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okasaki-clojure

An implementation of some data structures described in Okasaki's book "Purely Functional Data Structures".

I'm trying to follow almost directly the ML implementations using some sugar on David Nolen's [core.match] (https://github.com/clojure/core.match) library.

This sugar is defined in namespace datatype.core.

defdatatype

A datatype contains an object to name it, and a group of constructors with or without parameters.

For instance, a datatype for unbalanced binary search trees can be defined as:

(defdatatype
  ::UnbalancedBST
  Empty        
  (Node left root right)) 

Two kinds of constructors are possible:

• Constants, which are represented as symbols bound to the corresponding keyword (e.g. Empty to :user/Empty) both in the current namespace.

• Factories, which are represented as records with the same name as given and corresponding fields. Field names are mangled in order to not have conficts among different constructors using the same names. For instance, in the example, fields are named node-left, node-root and node-right.

defun

We can now define functions using pairs of patterns and actions:

(defun insert
    [x Empty] 
        (->Node Empty x Empty)
    [x [Node a y b]]
        (cond 
            (< x y) (->Node (insert x a) y b)
            (< y x) (->Node a y (insert x b))
            :else    t))

(defun member
    [_ Empty]
        false
    [x [Node a y b]]
        (cond
            (< x y) (recur x a)
            (< y x) (recur x b)
            :else   true))

As factory constructors are defined as records, we can use ->Node, :node-left, :node-root, :node-right on the actions.

as-patterns

A special as-pattern allows capturing the whole value of a parameter besides its parts. For instance:

(defun insert
    [x Empty] 
        (->Node Empty x Empty)
    [x ([Node a y b] :as t)]
        (cond 
            (< x y) (->Node (insert x a) y b)
            (< y x) (->Node a y (insert x b))
            :else  t))
            

or-patterns

We can define or-patterns to group conditions that correspond to the same action. They only work at the topmost level of a condition. For instance:

(defun ^:private balance
    (:or [Black [Node Red [Node Red a x b] y c] z d] 
         [Black [Node Red a x [Node Red b y c]] z d] 
         [Black a x [Node Red [Node Red b y c] z d]] 
         [Black a x [Node Red b y [Node Red c z d]]]) 
              (->Node Red (->Node Black a x b) y (->Node Black c z d))
    [c a x b]                                         
              (->Node c a x b))

caseof

We can also define conditional code using patterns by means of the caseof macro:

(caseof [t]
    [Empty] true
    [[Node _ _ _]] false)

$-notation

In chapter 4 of the book $-notation is presented to allow lazy-evaluation. We have defined symbol `$ with two complementary meanings, depending on the side of the rule where it appears:

• When used in the action part of the rule ($ expr) is completely equivalent to (delay expr)
• In a pattern, we have that ($ pattern) matched expr when pattern matches (force expr)

For instance, we can define Streams as delayed StreamCells and define:

(defun s-drop_
    [0 s] s
    [n ($ Nil)] ($ Nil)
    [n ($ [Cons x s])] (recur (dec n) s))

defunlazy

In order to simplify the construction of lazy-functions (which return delayed objects) we have defined the equivalent macro using the definition given in the book:

fun lazy f p = e 

is equivalent to

fun f x = $case x of p => force e

For instance, we can define now:

(defunlazy s-append
   [($ Nil) t] t
   [($ [Cons x s]) t] ($ (->Cons x (s-append s t))))

and the streams are not evaluated when applied but when accessed.

Some simplifications

An alternartive to $-notation has been defined to simplify some lazy definitions and datatypes.

Lazy constructors

Some constructors can be lazy: they are evaluated only if needed.

To define a constructor as lazy, we associate the :lazy metadata as true with it.

For instance, we can define the Streams type as

(defdatatype
   ::Streams
   Nil
   (^:datatype.core/lazy Cons elem stream))

which would define the Cons constructor to be lazy.

Using it we can define a function that returns the infinite stream of naturals

(defn nats
    ([]  (nats 0))
    ([n] (->Cons n (nats (inc n)))))
          

deflazy

When we want to define all the constructors of a datatype as lazy, we can use deflazy as a shortcut.

(deflazy
    ::Streams
    Nil
    (Cons elem stream))

and now both constructors would be defined as lazy.

(c) Juan Manuel Gimeno Illa