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# A history of APL in 50 functions
# But they're BQN functions
# Still about APL history though
# Also not all the functions are here yet
# https://www.jsoftware.com/papers/50/
# TODO: 13, 16, 18, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 41
# Some of the above are purely speculative snippets on the website, and may not be added to this example.
# Utilities
Words ← (⊢-˜¬×+`)∘=⟜' '⊸⊔ # Split string on spaces
Rand ← •rand.Range
# 0
ArrayLogic ← (>-<)⟜0
! 1‿¯1‿0‿1 ≡ ArrayLogic 5‿¯2.7‿0‿6
# 1
Average ← +´÷≠
! 3 ≡ Average 2‿1‿6
# 2
# x⌹x=x is an alternate way to get the average of vector x.
# There's no reason to do it this way in BQN (or APL, really).
# 3
IndexOfSelfie ← ⊐˜
! 0‿1‿2‿3‿4‿1 ≡ IndexOfSelfie "Selfie"
# 4
BarChart ← ".⎕"⊏˜>⌜⟜(↕⌈´)
! {
bc ← BarChart 3‿1‿4‿1‿5‿9
bc ≡ 6‿9⥊"⎕⎕⎕......⎕........⎕⎕⎕⎕.....⎕........⎕⎕⎕⎕⎕....⎕⎕⎕⎕⎕⎕⎕⎕⎕"
}
# 5
ParenthesesNesting ← +`1‿¯1‿0⊏˜"()"⊐⊢
! {
pn ← ParenthesesNesting "⍵((∇<S),=S,(∇>S))⍵⌷⍨?≢⍵"
pn ≡ 0‿1‿2‿2‿2‿2‿1‿1‿1‿1‿1‿2‿2‿2‿2‿1‿0‿0‿0‿0‿0‿0‿0
}
# 6
PerfectShuffle ← ⊢⊏˜·⍋∘⍒≠⥊0‿1˜
! "IAJBKCLDMENFOGPHQ" ≡ PerfectShuffle "ABCDEFGHIJKLMNOPQ"
# 7
_quicksort ← {Cmp←𝔽 ⋄ S←{p←𝕩Cmp˜𝕩⊏˜Rand≠𝕩 ⋄ ∾1‿0‿1𝕩{S⍟𝕨𝕗/˜0𝕏p}¨<‿=‿>}⍟(1<≠)}
! 1‿2‿2‿3‿5‿7‿9‿10‿10‿10‿11‿14‿16 ≡ -_quicksort 2‿2‿7‿10‿10‿11‿3‿10‿14‿5‿9‿1‿16
! {
Cmp ← ≢◶0‿(-´∘⍋∾○<)¨
srt ← Words "Anna Fi JD Jay Jd John Morten Roger Scott Zeus"
srt ≡ Cmp _quicksort Words "Fi Jay John Morten Roger JD Jd Anna Scott Zeus"
}
# 8
PascalsTriangle ← {>𝕩↑¨0(∾+∾˜)⍟(↕𝕩)⥊1}
! (>⟨1‿0‿0‿0⋄1‿1‿0‿0⋄1‿2‿1‿0⋄1‿3‿3‿1⟩) ≡ PascalsTriangle 4
! 1‿1‿2‿3‿5‿8‿13‿21‿34‿55 ≡ { 𝕩 ↑ +´¨ (+⌜˜↕𝕩) ⊔○⥊ PascalsTriangle 𝕩 } 10
# 9
GoldenRatio ← +⟜÷´¨∘(1↓↑) ⥊⟜1
