Uiua

Uiua
Developer	Kai Schmidt
Released	2023
Latest release	0.12.3 / 2024
Array model	flat with boxes
Index origin	0
Function styles	stack-based tacit
Numeric types	64-bit float, complex float
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

Functions with unique glyphs

In Uiua, we can type in the name of the function or a part of it and when formatted, it will format the name to its corresponding Unicode Symbol, so that we don't have to use non-ASCII keyboards (or Alt-codes).

The tables below are based on Uiua version 0.12.0, with since-removed primitives taken out.

Stack

Work with the stack

Table of Stack functions 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
⟜	On	Monadic modifier	Call a function but keep its top argument on top of the stack
⤙	But	Monadic modifier	Call a function but keep its last argument on top of the stack
⤚	With	Monadic modifier	Call a function but keep its top argument below the outputs on the stack
⊸	By	Monadic modifier	Duplicate a function's last argument before calling it
¨	Backward	Monadic modifier	Call a function with its arguments reversed
?	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 functions 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

Table of Dyadic Pervasive functions 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 functions 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 the first occurrences of items in an array
□	Box	Turn an array into a box

Dyadic Array

Operate on two arrays

Table of Dyadic Array functions 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
⮌	Orient	Change the order of the axes of an array.
◫	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
⟔	Coordinate	Find the first deep index of an 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.
≡	Rows	Monadic modifier	Apply a function to each row of an array or arrays.
⊞	Table	Monadic 2-argument modifier	Apply a function to each combination of rows of two arrays.
⍚	Inventory	Monadic modifier	Apply a function to each unboxed row of an array and re-box the results.
⍥	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
⊜	Partition	Group sequential sections of an array
◹	Triangle	Apply a function to each shrinking row of an array

Inversion Modifiers

Work with the inverses of functions

Table of Inversion Modifiers in Uiua

Glyph	Name	Type	Definition
°	Un	Monadic modifier	Invert the behavior of a function
⍜	Under	Dyadic modifier	Apply a function under another

🌎 Planet 🪐

Advanced stack manipulation

Table of Planet 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
⊃	Fork	Dyadic modifier	Call two functions on the same values
⊓	Bracket	Dyadic modifier	Call two functions on two distinct sets of values

Other Modifiers

Table of Other Modifiers in Uiua

Glyph	Name	Type	Definition
◇	Content	Monadic Modifier	Unbox the arguments to a function before calling it
⬚	Fill	Dyadic Modifier	Set the fill value for a function
⨬	Switch	Dyadic modifier	Call the function at the given index

External links

• Uiua on the concatenative language wiki
• Uiua, a Stack-based Array language on the Array Cast
• Documentation
• Interpreter
• Basic tutorial