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Please pledge or share!*

One of the first problems I attempted to solve with Typed Clojure turned out to be one of the most difficult: statically verifying Red/Black tree rebalancing invariants.

This post describes how close I am to checking these invariants. Note that the code examples might not be currently checkable.

I was lucky enough to have Rowan Davies supervise my Honours project on Typed Clojure. He introduced me to SML CIDRE, a refinement type checker for SML. Rowan’s perspective on types was an interesting counterpoint to mine, but perhaps even more interesting were the parallels between SML CIDRE and Typed Clojure: SML CIDRE is a type system layered on top of SML, and Typed Clojure is the same for Clojure.

The example that opened my eyes to the ability of SML CIDRE is one
that checks
a Red/Black tree implementation.
SML CIDRE can *statically* verify the colour invariant of Red/Black tree rebalancing operations.
This is quite remarkable.

Let’s explore the problem of verifying a Clojure Red/Black tree implementation with Typed Clojure.

An idiomatic Red/Black tree implementation in SML is defined using datatypes.
In that spirit, the Clojure port will be implemented with *plain hash maps*.

## Basic Types

We need three basic types in a Red/Black tree implementation: Red nodes, Black nodes, and Leaf nodes.

In Typed Clojure, they are simply type aliases of heterogeneous maps, differentiated
by the `:tree`

entry.

```
(defalias EntryT
"The payload"
'{:key Number
:datum Number})
(defalias Empty
"A terminating node."
'{:tree ':Empty})
(defalias Red
"A red node"
(TFn [[l :variance :covariant]
[r :variance :covariant]]
'{:tree ':Red
:entry EntryT
:left l
:right r}))
(defalias Black
"A black node"
(TFn [[l :variance :covariant]
[r :variance :covariant]]
'{:tree ':Black
:entry EntryT
:left l
:right r}))
```

We also define `EntryT`

, the payload type.

## Refinement types

A *refinement type* in SML CIDRE is an additional layer of types over SML types for expressing
logical properities and invariants.
They are realised via *datasorts* (for example, search for ‘datasort’ in the
Red/Black tree implementation).
Refinement types are used to model and enforce the colour invariant of Red/Black trees
by defining a set of type refinements, and then annotating the rebalancing functions with their expected types.

We can emulate something similar in Typed Clojure with heterogeneous maps, by taking advantage of subtyping between heterogeneous maps.

```
(defalias rbt
"Trees with only black children for red nodes"
(U
Empty
(Black rbt rbt)
(Red bt bt)))
(defalias bt
"Like rbt but additionally the root node is black"
(U
Empty
(Black rbt rbt)))
(defalias red
"Trees with a red root"
(Red bt bt))
(defalias badRoot
"Invariant possibly violated at the root"
(U
Empty
(Black rbt bt)
(Red rbt bt)
(Red bt rbt)))
(defalias badLeft
"Invariant possibly violated at the left child"
(U
Empty
(Black rbt rbt)
(Red bt bt)
(Black badRoot rbt)))
(defalias badRight
"Invariant possibly violated at the right child"
(U
Empty
(Black rbt rbt)
(Red bt bt)
(Black rbt badRoot)))
```

It is immediately interesting that these type aliases are not special as far as
Typed Clojure types go. All of the types can be explained using unions (`U`

) and heterogeneous
maps.

# Rebalancing operations

We can use our type aliases to annotate our rebalancing operators. For example, here is
`restore-right`

, that takes a `badRight`

(a tree violated at the right child), and returns
an `rbt`

(a tree with only black children for red nodes).

```
(ann restore-right [badRight -> rbt])
(defn restore-right [tmap]
(cond
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :left :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :left :tree)))
(let [{lt :left rt :right e :entry} tmap]
;re-color
{:tree :Red
:entry e
:left (assoc lt
:tree :Black)
:right (assoc rt
:tree :Black)})
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :left :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :right :tree)))
(let [{lt :left rt :right e :entry} tmap]
{:tree :Red
:entry e
:left (assoc lt
:tree :Black)
:right (assoc rt
:tree :Black)})
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :left :tree)))
(let [{e :entry
l :left
{re :entry
{rle :entry
rll :left
rlr :right}
:left
rr :right}
:right} tmap]
;l is black, deep rotate
{:tree :Black
:entry rle
:left {:tree :Red
:entry e
:left l
:right rll}
:right {:tree :Red
:entry re
:left rlr
:right rr}})
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :right :tree)))
(let [{e :entry
l :left
{re :entry
rl :left
rr :right}
:right} tmap]
;l is black, shallow rotate
{:tree :Black
:entry re
:left {:tree :Red
:entry e
:left l
:right rl}
:right rr})
:else tmap))
```

To check this implementation, Typed Clojure needs to understand the notion of a *path*
down a map, and how to update it.

