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Fun with Total Array Ordering (TAO) ⍒⍋ 1 Adám Brudzewsky (material by Jay Foad) 大道至簡 Fun with Total Array Ordering (TAO) ⍒⍋ Fun with Total Array Ordering (TAO) 2 Grading ⍋'APL' 1 3 2 Fun with Total Array Ordering (TAO) 3 Grading ⍋'APL' 1 3 2 Fun with Total Array Ordering (TAO) 4 Grading [1,2,3] ⍋'APL' 1 3 2 'A' 'L' 'P' ⍨ Fun with Total Array Ordering (TAO) 5 Major Cells: Vector ┌→──┐ │APL│ └───┘ Fun with Total Array Ordering (TAO) 6 Major Cells: Matrix ┌→──┐ ↓AAA│ │PPP│ │LLL│ └───┘ Fun with Total Array Ordering (TAO) 7 Major Cells: Cube ┌→──┐ ↓AAA│ │AAA│──┐ │AAA│PP│ └───┘PP│──┐ │PPP│LL│ └───┘LL│ │LLL│ └───┘ Fun with Total Array Ordering (TAO) 8 Major Cells: mat ┌→─────┐ ↓John │ │Roger │ │Adam │ │Morten│ │Jay │ └──────┘ Fun with Total Array Ordering (TAO) 9 Major Cells: mat ⍋mat ┌→─────┐ 3 5 1 4 2 ↓John │ │Roger │ │Adam │ │Morten│ │Jay │ └──────┘ Fun with Total Array Ordering (TAO) 10 Major Cells: mat ⍋mat ┌→─────┐ 3 5 1 4 2 ↓John │ │Roger │ │Adam │ │Morten│ │Jay │ └──────┘ Fun with Total Array Ordering (TAO) 11 Major Cells: mat mat[⍋mat;] ┌→─────┐ ┌→─────┐ ↓John │ ↓Adam │ │Roger │ │Jay │ │Adam │ │John │ │Morten│ │Morten│ │Jay │ │Roger │ └──────┘ └──────┘ Fun with Total Array Ordering (TAO) 12 Sorting mat[⍋mat;] {(⊂⍋⍵)⌷⍵} mat Fun with Total Array Ordering (TAO) 13 Sorting mat[⍋mat;] {(⊂⍋⍵)⌷⍵} mat Fun with Total Array Ordering (TAO) 14 Sorting mat[⍋mat;] {(⊂⍒⍵)⌷⍵} mat Fun with Total Array Ordering (TAO) 15 Sorting mat[⍋mat;] Sort ← {(⊂⍋⍵)⌷⍵} Fun with Total Array Ordering (TAO) 16 Sorting mat[⍋mat;] Sort ← {(⊂⍋⍵)⌷⍵} Sort mat Fun with Total Array Ordering (TAO) 17 Sorting mat[⍋mat;] Sort ← {(⊂⍋⍵)⌷⍵} Sort mat Adam Jay John Morten Roger Fun with Total Array Ordering (TAO) 18 Sorting n ← 'Roger' 'Adam' 'Jay' Fun with Total Array Ordering (TAO) 19 Sorting n ← 'Roger' 'Adam' 'Jay' Sort n ⍝ 16.0 Fun with Total Array Ordering (TAO) 20 Sorting n ← 'Roger' 'Adam' 'Jay' Sort n ⍝ 16.0 DOMAIN ERROR Fun with Total Array Ordering (TAO) 21 Sorting n ← 'Roger' 'Adam' 'Jay' Sort n ⍝ 16.0 DOMAIN ERROR Sort n ⍝ 17.0 Fun with Total Array Ordering (TAO) 22 Sorting n ← 'Roger' 'Adam' 'Jay' Sort n ⍝ 16.0 DOMAIN ERROR Sort n ⍝ 17.0 ┌────┬───┬─────┐ │Adam│Jay│Roger│ └────┴───┴─────┘ Fun with Total Array Ordering (TAO) 23 A bit of history 'Morten' 'Roger ' Fun with Total Array Ordering (TAO) 24 A bit of history 'Morten' 'Roger▉' Fun with Total Array Ordering (TAO) 25 A