! 1e¯5 ∧´∘> | (GoldenRatio 16) - 1‿2‿1.5‿1.66667‿1.6‿1.625‿1.61538‿1.61905‿1.61765‿1.61818‿1.61798‿1.61806‿1.61803‿1.61804‿1.61803‿1.61803
# 10
NewtonsMethod ← (2 ÷˜ ⊢ + ÷)´¨∘(1↓↑) ⥊
! 1e¯5 ∧´∘> | (7 NewtonsMethod 2) - 2‿1.5‿1.41667‿1.41422‿1.41421‿1.41421‿1.41421
# 11
InnerProduct ← +˝∘×⎉1‿2
! (>⟨2‿3⋄4‿1⟩) ≡ (>⟨1‿0⋄¯1‿1⟩) InnerProduct >⟨2‿3⋄6‿4⟩
# 12
CayleysTheorem ← {R←⥊⊐⊢ ⋄ (R𝕩) ≡ R ⊏˜⌜˜ <˘R𝕩}
! {
t ← 2‿2⊸⥊¨ (⌽2|⌊∘÷⟜2⍟(↕4))¨ 9‿6‿7‿11‿13‿14
g ← ≠˝∘∧⎉1‿2⌜˜t
CayleysTheorem g
}
# 14
IntervalIndex ← 1-˜≠∘⊣(⊣↓⊢⍋⊸⊏+`∘>)⍋∘∾ # Or ⍋, of course
! 0‿2‿3‿2‿4‿1‿0‿2‿¯1 ≡ ¯1‿2‿3‿7‿8.5 IntervalIndex 0‿4‿7‿6‿11‿2‿1‿3‿¯5
! 0‿1‿0‿¯1‿4‿4 ≡ "Fi Jay John Morten Roger" IntervalIndex○Words "JD Jd Geoff Anna Scott Zeus"
# 15
CentralLimit ← { 41 ↑ /⁼ (5×↕40) ⍋ +˝ Rand 𝕩⥊21 }
# 5‿8 ⥊ CentralLimit 10 1e3
# 17
SeventeenTwentyNine ← {F:
c ← 3⋆˜1+↕200
t ← (<⌜˜↕200) × +⌜˜c
d ← ⍷⊸⊐⊸⊔ ⥊t
⌊´ ⊑¨ (2=≠¨)⊸/ d
}
# ! 1729 ≡ SeventeenTwentyNine⟨⟩ # slow
# 19
Permutations ← {𝕊0:1‿0⥊0; ∾˝(0∾˘1+𝕊𝕩-1)⊸⊏˘⍒˘=⌜˜↕𝕩}
! (6‿3⥊0‿1‿2‿0‿2‿1‿1‿0‿2‿1‿2‿0‿2‿0‿1‿2‿1‿0) ≡ Permutations 3
# 20
InversePermutation ← ⍋ # Or ⊐⟜(↕≠) or ∾∘⊔
! (↕10) ≡ InversePermutation⊸⊏ 1‿4‿5‿2‿6‿8‿3‿7‿0‿9
# 21
⟨IndexFromPermutation⇐Ifp,PermutationFromIndex⇐Pfi⟩ ← {
Rfd ← ⊢+´∘>¨¯1↓↓
Dfr ← (⍋∘⍋∾)´
Ifp ⇐ {𝕊⁼:𝕨Pfi𝕩; (⌽×`1+»↕≠𝕩) +´∘× Rfd𝕩}
Pfi ⇐ {𝕊⁼:𝕨Ifp𝕩; w𝕊𝕩: Dfr (1+↕w) (⌽⊣|¯1↓·⌊∘÷`∾˜) 𝕩}
}
! {
p ← 1‿3‿0‿7‿6‿5‿4‿9‿8‿2
! p ≡ 10 PermutationFromIndex i ← IndexFromPermutation p
! p ≡ ⊢⌾(10⊸IndexFromPermutation) p
! i ≡ ⊢⌾(10⊸PermutationFromIndex) i
}
# 22
Combinations ← {𝕨(=∨0=⊣)◶⟨(0∾˘𝕊⌾(-⟜1))∾1+𝕊⟜(-⟜1), ≍∘↕⊣⟩𝕩}
! (3 Combinations 5) ≡ 10‿3⥊0‿1‿2‿0‿1‿3‿0‿1‿4‿0‿2‿3‿0‿2‿4‿0‿3‿4‿1‿2‿3‿1‿2‿4‿1‿3‿4‿2‿3‿4
# 23
⟨IndexFromCombination⇐Ic,CombinationFromIndex⇐Ci⟩ ← {
C ← ((-˜+↕∘⊣)÷○(×´1⊸+)↕∘⊣)⚇0 # Combination function (APL's dyadic !)
