BasedPL

BasedPL, the Based-array Programming Language, is an APL-derived array language, borrowing ideas from J and BQN, with an emphasis on simple, consistent notation. Most of this documentation calls it BPL. 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.
• Brackets write vectors, and spaces group: [a b c] is a vector, and a+b × c+d is (a+b)×(c+d). A run that ends in a function is a train, so +/÷≢ x is the mean of x, and 2× binds 2. An array next to an argument selects from it: v 0 is the first item.
• Leading-axis broadcasting, including expansion of length-1 axes, 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.

Positions and axes count from 0, as in BQN and Python. Approximate comparisons use tolerance 1E¯14. See the language principles for why BPL works this way, the glyph reference for Dyalog differences and the language rules for the array model.

Install and try it

pip install basedpl
bpl

At the prompt, define a mean and apply it:

avg←+/÷≢
avg 2 4 9

5

Use bpl -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 the glyph reference, which lists each glyph’s key.

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

Numbers separated by 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 0…9:

+/⍳10

45

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 of 1 to 50 (|⊗⍨1+⍳50), count the zeros down each column (+⌿0=), and find the positions (⍸) whose count is two. Positions count from 0, so 1+ turns them back into numbers:

1+⍸2=+⌿0=|⊗⍨1+⍳50

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

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":["NY" "LA"] "month":["Jan" "Feb" "Mar"]]
sales←axes:[10 20 30 ⋄ 40 50 60]
"LA" "Feb"⌷sales
+/⍠"month" sales

50

["city":2]⍴["NY":60 "LA":150]

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

Function operators

A vector of counts keeps the history: one state for each count, where count 0 is the initial value. Here, double up to four times:

2×⍣(⍳5) 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

[2 2 2 3 3 5]ₓ

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

f←[1 2 3]ₓ⌻ ⋄ 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 sales

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

BPL 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 BPL library contains more algorithms, codecs, interpreters and puzzles.

Install from source

Requires Python ≥3.10 and Rust ≥1.98. On x86-64, it also requires a CPU with AVX2 (x86-64-v3). In your Python environment:

git clone https://github.com/AnswerDotAI/basedpl.git
cd basedpl
pip install .

For a standalone executable without Python, run cargo install --path .. Cargo installs it in its bin directory, normally ~/.cargo/bin. Put that directory on your PATH.

Contributing

Install the development and documentation tools with pip install -e '.[dev]'. The main commands are:

cargo test
cargo run -- -e '2×3+4'
cargo fastfmt
maturin develop
pytest
ship-rs-build

Rebuild with maturin develop after Rust changes, before checking the installed extension. Cargo tests alone don’t update the editable Python installation. Use cargo fastfmt, not cargo fmt.

Before a release, run python scripts/prep.py from the repository root. It checks the glyph reference against the interpreter’s glyph table, writes the syntax highlighters’ glyph lists, cleans and exports the notebooks, and renders this README.

Use lowercase j in complex literals throughout tests and examples, including adapted reference cases. Reserve uppercase J for explicit input-alias tests. Keep archived upstream source unchanged.

Write literal matrices in array notation, [10 20 30 ⋄ 40 50 60], not as a reshape, 2 3⍴10 20 30 40 50 60. Keep ⍴ where the example is about reshape.

Brackets are the usual way to write a list. A bare literal list, such as 1 2 3, is a shorthand. End an operand with a space, not with parentheses or ⊢. Where a literal operand would run into the argument, write the argument in parentheses: f⍤1(2 3). Report any case where neither a space nor parentheses works.

A name applied to a literal touches it, as a glyph does: fib10, v¯1 and 1+⌽f5. Write it that way where it’s shorter and needs no new parentheses. Keep the space where the literal would join the next part of the expression, or where an array before the name is its left argument.