bit of history A ← 'Morten' B ← 'Roger▉' Fun with Total Array Ordering (TAO) 26 A bit of history A ← 'Morten' B ← 'Roger▉' ⍋ A ,[0.5] B 1 2 Fun with Total Array Ordering (TAO) 27 A bit of history A ← 'Morten' B ← 'Roger▉' ⍋ A ,[0.5] B 1 2 'Morten' 'Roger▉' Fun with Total Array Ordering (TAO) 28 Total Array Ordering some other array array ? Fun with Total Array Ordering (TAO) 29 Total Array Ordering How would you order: ⎕ ← 0 'Jay' (2 3⍴'DYALOG') (⍳9) ┌→──────────────────────────────────┐ │ ┌→──┐ ┌→──┐ ┌→────────────────┐ │ │ 0 │Jay│ ↓DYA│ │1 2 3 4 5 6 7 8 9│ │ │ └───┘ │LOG│ └~────────────────┘ │ │ └───┘ │ └∊──────────────────────────────────┘ Fun with Total Array Ordering (TAO) 30 Total Array Ordering How would you order: ⎕ ← 0 'Jay' (2 3⍴'DYALOG') (⍳9) ┌→──────────────────────────────────┐ │ ┌→──┐ ┌→──┐ ┌→────────────────┐ │ │ 0 │Jay│ ↓DYA│ │1 2 3 4 5 6 7 8 9│ │ │ └───┘ │LOG│ └~────────────────┘ │ │ └───┘ │ └∊──────────────────────────────────┘ Fun with Total Array Ordering (TAO) 31 Total Array Ordering Fun with Total Array Ordering (TAO) 32 TAO Fun with Total Array Ordering (TAO) 33 TAO Rules Fun with Total Array Ordering (TAO) 34 TAO some other array array ? Fun with Total Array Ordering (TAO) 35 TAO some other array array ? Fun with Total Array Ordering (TAO) 36 TAO some other array array ? no refs no refs Fun with Total Array Ordering (TAO) 37 TAO Rules Fun with Total Array Ordering (TAO) 38 TAO Rules Compatibility: simple arrays Fun with Total Array Ordering (TAO) 39 TAO Rules Compatibility: simple arrays Consistency: (⍺ ⍵)∧(⍵ ⍺) ⍺ ⍵ Fun with Total Array Ordering (TAO) 40 TAO Rules Compatibility: simple arrays Consistency: (⍺ ⍵)∧(⍵ ⍺) ⍺ ⍵ ⍺≡⍵ Fun with Total Array Ordering (TAO) 41 TAO Rules Compatibility: simple arrays Consistency: (⍺ ⍵)∧(⍵ ⍺) ⍺ ⍵ ⍺≡⍵ ⍬≢'' Fun with Total Array Ordering (TAO) 42 TAO Rules: simple scalars Fun with Total Array Ordering (TAO) 43 TAO Rules: simple scalars character vs character: ✓ Fun with Total Array Ordering (TAO) 44 TAO Rules: simple scalars character vs character: ✓ real number vs real number: ✓ Fun with Total Array Ordering (TAO) 45 TAO Rules: simple scalars character vs character: ✓ real number vs real number: ✓ complex: 1 1J1 2J¯1 2 Fun with Total Array Ordering (TAO) 46 TAO Rules: simple scalars character vs character: ✓ real number vs real number: ✓ complex: 1 1J1 2J¯1 2 number vs character: 100 'A' Fun with Total Array Ordering (TAO) 47 TAO Rules: simple