Uiua

Uiua
Developer	Kai Schmidt
Released	2023
Latest release	0.8.0 / 2024
Array model	flat with boxes
Index origin	0
Function styles	stack-based tacit
Numeric types	64-bit float
Unicode support	full
File extensions	.ua
Implementation languages	Rust
Source code	GitHub
Platforms	native, in-browser
License	MIT
Website	uiua.org
Documentation	Documentation
Run online	Pad
Influenced by	Forth, BQN, J

Uiua is a stack-based array language emphasizing tacit programming, designed by Kai Schmidt. Initially influenced strongly by BQN, Uiua has added original stack-based features and glyphs and moved to the boxed array model of SHARP APL and J. It also makes functions second-class rather than first-class to facilitate features that depend on the number of arguments functions take.

Overview

Uiua uses concatenative evaluation (a context-free grammar) with a right-to-left ordering as in Polish notation. The language supports tacit programming using stack manipulation primitives, and all complex functions must be defined this way as there is no explicit function form that allows local variables. Functions have a fixed number of input and output values, meaning the overloading of ambivalent functions is removed. Because of this, Uiua often splits APL primitives into two functions. Its primitives use Unicode glyphs including many not found in other languages. To avoid typing glyphs necessitating a keyboard layout containing these, each primitive can also be spelled using a name that consists of lowercase letters (user-defined names must have at least one uppercase letter). By default, the language formats source files when run to convert these names into the corresponding glyphs.

Like the SHARP APL family, Uiua supports only flat arrays with a homogeneous type, and has a dedicated box type for array nesting. However, arithmetic is pervasive as in the nested array model. Arithmetic also follows leading axis agreement, extended so that a length-1 axis is matched with an axis of any length by replicating along it (like the broadcasting system of Julia and NumPy). It supports character arithmetic as in BQN, as well as other primitives characteristic of BQN such as structural Under, Classify, and a Transpose that rotates the axes of its argument by one.

Primitives

Function with unique glyph

In the web version of Uiua, we can type in the name of the function or a part of it and when run, the interpreter will format the name to Unicode Symbol, so that we don't need use of non-ASCII keyboards.

All table below are based on Uiua version 0.8.0-1/28/2024.

Yellow background text mean that the glyph is deprecated and will be removed in a future version

And 🧪 emoji mean that the glyph is experimental and may be changed or removed in the future.

Stack

Work with the stack

Table of Stack function in Uiua

Glyph	Name	Type	Definition
.	Duplicate	Monadic 2-output function	Duplicate the top value on the stack
,	Over	Dyadic 3-output function	Duplicate the second-to-top value to the top of the stack
:	Flip	Dyadic 2-output function	Swap the top two values on the stack
◌(;)	Pop	Monadic 0-output function	Discard the top stack value
?	Stack	Noadic 0-output function	Debug print all stack values without popping them
⸮	Trace	Monadic function	Debug print the top value on the stack without popping it

Constants

Push a constant value onto the stack

Table of Constants in Uiua

Glyph	Name	Definition
η	Eta	The number of radians in a quarter circle
π	Pi	The ratio of a circle's circumference to its diameter
τ	Tau	The ratio of a circle's circumference to its radius
∞	Infinity	The biggest number

Monadic Pervasive

Operate on every element in an array

Table of Monadic Pervasive in Uiua

Glyph	Name	Definition
¬	Not	Logical not
±	Sign	Numerical sign (1, ¯1, or 0)
¯	Negate	Negate a number
⌵	Absolute value	Get the absolute value of a number
√	Sqrt	Take the square root of a number
○	Sine	Get the sine of a number
⌊	Floor	Round to the nearest integer towards ¯∞
⌈	Ceiling	Round to the nearest integer towards ∞
⁅	Round	Round to the nearest integer

Dyadic Pervasive

Operate on every pair of elements in two arrays (Note that True is 1 and False is 0)

Table of Monadic Pervasive in Uiua

Glyph	Name	Definition	Formats from:
=	Equals	Compare for equality	=
≠	Not equals	Compare for inequality	!=
<	Less than	Compare for less than	<
>	Greater than	Compare for greater than	>
≤	Less or equal	Compare for less than or equal	<=
≥	Greater or equal	Compare for greater than or equal	>=
+	Add	Add values	+
-	Subtract	Subtract values	-
×	Multiply	Multiply values	*
÷	Divide	Divide values	%
◿	Modulus	Modulo values	mod
ⁿ	Power	Raise a value to a power	pow
ₙ	Logarithm	Get the based logarithm of a number	log
↧	Minimum/Logical And	Take the minimum of two arrays (Can use for logical AND.)	min
↥	Maximum/Logical Or	Take the maximum of two arrays (Can use for logical OR.)	max
∠	Atangent	Take the arctangent of two numbers	atan
ℂ	Complex	Make a complex number from imaginary and real parts	com

Monadic Array

Operate on a single array

Table of Monadic Array in Uiua

Glyph	Name	Definition
⧻	Length	Get the number of rows in an array
△	Shape	Get the dimensions of an array
⇡	Range	Make an array of all natural numbers less than a number
⊢	First	Get the first row of an array
⇌	Reverse	Reverse the rows of an array
♭	Deshape	Make an array 1-dimensional
¤	Fix	Add a length-1 axis to an array
⋯	Bits	Encode an array as bits (LSB-first)
⍉	Transpose	Rotate the shape of an array
⍏	Rise	Get the indices into an array if it were sorted ascending
⍖	Fall	Get the indices into an array if it were sorted descending
⊚	Where	Get indices where array values are not equal to zero
⊛	Classify	Assign a unique index to each unique element in an array
◴	Deduplicate	Remove duplicate elements from an array
◰	Unique	Get a mask of first occurrences of items in an array
□	Box	Turn an array into a box This is Uiua's primary way to create nested or mixed-type arrays. Normally, arrays can only be created if their rows have the same shape and type. ⬚ fill can help you with the shape part, but it is not always wanted, and it can't help with the type part. You can go to (To See more information about the □ box function)

