# Glyph reference Each name below links to its definitions and examples. Examples show an equivalent APL value after `⍝`. A monad takes one argument; a dyad takes two. Operators take functions or arrays and derive functions. Notes give glyph-specific differences from Dyalog APL. ## Functions
Glyph Monad Dyad Note
+ Add Conjugate Add Character offsets as BQN: 'a'+3 is 'd'
- Dash Negate Subtract Character difference as BQN: 'd'-'a' is 3x
× Mul Direction Multiply
÷ Div Reciprocal Divide Exact rationals as J: 1x÷3x is 1r3
⌊ Floor Floor Minimum ⌊/⍬ is ∞
⌈ Ceiling Ceiling Maximum ⌈/⍬ is ¯∞
\| Stile Magnitude Residue
* Star Exponential Exponent
⍟ Log Natural log Logarithm
○ Circle Unit circle Circular functions Monad is *0j1×Y, a unit-circle point. Pi times is π
π Pi Pi times Pi fraction Monad takes over Pi times from ○. XπY is Xπ÷Y
√ Root Square root Nth root As BQN’s √, with complex results: √¯4 is 0j2
! Factorial Factorial Binomial
ℙ Prime Nth prime (one-based) Prime operations As J’s p: with the same codes, but one-based
⨸ Factor Prime factors Exponents / factor table As J’s q:, with ∞ and ¯∞ for J’s _ and __
⊛ Polynomial Roots / coefficients Evaluate As J’s p.: coefficients are constant-first
? Question Roll Deal
∨ Or Real / imaginary parts OR / GCD Monad as J’s +.
∧ And Magnitude / angle AND / LCM Monad as J’s *.
⍲ Nand Square NAND Monad as J’s *:
⍱ Nor Double NOR Monad as J’s +:
~ Tilde NOT Without
= Equal Self-classify Equal Monad as J: one mask row per distinct major cell
≠ Not equal Unique mask Not equal
< Less — Less
≤ Less or equal Decrement Less or equal Monad as J’s <:
> Greater — Greater
≥ Greater or equal Increment Greater or equal Monad as J’s >:
≡ Match Depth Match
≢ Tally Tally Not match
⍴ Rho Shape Reshape ⍬⍴Y gives a rank-0 array: ⍬⍴1 2 is ⊂1
, Comma Ravel Catenate
⍪ Table Table Catenate first
⌽ Reverse Reverse last Rotate last
⊖ Reverse first Reverse first Rotate first
⍉ Transpose Transpose Reorder / diagonal axes
↑ Take First Take Monad as Dyalog ⎕ML≥2, but takes a major cell: ↑2 3⍴⍳6 is 1 2 3
↓ Drop Split Drop
↕ Windows — Leading-axis windows As BQN’s ↕, plus movements and padding (negative sizes) as in ⌺. Replaces windowed reduce: +/2↕1 2 3 4 is 3 5 7
⊂ Enclose Enclose Partitioned enclose Encloses atoms too, as BQN: (⊂3)≡3 is 0x
⊆ Nest Nest Partition
⊃ Mix Mix Pick Monad as Dyalog ⎕ML≥2. Pick takes cells: 2⊃2 3⍴⍳6 is 4 5 6. A string picks by key
⌷ Squad Identity Index
⍳ Iota Index generator Index of ⍳[axes]Y returns axis selectors
⍸ Where Where Interval index
∊ Member Enlist Membership
∪ Union Unique Union
∩ Intersection — Intersection
⍷ Find — Find
⍋ Grade up Grade up Grade with collation
⍒ Grade down Grade down Grade with collation
/ Slash — Replicate last
⌿ Slash bar — Replicate first
\ Backslash — Expand last
⍀ Backslash bar — Expand first
⊤ Encode Binary encode Encode Monad as J’s #:, but digits run along the first axis
⊥ Decode Binary decode Decode Monad as J’s #., but digits run along the first axis
⌹ Domino Matrix inverse Matrix divide
⊣ Left Identity Left
⊢ Right Identity Right
⍎ Execute Execute Keyed lookup: X⍎Y is Y⊃X Dyad runs no code. Dyalog executes Y in namespace X