scalars character vs character: ✓ real number vs real number: ✓ complex: 1 1J1 2J¯1 2 number vs character: 100 'A' null: ⎕NULL ¯100 Fun with Total Array Ordering (TAO) 48 TAO Rules: simple scalars ⎕NULL ¯1 0J¯2 0 0J2 1 'A' Fun with Total Array Ordering (TAO) 49 TAO Rules: same shape Fun with Total Array Ordering (TAO) 50 TAO Rules: same shape ┌→───┐ ┌→───┐ │Jay▉│ │John│ └────┘ └────┘ Fun with Total Array Ordering (TAO) 51 TAO Rules: same shape ┌→───┐ ┌→───┐ │Jay▉│ │John│ └────┘ └────┘ Fun with Total Array Ordering (TAO) 52 TAO Rules: same shape ┌→───┐ ┌→───┐ │Jay▉│ │John│ └────┘ └────┘ ┌→─────────┐ ┌→─────────┐ │ ┌→─┐ │ │ ┌→─┐ │ │ │JF│ 3 1 │ │ │JD│ 2 7 │ │ └──┘ │ │ └──┘ │ └∊─────────┘ └∊─────────┘ Fun with Total Array Ordering (TAO) 53 TAO Rules: same shape ┌→───┐ ┌→───┐ │Jay▉│ │John│ └────┘ └────┘ ┌→─────────┐ ┌→─────────┐ │ ┌→─┐ │ │ ┌→─┐ │ │ │JF│ 3 1 │ │ │JD│ 2 7 │ │ └──┘ │ │ └──┘ │ └∊─────────┘ └∊─────────┘ Fun with Total Array Ordering (TAO) 54 TAO Rules: same shape ┌→───┐ ┌→───┐ │Jay▉│ │John│ └────┘ └────┘ ┌→─────────┐ ┌→─────────┐ │ ┌→─┐ │ │ ┌→─┐ │ │ │JF│ 3 1 │ │ │JD│ 2 7 │ │ └──┘ │ │ └──┘ │ └∊─────────┘ └∊─────────┘ Fun with Total Array Ordering (TAO) 55 TAO Rules: different shape CAN CAR CARPENTER CARPET CAT Fun with Total Array Ordering (TAO) 56 TAO Rules: different shape CAN CAR CARPENTER CARPET CAT Fun with Total Array Ordering (TAO) 57 TAO Rules: different shape CAR CARPENTER Fun with Total Array Ordering (TAO) 58 TAO Rules: different shape CAR CARPENTER Fun with Total Array Ordering (TAO) 59 TAO Rules: different shape CAR CARPENTER Fun with Total Array Ordering (TAO) 60 TAO Rules: different shape CAN CAR CARPENTER CARPET CAT Fun with Total Array Ordering (TAO) 61 TAO Rules: different shape CAN CAR CARPENTER CARPET CAT Fun with Total Array Ordering (TAO) 62 TAO Rules: different shape ↑'CAR' 'CARPENTER' Fun with Total Array Ordering (TAO) 63 TAO Rules: different shape ↑'CAR' 'CARPENTER' CAR▉▉▉▉▉▉ CARPENTER Fun with Total Array Ordering (TAO) 64 TAO Rules: different shape ↑'CAR' 'CAR␀␀␀␀␀␀' CAR▉▉▉▉▉▉ CAR␀␀␀␀␀␀ Fun with Total Array Ordering (TAO) 65 TAO Rules: different shape ⍋↑'CAR' 'CAR␀␀␀␀␀␀' 2 1 Fun with Total Array Ordering (TAO) 66 TAO Rules: different shape ⍋↑'CAR' 'CAR␀␀␀␀␀␀' 2 1 Fun with Total Array Ordering (TAO) 67 TAO Rules: different shape 1 2 3 1 2 3 4 Fun with Total Array Ordering (TAO) 68 TAO Rules: different shape ¯1 ¯2 ¯3 ¯1 ¯2 ¯3 ¯4 Fun with Total Array Ordering (TAO) 69 TAO Rules: different shape ¯1 ¯2 ¯3 ¯1 ¯2 ¯3 ¯4 Fun with Total Array Ordering (TAO) 70 TAO Rules: different shape ↑(¯1 ¯2 ¯3) (¯1 ¯2 ¯3 ¯4) Fun with Total Array Ordering (TAO) 71 TAO Rules: different shape ↑(¯1 ¯2 ¯3) (¯1 ¯2 ¯3 ¯4) ¯1 ¯2 ¯3 0 ¯1 ¯2 ¯3 ¯4 Fun with Total Array Ordering (TAO) 72 TAO Rules: different shape ⍋↑(¯1 ¯2 ¯3) (¯1 ¯2 ¯3 ¯4) Fun with Total Array Ordering (TAO) 73 TAO Rules: different shape ⍋↑(¯1 ¯2 ¯3) (¯1 ¯2 ¯3 ¯4) 2 1 Fun with Total Array Ordering (TAO) 74 TAO Rules: different shape ⍋↑(¯1 ¯2 ¯3) (¯1 ¯2 ¯3 ¯4) 2 1 Fun with Total Array Ordering (TAO) 75 TAO Rules: different shape ¯∞ Fun with Total Array Ordering (TAO) 76 TAO Rules: different shape ¯∞ ⎕NULL ¯1 0J¯2 0 0J2 1 'A' Fun with Total Array Ordering (TAO) 77 TAO Rules: different shape CAR¯∞¯∞¯∞¯∞¯∞¯∞ CARPENTER Fun with Total Array Ordering (TAO) 78 TAO Rules: different shape CAR¯∞¯∞¯∞¯∞¯∞¯∞ CARPENTER Fun with Total Array Ordering (TAO) 79 TAO Rules: different shape 1 2 1 2 3 1 2 1 2 3 1 2 Fun with Total Array Ordering (TAO) 80 TAO Rules: different shape 1 2 ¯∞ 1 2 3 1 2 ¯∞ 1 2 3 1 2 ¯∞ ¯∞ ¯∞ ¯∞ Fun with Total Array Ordering (TAO) 81 TAO Rules: different shape 1 2 ¯∞ 1 2 3 1 2 ¯∞ 1 2 3 1 2 ¯∞ ¯∞ ¯∞ ¯∞ Fun with Total Array Ordering (TAO) 82 TAO Rules: different shape 1 2 ¯∞ 1 2 3 1 2 ¯∞ 1 2 3 1 2 ¯∞ ¯∞ ¯∞ ¯∞ Fun with Total Array Ordering (TAO) 83 TAO Rules: different rank ┌→──┐ ┌→────┐ │APL│ ↓Adam │ └───┘ │Jay │ │Roger│ └─────┘ Fun with Total Array Ordering (TAO) 84 TAO Rules: different rank ┌→──┐ ┌→────┐ ↓APL│ ↓Adam │ └───┘ │Jay │ │Roger│ └─────┘ Fun with Total Array Ordering (TAO) 85 TAO Rules: different rank ┌→────┐ ┌→────┐ ↓APL¯∞¯∞│ ↓Adam │ │¯∞¯∞¯∞¯∞¯∞│ │Jay │ │¯∞¯∞¯∞¯∞¯∞│ │Roger│ └─────┘ └─────┘ Fun with Total Array Ordering (TAO) 86 TAO Rules: different rank ┌→────┐ ┌→────┐ ↓APL¯∞¯∞│ ↓Adam │ │¯∞¯∞¯∞¯∞¯∞│ │Jay │ │¯∞¯∞¯∞¯∞¯∞│ │Roger│ └─────┘ └─────┘ Fun with Total Array Ordering (TAO) 87 TAO Rules: different rank ┌→──┐ ┌→──┐ │APL│ ↓APL│ └───┘ └───┘ Fun with Total Array Ordering (TAO) 88 TAO Rules: different rank ┌→──┐ ┌→──┐ ↓APL│ ↓APL│ └───┘ └───┘ Fun with Total Array Ordering (TAO) 89 TAO Rules: different rank ┌→──┐ ┌→──┐ │APL│ ↓APL│ └───┘ └───┘ 1 2 Fun with Total Array Ordering (TAO) 90 TAO Rules: different rank scalar vector 'A' ,'A' A A vector 1-row matrix 3⍴'APL' 1 3⍴'APL' APL APL Fun with Total Array Ordering (TAO) 91 TAO Rules: recap simple scalars
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