Ic ⇐ {
𝕊⁼: 𝕨Ci𝕩;
m‿n𝕊𝕩:
⊑-˝(m-↕m) +˝∘(C˘) n-(»1+𝕩)∾˘𝕩
}
Ci ⇐ {
𝕊⁼: 𝕨Ic𝕩;
0‿n𝕊𝕩: ⟨⟩;
m‿n𝕊𝕩:
v←+`(m-1)C(1-m)↓⌽↕n
k←(v>𝕩)⊐1
k∾(1+k)+(𝕨-1∾1+k)𝕊(𝕩-k⊏0∾v)
}
}
! 11 ≡ 4‿6 IndexFromCombination 1‿2‿3‿5
! 1‿2‿3‿5 ≡ 4‿6 CombinationFromIndex 11
! 11 ≡ 4‿6 CombinationFromIndex⁼ 4‿6 CombinationFromIndex 11
# 24
SymmetricArray ← ⍉⊸≡ ∧ 1⊸⍉⊸≡
! SymmetricArray +⌜´ 3⥊<2‿3‿7‿11
# 25
NQueens ← {
Arr ← {>(⊑⊸⊏⟜𝕩∾1⊸⊑)¨/○⥊⟜(↕≢)¬(↕𝕨)∊⎉1𝕩⥊∘+⎉1‿2(-⟜↕¯1⊑≢𝕩)×⌜¯1‿0‿1}
𝕩 Arr⍟(𝕩-1) ≍˘↕𝕩
}
QueenCheck ← {
! 1=≠≢𝕩
! (↕≠𝕩)≡∧𝕩
! ∧´⥊(=⌜˜↕≠𝕩)≥(|-⌜˜𝕩)=|-⌜˜↕≠𝕩
}
QueenCheck˘ NQueens 8
# 26
KnightsTour ← {
Kmoves ← {
t ← (⥊↕𝕩‿𝕩)+⌜<˘8‿2⥊2‿1‿2‿¯1‿1‿2‿1‿¯2‿¯1‿2‿¯1‿¯2‿¯2‿1‿¯2‿¯1
(∧˝⎉1(>t)∊↕𝕩) /¨○<˘ 𝕩{+⟜(𝕨⊸×)´⌽𝕩}¨t
}
m ← >↑˜¨⟜(⌈´≠¨) Kmoves 𝕩
b ← (𝕩×𝕩)⥊1
F ← {b↩0⌾(𝕩⊸⊑)b ⋄ (⊑⍋+˝˘(j⊏m)⊏b)⊑j←⊏⟜b⊸/𝕩⊏m}
𝕩‿𝕩⥊⍋F⍟(↕𝕩⋆2) 0
}
! (KnightsTour 6) ≡ 6‿6⥊0‿9‿20‿35‿6‿11‿21‿32‿7‿10‿19‿26‿8‿1‿34‿25‿12‿5‿33‿22‿31‿16‿27‿18‿2‿15‿24‿29‿4‿13‿23‿30‿3‿14‿17‿28
# 34
# Fails when 𝕩 is 1; the APL version did this too!
_pow ← {𝔽´𝔽˜⍟(/2|⌊∘÷⟜2⍟(↕·⌈2⋆⁼⊢)𝕩)𝕨}
! 847288609443 ≡ 3 ×_pow 25
# 38
Hanoi ← {¯1↓(⥊⊢≍⎉0˜≠⥊⊑⊑⟨1‿5‿2⋄0‿3‿4⟩˜)⍟𝕩⥊1}
! 0‿1‿3‿0‿4‿5‿0 ≡ Hanoi 3
# 39
Ack ← {
0 𝕊 𝕩: 1+𝕩;
𝕨 𝕊 0: (𝕨-1) 𝕊 1;