Dyadic Array

Operate on two arrays

Table of Monadic Array in Uiua

Glyph	Name	Definition
≍	Match	Check if two arrays are exactly the same
⊟	Couple	Combine two arrays as rows of a new array
⊂	Join	Append two arrays end-to-end
⊏	Select	Select multiple rows from an array
⊡	Pick	Index a row or elements from an array
↯	Reshape	Change the shape of an array
☇	Rerank	Change the rank of an array's rows
↙	Take	Take the first n elements of an array
↘	Drop	Drop the first n elements of an array
↻	Rotate	Rotate the elements of an array by n
◫	Windows	The n-wise windows of an array
▽	Keep	Discard or copy some rows of an array
⌕	Find	Find the occurrence's of one array in another
∊	Member	Check if each row of one array exists in another
⊗	Indexof	Find the first index of each row of one array in another

Iterating Modifiers

Iterate and apply a function to an array or arrays

Table of Iterating Modifiers in Uiua

Glyph	Name	Type	Definition
∵	Each	Monadic modifier	Apply a function to each element of an array or arrays. This is the element-wise version of ≡ rows.
≡	Rows	Monadic modifier	Apply a function to each row of an array or arrays. This is the row-wise version of ∵ each.
⊞	Table	Monadic 2-argument modifier	Apply a function to each combination of rows of two arrays. This is often what you want instead of ∵ each.
⊠	Cross	Monadic 2-argument modifier	Apply a function to each combination of rows of arrays. This was the row-wise version of ⊞ Table
⍥	Repeat	Monadic modifier	Repeat a function a number of times
⍢	Do	Dyadic modifier	Repeat a function while a condition holds

Aggregating Modifiers

Apply a function to aggregate an array

Table of Aggregating Modifiers in Uiua

Glyph	Name	Definition
/	Reduce	Apply a reducing function to an array. For reducing with an initial value, see ∧ fold
∧	Fold	Apply a function to aggregate arrays.
\	Scan	Reduce, but keep intermediate values
⊕	Group	Group elements of an array into buckets by index Takes a function and two arrays. The arrays must be the same ⧻ length. The first array must be rank 1 and contain integers. Rows in the second array will be grouped into buckets by the indices in the first array. Keys <0 will be omitted. The function then processes each group in order. The result depends on what the function is. If the function takes 0 or 1 arguments, then ⊕ group behaves like ≡ rows. This is called iterating ⊕ group. Expects as many arguments as its function takes. The function must take at least 1 more argument than it returns outputs. Arguments that are lower on the stack that will be used as accumulators. Arguments that are higher on the stack will be iterated over. The function will be repeatdely called with the rows of the iterated arrays followed by the accumulators. On each iteration, the returned values will be used as the new accumulators.
⊜	Partition	Group sequential sections of an array The most common use of ⊜ partition is to split an array by a delimiter. Takes a function and two arrays. The arrays must be the same ⧻ length. The first array must be rank 1 and contain integers. Consecutive rows in the second array that line up with groups of the same key in the first array will be grouped together. Keys ≤0 will be omitted. The function then processes each group in order. The result depends on what the function is. If the function takes 0 or 1 arguments, then ⊜ partition behaves like ≡ rows. This is called iterating ⊜ partition.

Inversion Modifiers

Work with the inverses of functions

Table of Iterating Modifiers in Uiua

Glyph	Name	Type	Definition
°	Un	Monadic modifier	Invert the behavior of a function Most functions are not invertible.
⌅	🧪Rectify	Monadic modifier	🧪Set a function as its own inverse
⍜	Under	Dyadic modifier	Apply a function under another This is a more powerful version of ° un. Conceptually, ⍜ under transforms a value, modifies it, then reverses the transformation. ⍜ under takes 2 functions f and g and another argument x. It applies f to x, then applies g to the result. It then applies the inverse of f to the result of g. Any function that can be ° uned can be used with ⍜ under. Some functions that can't be ° uned can still be used with ⍜ under. You can go to (To See more information about the ⍜ under function)

🌎 Planet 🪐

Advanced stack manipulation

Table of Iterating Modifiers in Uiua

Glyph	Name	Type	Definition
∘	Identity	Monadic function	Do nothing with one value
⋅	Gap	Monadic modifier	Discard the top stack value then call a function
⊙	Dip	Monadic modifier	Temporarily pop the top value off the stack and call a function
∩	Both	Monadic 2-argument modifier	Call a function on two sets of values For monadic functions, ∩ both calls its function on each of the top 2 values on the stack.
⊃	Fork	Dyadic modifier	Call two functions on the same values ⊃ fork is one of the most important functions for working with the stack.
⪾	🧪Cascade	Dyadic modifier	🧪Call one function after another, reusing some values ⪾ cascade's second function is called, then its first argument(s) are reused, along with its output, as arguments to the first function.
⊓	Bracket	Dyadic modifier	Call two functions on two distinct sets of values
⋔	🧪All	Dyadic modifier	🧪Call a function on many distinct sets of values For just 2 sets of values, ∩ both is often simpler. The second function will be called, then the first function will be called on groups of values. To do something similar to ∩ both on more than 2 sets of values, you can use ⊙ dip and ∘ identity to select values.

External links

• Uiua on the concatenative language wiki
• Uiua, a Stack based Array language on the Array Cast
• Doc for the Documentation
• Interpreter for the interpreter
• Basic to start learning Uiua