Typed Clojure needs to know that in the following branch

```
(if (= :Red (-> tmap :right :right :tree))
e1
e2)
```

it is recognised that

`tmap`

has a`:Red`

key down the path`[:right :right :tree]`

when checking`e1`

`tmap`

*does not*have a`:Red`

key down the path`[:right :right :tree]`

when checking`e2`

This style of updating works in Typed Clojure *today*.

## The problem

What makes checking `restore-right`

so difficult is the sheer number of types it must reason
about. The rebalancing operators generate many type calculations, mainly type intersections.
In fact, the most interesting aspect of this example is the huge number of intersection
calculations with respect to the size of the code base (or *any* code base, for that matter).

To better understand the explosion of type calculations, let’s see what Typed Clojure infers from each branch.

The first branch isn’t horrible:

```
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :left :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :left :tree)))
```

infers the proposition set:

```
{:then (& (is ':Black tmap [(Key :tree)])
(is ':Red tmap [(Key :left) (Key :tree)])
(is ':Red tmap [(Key :right) (Key :tree)])
(is ':Red tmap [(Key :right) (Key :left) (Key :tree)]))
:else (| (! ':Black tmap [(Key :tree)])
(! ':Red tmap [(Key :left) (Key :tree)])
(! ':Red tmap [(Key :right) (Key :tree)])
(! ':Red tmap [(Key :right) (Key :left) (Key :tree)]))}
```

The `:then`

entry states what is true in the then branch, and the `:else`

branch states what is true in the else branch. It should be easy to guess
what each proposition means by referring back to the original test expression.

For example, the expression

```
(= :Black (-> tmap :tree))
```

has the proposition set

```
{:then (is ':Black tmap [(Key :tree)])
:else (! ':Black tmap [(Key :tree)])}
```

At the second branch, we’re starting to see a problem. Because we have to take the results of previous branches into account, we are inferring much more information:

```
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :left :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :right :tree)))
```

infers the proposition set:

```
{:then
(&
(! (Value :Red) tmap [(Key :right) (Key :left) (Key :tree)])
(is (Value :Red) tmap [(Key :right) (Key :tree)])
(is (Value :Red) tmap [(Key :left) (Key :tree)])
(is (Value :Black) tmap [(Key :tree)])
(is (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])),
:else
(&
(|
(! (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :left) (Key :tree)])
(! (Value :Black) tmap [(Key :tree)]))
(|
(! (Value :Red) tmap [(Key :right) (Key :left) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :left) (Key :tree)])
(! (Value :Black) tmap [(Key :tree)])))}
```

If we go down the second then branch, this information will update `tmap`

to:

```
(U Empty
(Black rbt rbt)
(Red bt bt)
(Black rbt (Red bt rbt)))
```

Just for fun, here’s the 4th branch:

```
(and (= :Black (-> tmap :tree))
(= :Red (-> tmap :left :tree))
(= :Red (-> tmap :right :tree))
(= :Red (-> tmap :right :right :tree)))
```

infers the proposition set:

```
{:then
(&
(|
(! (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :left) (Key :tree)]))
(! (Value :Red) tmap [(Key :right) (Key :left) (Key :tree)])
(is (Value :Red) tmap [(Key :right) (Key :tree)])
(is (Value :Black) tmap [(Key :tree)])
(is (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])
(|
(! (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :left) (Key :tree)]))),
:else
(&
(|
(! (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Black) tmap [(Key :tree)]))
(|
(! (Value :Red) tmap [(Key :right) (Key :left) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Black) tmap [(Key :tree)]))
(|
(! (Value :Red) tmap [(Key :right) (Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :left) (Key :tree)])
(! (Value :Black) tmap [(Key :tree)]))
(|
(! (Value :Red) tmap [(Key :right) (Key :left) (Key :tree)])
(! (Value :Red) tmap [(Key :right) (Key :tree)])
(! (Value :Red) tmap [(Key :left) (Key :tree)])
(! (Value :Black) tmap [(Key :tree)])))}
```

In the final clause, always returns the original tree, we should have refined the type of
`tmap`

to an `rbt`

*by taking into account all of the previous branches*.

## Solutions

SML CIDRE checks this Red/Black tree code in a matter of milliseconds. The checker precalculates all possible type intersections before type checking, which delivers massive performance improvements during checking.

Typed Clojure seems to choke under these intensive calculations.

To get this implementation to work, I suspect it involves:

- tons of caching
- aggressive simplification and rearrangement of types

## Why?

While the Red/Black trees are fun, the most notable characteristic of the checking it is the sheer amount of types generated. We probably won’t be using this implementation in our own code.

If we can get this Red/Black tree implementation checking quickly and accurately there are several interesting results:

- we can investigate whether it can check in Typed Racket
- large conditionals with many branches will type check much faster, even in “normal” code
- Typed Clojure could be a great way to further explore these kinds of problems

*I am crowdfunding 12 months of full time development on Typed Clojure.
If you want to see awesome problems like these solved in Typed Clojure, please pledge or share!*