# bAsedPL


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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](https://answerdotai.github.io/basedpl/rules.html#arrays-nesting-and-fill),
  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](https://answerdotai.github.io/basedpl/glyphs.html)
for Dyalog differences and the [language
rules](https://answerdotai.github.io/basedpl/rules.html) for the array
model.

## Install and try it

``` sh
pip install basedpl
bapl
```

At the prompt, define a mean and apply it:

``` apl
avg←+/÷≢
avg 2 4 9
```

    5

Use `bapl -e 'avg←+/÷≢ ⋄ avg 2 4 9'` for a shell command. The
[command-line guide](https://answerdotai.github.io/basedpl/cli.html)
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](https://answerdotai.github.io/basedpl/repl.html) and
[Keyboard](https://answerdotai.github.io/basedpl/keyboard.html).

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](https://answerdotai.github.io/basedpl/notebooks.html).

## 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:

``` apl
10+1 2 3
```

    11 12 13

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

``` apl
+/⍳10
```

    55

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

``` apl
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:

``` apl
⍸2=+⌿0=|⌝⍨⍳50
```

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

[Getting
started](https://answerdotai.github.io/basedpl/getting-started.html#example-algorithms)
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:

``` apl
1r3+1r6
```

    1r2

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

``` apl
√¯4
```

    0j2

See [numbers](https://answerdotai.github.io/basedpl/rules.html#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:

``` apl
m←[1 2 3 ⋄ 4 5 6]
m+10 20
```

    11 12 13
    24 25 26

See [array
notation](https://answerdotai.github.io/basedpl/glyphs/brackets.html)
and
[broadcasting](https://answerdotai.github.io/basedpl/rules.html#agreement-and-pervasion).

### 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:

``` apl
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](https://answerdotai.github.io/basedpl/keyed.html).

### Function operators

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

``` apl
(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:

``` apl
⌊⌾(10∘×)⊢1.25 2.78
```

    1.2 2.7

Explore [iteration and
inverses](https://answerdotai.github.io/basedpl/glyphs/power.html),
[Under](https://answerdotai.github.io/basedpl/glyphs/under.html),
[windows](https://answerdotai.github.io/basedpl/glyphs/windows.html) and
[function
selection](https://answerdotai.github.io/basedpl/glyphs/agenda.html).

### Mathematical tools

[Primes](https://answerdotai.github.io/basedpl/glyphs/prime.html) and
[factorisation](https://answerdotai.github.io/basedpl/glyphs/factor.html)
are built in:

``` apl
⨸360x
```

    2ₓ 2ₓ 2ₓ 3ₓ 3ₓ 5ₓ

[Polynomials](https://answerdotai.github.io/basedpl/glyphs/polynomial.html)
support coefficients, roots and evaluation. Polynomial functions can be
[differentiated](https://answerdotai.github.io/basedpl/glyphs/derivative.html):
for f(x) = 1 + 2x + 3x², f′(2) = 14.

``` apl
f←1x 2x 3x∘⊛ ⋄ f∂2x
```

    14ₓ

[Probability
distributions](https://answerdotai.github.io/basedpl/distributions.html)
provide sampling, density, CDF and quantiles. Two fair coin tosses give
these probabilities for 0, 1 and 2 heads:

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

    0.25 0.5 0.25

[Matrix
division](https://answerdotai.github.io/basedpl/glyphs/domino.html)
handles linear systems and least squares.

### Data and text

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

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

    21

CSV headers likewise name column vectors. [Files, CSV and
JSON](https://answerdotai.github.io/basedpl/data.html) covers reading,
transforming and writing data.
[Regex](https://answerdotai.github.io/basedpl/regex.html) 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](https://answerdotai.github.io/basedpl/plot.html).

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

![](index_files/figure-commonmark/cell-17-output-1.svg)

Build SVG from element functions and keyed attributes. Notebooks display
the picture directly. The same element trees serialize to XML. See [XML
and SVG](https://answerdotai.github.io/basedpl/xml.html).

``` apl
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)
```

![](index_files/figure-commonmark/cell-18-output-1.svg)

## Python

APL functions are Python callables:

``` python
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](https://answerdotai.github.io/basedpl/python.html).

For other frontends, the [process
interfaces](https://answerdotai.github.io/basedpl/processes.html)
provide JSON messages and interruptible workers. The [APL
library](https://github.com/AnswerDotAI/basedpl/tree/main/lib) contains
more algorithms, codecs, interpreters and puzzles. See
[DEV.md](https://github.com/AnswerDotAI/basedpl/blob/main/DEV.md) for
source installation and contributing.