(𝕨-1) 𝕊 𝕨 𝕊 𝕩-1
}
! 9 ≡ 2 Ack 3
! 29 ≡ 3 Ack 2
# 40
# If _H solves the halting problem (𝔽 _H 𝕩 is 1 iff 𝔽𝕩 halts) then the
# following function causes a paradox.
# F←{𝕊 _H 𝕩 ? 𝕊 𝕩; ⟨⟩}
# 42
Minors ← {(/¨≠⟜<↕≠𝕩)(⍉⊏)⎉0‿∞ 𝕩}⎉2⍟2
Minors1 ← {𝕊 mat: >{mat ⍉∘{𝕩/˜¬𝕨=↕≠𝕩}´ ⌽𝕩}¨↕≢𝕩}
! (Minors ≡ Minors1) 3‿4⥊↕12
! (Minors ≡ Minors1) 4‿4⥊'A'+↕26
# 43
S1 ← {
𝕊 0: ⋈1;
(0∾t)+(t∾0)× 𝕩-1⊣t←𝕊 𝕩-1
}
S2 ← {
𝕊 0: ⋈1;
(0∾t)+(t∾0)×↕𝕩+1⊣t←𝕊 𝕩-1
}
! ⟨ 0, 6, 11, 6, 1 ⟩ ≡ S1 4
! ⟨ 0, 1, 7, 6, 1 ⟩ ≡ S2 4
# 44
XEA ← {(𝕨∾1‿0) {0=⊑𝕩 ? 𝕨; 𝕩 𝕊 𝕨-𝕩×⌊(⊑𝕨)÷⊑𝕩} (𝕩∾0‿1)}
CRT ← {
m‿r 𝕊 n‿s:
gcd‿a‿b ← m XEA n
lcm ← m×n÷gcd
c ← lcm|gcd÷˜(r×b×n)+(s×a×m)
! (r=m|c)∧(s=n|c)
lcm∾c
}
! 2‿¯1‿1 ≡ 4 XEA 6
! 12‿9 ≡ 4‿1 CRT 6‿3
# 46
Sieve←{
4≥𝕩 ? 𝕩⥊0‿0‿1‿1;
𝕊 n:
r←⌊√n
p ← 2‿3‿5‿7‿11‿13‿17‿19‿23‿29‿31‿37‿41‿43
p ↩ (1+(×`p)⊸⍋⌾<n-1)↑p
b ← 0⌾(1⊸⊑) 1 {m←n⌊𝕨×≠𝕩 ⋄ (m⥊𝕩)>m⥊𝕨↑1}´ ⌽p
{
r≥q←b⊸⊐⌾<1 ?
b 0¨⌾((q∾q×/b↑˜⌈n÷q)⊸⊏)↩
𝕊 𝕩∾q
;
b 1¨⌾(𝕩⊸⊏)↩
}p
}
! 2‿3‿5‿7‿11‿13‿17‿19‿23‿29 ≡ /Sieve 30
! 78498 ≡ +´Sieve 1e6
# 47
Pg ← •rand.Deal{
d ← 𝕩+𝕩
s ← 𝕩×𝕩
t ← (𝕩∾𝕩)↑(𝕩∾d)⥊(1-d)↑(𝕩𝔽26)⊏"abcdefghijklmnopqrstuvwxyz"
p ← ⍋(s𝔽s)+(-´¨1↓↑𝕩-˜⌽↕d)/s×↕d
(⌽↕𝕩)⌽˘((d-1)⥊1‿0)⊸{𝕩⌾(𝕨⊸/)' '¨𝕨}˘(𝕩∾𝕩)⥊p⊏⥊t
}
#•Show Pg¨ 4‿5‿6‿7
TestPg ← {n𝕊p:
! (>(«∨»)⍟(↕n)1(⊢∾∾)0⥊˜n-1) ≡ ' '≠p
rows ← (<' '⊸≠⊸/)˘p
! ∧´ (<"") (»(∧≡↕∘≠)∘⊒¨⊢) rows
! ∧´∊ ¯1⊑rows
}
TestPg⟜Pg¨ 4‿5‿6‿7
# 48
Stick ← {
c ← 𝕩 •rand.Range 3 # where the car is hidden
i ← 𝕩 •rand.Range 3 # your initial choice of door
+´c=i # number of cars that you win
}
Change ← {
c ← 𝕩 •rand.Range 3 # where the car is hidden
i ← 𝕩 •rand.Range 3 # your initial choice of door
j ← (c×i≠c)+(3|i+1+•rand.Range¨𝕩⥊2)×i=c # your changed choice
+´c=j # number of cars that you win
}
#•Show Stick‿Change {𝕎𝕩}¨ 1e6
# 49
# +´(1+↕∞)⋆-s ←→ ×´÷1-(⍭↕∞)⋆-s
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