bAsedPL

bAsedPL (“Based-array APL”) is an APL-derived array language, borrowing ideas from J and BQN, with an emphasis on simple, consistent notation. It is implemented in Rust, with a native executable, Jupyter kernel and Python API.

For APL users, the main choices are:

• Based arrays, as in BQN: numbers, characters and functions are atoms; enclosure always adds a layer.
• Leading-axis broadcasting, including unit-axis expansion, plus string keys and names on axes.
• Exact integers and rationals alongside approximate real and complex numbers.
• Dfns, trains and operators, with additions such as Under, iteration histories, windows and function arrays.

Indices start at 1; approximate comparisons use tolerance 1E¯14. See the glyph reference for Dyalog differences and the language rules for the array model.

Install and try it

pip install basedpl
bapl

At the prompt, define a mean and apply it:

avg←+/÷≢
avg 2 4 9

5

Use bapl -e 'avg←+/÷≢ ⋄ avg 2 4 9' for a shell command. The command-line guide covers source files and pipes.

Interactive use

In the REPL, type a backtick followed by a glyph name: `iota becomes ⍳ when you press Tab or type a non-letter. Abbreviations and Alt-key shortcuts are available. ]help + shows help; ]box on -style=max -trains=tree -fns=on enables boxed arrays and function trees. See REPL and Keyboard.

In Jupyter, select the installed bAsedPL kernel. Cells share definitions and support completion, Shift-Tab help and interruption. You can also use %%apl cells in a Python notebook. See Using bAsedPL notebooks.

New to APL?

APL is a language built around operations on whole arrays and notation for combining functions. Here is a taste of that style in bAsedPL.

Spaces form a vector. Arithmetic applies to every element:

10+1 2 3

11 12 13

Operators modify or combine functions. Reduce (/) turns addition into summation; ⍳10 generates 1…10:

+/⍳10

55

Functions can also be combined without naming their arguments. In avg←+/÷≢, sum (+/) divided by tally (≢) defines the mean:

avg 1 2 3 4

2.5

To see how these ideas express an algorithm, start from “a prime has exactly two positive divisors”. Form all remainders (|⌝⍨), count the zeros down each column (+⌿0=), and find the positions (⍸) whose count is two:

⍸2=+⌿0=|⌝⍨⍳50

2ₓ 3ₓ 5ₓ 7ₓ 11ₓ 13ₓ 17ₓ 19ₓ 23ₓ 29ₓ 31ₓ 37ₓ 41ₓ 43ₓ 47ₓ

Getting started builds this expression step by step, displaying the divisibility matrix along the way.

What’s distinctive?

Numbers

Bare numbers are approximate. Use x for exact integers and r for exact fractions:

1r3+1r6

1r2

Complex numbers use j between real and imaginary parts. Functions such as square root extend into the complex domain:

√¯4

0j2

See numbers for conversion and mixed arithmetic.

Array literals and broadcasting

Write matrix rows directly in an array literal. Leading-axis agreement lets a vector supply one offset per row:

m←[1 2 3 ⋄ 4 5 6]
m+10 20

11 12 13
24 25 26

See array notation and broadcasting.

Keys and named axes

Axes can have names, and positions along them can have string keys. Describe the axes once, then select by key or reduce by axis name:

axes←('city':'London' 'Paris' ⋄ 'month':'Jan' 'Feb' 'Mar')
sales←axes:[10 20 30 ⋄ 40 50 60]
sales['Paris';'Feb']
+/['month']sales

50

('London':60 ⋄ 'Paris':150)

Keys and names travel with axes through operations such as transpose. Arithmetic aligns matching names and keys. See Axis keys.

Function operators

Enclose an iteration count to keep the history, including the initial value. Here, double four times:

(2∘×)⍣[4]⊢1

1 2 4 8 16

Under (⌾) transforms the argument, applies a function, then reverses the transformation. Scale by ten, floor, and scale back to round down to tenths:

⌊⌾(10∘×)⊢1.25 2.78

1.2 2.7

Explore iteration and inverses, Under, windows and function selection.

Mathematical tools

Primes and factorisation are built in:

⨸360x

2ₓ 2ₓ 2ₓ 3ₓ 3ₓ 5ₓ

Polynomials support coefficients, roots and evaluation. Polynomial functions can be differentiated: for f(x) = 1 + 2x + 3x², f′(2) = 14.

f←1x 2x 3x∘⊛ ⋄ f∂2x

14ₓ

Probability distributions provide sampling, density, CDF and quantiles. Two fair coin tosses give these probabilities for 0, 1 and 2 heads:

coin←•binomial 2 0.5
coin.density 0 1 2

0.25 0.5 0.25

Matrix division handles linear systems and least squares.

Data and text

JSON objects become keyed arrays, with dot access to their fields:

order←•json '{"price":10.5,"qty":2}'
order.price×order.qty

21

CSV headers likewise name column vectors. Files, CSV and JSON covers reading, transforming and writing data. Regex supplies matching, captures and replacement through Rust’s regex engine.

Drawing

•plot draws charts from arrays. Keys label the axes and name the lines. See Plots.

('legend':'end') •plot ('city':'London' 'Paris' ⋄ 'month':'Jan' 'Feb' 'Mar'):[10 20 30 ⋄ 40 50 60]

Build SVG from element functions and keyed attributes. Notebooks display the picture directly. The same element trees serialize to XML. See XML and SVG.

circle←•element 'circle'
text←•element 'text'
c←('cx':50 ⋄ 'cy':40 ⋄ 'r':25 ⋄ 'fill':'orange') circle ''
t←('x':50 ⋄ 'y':85 ⋄ 'text-anchor':'middle') text 'Hello, SVG'
('width':240 ⋄ 'height':240) •svg (c ⋄ t)

Python

APL functions are Python callables:

from basedpl import fn

mean = fn('+/÷≢')
mean([1, 2, 3])

Arrays have .py, .np and .df conversions for Python values, NumPy and pandas. Functions also have Python names and composition operators. See the Python tutorial.

For other frontends, the process interfaces provide JSON messages and interruptible workers. The APL library contains more algorithms, codecs, interpreters and puzzles. See DEV.md for source installation and contributing.