⍕ Format Format Format by specification
## Operators `f`, `g` are functions; `a`, `n`, `r` are arrays or numbers.
Glyph Form Meaning Note
/ Slash f/ Reduce / seeded reduce last Seed as BQN’s ´: 10 -/1 2 3 is ¯8. For windows use ↕
⌿ Slash bar f⌿ Reduce / seeded reduce first Seed and windows as /
\ Backslash f\ Scan / seeded scan last Accumulates left to right as BQN’s Scan: -\1 2 3 is 1 ¯1 ¯4
⍀ Backslash bar f⍀ Scan / seeded scan first Left to right as \
¨ Each f¨ Each
⍨ Commute f⍨, a⍨ Self / commute / constant
∘ Compose f∘g, a∘f, f∘a Compose / bind
⍤ Rank f⍤g, f⍤r Atop / rank
⍥ Over f⍥g Over
⍛ Behind f⍛g Behind
. Dot f.g Inner product Outer product is g⌝. After an array, T.name is 'name'⊃T
⌝ Outer product g⌝ Outer product As BQN’s ⌜; ∘.g is a SYNTAX ERROR
⌸ Key f⌸ Key
⍣ Power f⍣n, f⍣g, f⍣[p] Iterate / invert / repeat until; enclose count or predicate for history Array counts as J’s ^:: (1∘+)⍣3 ¯2 0 3⊢10 is 13 8 10 13
⇄ Inverse pair f⇄g Attach an explicit inverse As J’s :.
⌾ Under f⌾g Transform, apply, inverse-transform As J’s &.
∂ Derivative f∂ Gradient / vector–Jacobian product
◶ Agenda selector◶cases Select and call one function As BQN’s ◶, but one-based
@ At f@a, a@g Functional amend
⌺ Stencil f⌺a Stencil
## Syntax and literals
Form Meaning
˘ Strand Form a vector of values: 1˘+˘'abc'
← Assign Assignment, including modified/indexed/selective forms
→ Pipe Left-to-right function application
(…) Parentheses Grouping / nested array literals / trains
[…] Brackets Enclosure, array literals, indexing and axes
; Semicolon Index-axis separator
{…} Braces Defined function or operator
⍺ Alpha, ⍵ Omega Left / right argument
⍶ Alpha underbar, ⍹ Omega underbar Left / right operand
∇ Del, ⍢ Del diaeresis Function / operator self-reference
: Colon, :: Error guard Unkey / key axes; Boolean guard / error guard inside dfns
⋄ Diamond Statement or literal separator
⍝ Comment Comment
⎕← Quad Explicit output
• Bullet System name prefix
⍬ Zilde Empty numeric vector
'…' Quote Character literal
¯ Overbar Negative literal sign
∞ Infinity Real infinity
Numeric notation `x`, `r`, `j`, `E`: see [numbers](rules.qmd#numbers). ## System names Names are case-insensitive. `•a` and `•d` are constant arrays; the other names are functions, usable with operators and composition.
Name Meaning
•a Alphabet Uppercase Latin alphabet
•d Digits Decimal digits
•c Case Unicode case conversion
•csv CSV CSV text → keyed column vectors
•tocsv CSV Keyed column vectors → CSV text
•r Regex Compiled search, captures and replacement
•normal, •binomial, … Distributions Sampling, density, CDF and quantiles; 17 families
•json JSON JSON text → arrays and keyed vectors
•tojson JSON Arrays and keyed vectors → JSON text
•vfi Numeric input Parse numeric fields with a validity mask
•nget Read Read UTF-8 text or bytes
•nput Write Write UTF-8 text or bytes
•element, •xml, •svg XML and SVG Build, serialize and display element trees
•mime Rich display The MIME bundle that display uses
•plot Plots Charts from vectors, matrices and tables
•ucs Unicode Unicode code points / encodings
•load Load Evaluate an APL source file
•signal Signal Raise an ordinary APL error
•nc Name class Classify visible names
•nl Name list List names by class and prefix
•src Source Function/operator source
•ex Expunge Erase bindings
Index origin: 1. Comparison tolerance: `1E¯14`. ## Language-wide differences from Dyalog APL - [Atoms](rules.qmd#arrays-nesting-and-fill): numbers, characters and functions are atoms, distinct from scalars, as BQN. Enclosing always adds a layer: `(⊂3)≡3` is `0x`. - [Numbers](rules.qmd#numbers): bare numbers are approximate. `x` and `r` mark exact integers and rationals, as J: `1x÷3x` is `1r3`. Predicates, positions, tally and shape are exact. - [Agreement](rules.qmd#agreement-and-pervasion): scalar functions, Each and Rank align leading axes, and unit dimensions expand: `(2 3⍴⍳6)+10 20` is `[11 12 13 ⋄ 24 25 26]`. - [System names](#system-names): written `•name`, as BQN, not `⎕NAME`. Most Dyalog system functions and variables are not included. - [Axis keys](keyed.qmd) name data. `K:Y` evaluates its keys. `T.a` is `'a'⊃T`, and `X⍎Y` is `Y⊃X`.