lynn's house

Reader-style Uiua

Lately I’ve been playing with Uiua, an APL-like programming language that has no local variables on purpose. Instead, you describe how data flows through your function using combinators. One can also think of the language as stack-based, though the official documentation has recently started to discourage this perspective.

I appreciate the die-hard commitment to tacit, point-free programming on an aesthetic level, but on a practical level I find it often gets in the way. I’ve landed on a nice way to make programming in Uiua feel more like a regular stack-based language. We can write functions that keep the top-of-stack fixed to some ambient passed-in context, that we constantly either read parts from, or dip past.

Here’s an example of a quadratic discriminant function, Disc(c,b,a) = b²−4ac, side-by-side with what we might write if Uiua had local variables:

Disc ← (           # Disc(C_B_A) ← (
  ~ {C B A}        #
  ⊙ⁿ₂ ⟜B           #   ⁿ₂ B
  ⊙× ⊙4 ⊙× ⟜A ⟜C   #   × 4 × A C
  ⊙-               #   -
  ◌                #
)

We define the shape of the context using a local data definition ~. This context stays on top, and everything underneath it is the effective stack, which starts empty.

Now here’s how we write our code:

  1. Every time we want to read from our context, we use ⟜F to push F(context) to the top of the effective stack.
  2. To manipulate the effective stack any other way, we use ⊙ to dip past the context.
  3. At the end, we pop ◌ the context, leaving only the effective stack values.

I want to call this style reader style, because in a way, this is like programming in a Reader monad: ⟜f is like asks f and ⊙ is like pure or fmap or liftA2. For comparison, here is an equivalent Haskell definition of disc, which is also “point-free without really being point-free.”

import Control.Monad.Reader
import Control.Applicative

data Quadratic =
  Quadratic { c, b, a :: Float } deriving (Eq, Show)

disc :: Reader Quadratic Float
disc =
  liftA2 (-)
    (fmap (^2) (asks b))
    (liftA2 (*) (pure 4) $ liftA2 (*) (asks a) (asks c))

This style can and should be mixed freely with regular flavors of Uiua. For example, ⊙× ⊙4 ⊙× ⟜A ⟜C could be written ⟜(×₄×⊃A C), where we use a small, easy-to-read tacit function as our accessor function.

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