, a subtle consequence of
that is that we have been creating a DAG all along.`leaf`

to be distinct, it’s easy: we can write
```
data BTDistinct:
| leaf()
| node(v, l :: BTDistinct, r :: BTDistinct)
end
```

`leaf`

function everywhere:
```
c-tree :: BTDistinct =
node(5,
node(4, leaf(), leaf()),
node(4, leaf(), leaf()))
```

```
check:
leaf() is-not%(identical) leaf()
end
```

##### 15.1.6` `From Acyclicity to Cycles

`web-colors = link("white", link("grey", web-colors))`

`map2(color-table-row, table-row-content, web-colors)`

`color-table-row`

function to two arguments: the
current row from `table-row-content`

, and the current color from
`web-colors`

, proceeding in lockstep over the two lists.Unfortunately, there are many things wrong with this attempted definition.

Do Now!

Do you see what they are?

This will not even parse. The identifier

`web-colors`

is not bound on the right of the`=`

.- Earlier, we saw a solution to such a problem: use
`rec`

[Streams From Functions]. What happens if we write`rec web-colors = link("white", link("grey", web-colors))`

instead?Exercise

Why does

`rec`

work in the definition of`ones`

but not above? Assuming we have fixed the above problem, one of two things will happen. It depends on what the initial value of

`web-colors`

is. Because it is a dummy value, we do not get an arbitrarily long list of colors but rather a list of two colors followed by the dummy value. Indeed, this program will not even type-check.Suppose, however, that`web-colors`

were written instead as a function definition to delay its creation:`fun web-colors(): link("white", link("grey", web-colors())) end`

On its own this just defines a function. If, however, we use it—`web-colors()`

—it goes into an infinite loop constructing `link`

s.Even if all that were to work,

`map2`

would either (a) not terminate because its second argument is indefinitely long, or (b) report an error because the two arguments aren’t the same length.

When you get to cycles, even defining the datum becomes difficult because its definition depends on itself so it (seemingly) needs to already be defined in the process of being defined. We will return to cyclic data later in Cyclic Data, and to this specific example in Recursion and Cycles from Mutation.