Minimal Set

While TinyAPL is "tiny" in many ways, it is definitely not tiny in the amount of primitives it has. It might be interesting to find out what the minimal set of primitives is, i.e. the ones that can't be implemneted in terms of other primitives.

Aliases

Some of the primitives are pretty much simple applications of other primitives:

∡ Arctangent is ∡⍤⊕⍨
ᑣ Boxed is ⊂⍤F
⊕ Cartesian is +∘(0ᴊ1∘×)
⌈ Ceiling is -⍤⌊⍤-
¨ Each is Fᑣᑒ◠
ᐵ Each Left is F¨∘⊂
ᑈ Each Right is ⊂⍛(F¨)
⊇ From is ⌷⍨∘⊃⍨⍤0‿∞
, Laminate is ⍪⍥∧
⊇ Last is ⊃⍤⊖⍤,
↓ Major Cells is ⊂⍤¯1
↑ Mix is ⊃⍤0
⍲ Nand is ~⍤∧
⍱ Nor is ~⍤∨
~ Not is 1∘-
◡ On Cells is F⍤¯1
ᑒ On Contents is F⍥⊃
◠ On Scalars is F⍤0
⌓ On Simple Scalars is F⍥0
⊗ Polar is ×∘(0ᴊ1∘×⍛*)
ϼ Rank is ≠⍤⍴
⍪ Join is ⊃⍤(⍪¨⍆)
√ Root is *∘÷⍨
⊵ Sort Down is ⍒⍨
⊴ Sort Up is ⍋⍨
⍬ Empty Vector is ⟨⟩
∻ Empty Dictionary is ⟨:⟩
⍮ Pair and ⍮ Singleton are both ,⍥⊂

Default argument

Some primitives' monadic case is related to the corresponding dyad by either binding an argument or using a hook.

⍟ Natural Logarithm is (*1)∘⍟
- Negate is 0∘-
⊕ Pure Imaginary is 0∘⊕
÷ Reciprocal is 1∘÷
√ Square Root is 2∘√
⊗ Unit Polar is 1∘⊗
⋷ Histogram is (⍳⍤≢)⊸⋷

Combinators

All combinators can be rewritten using trains, as described in Combinators ↔ Trains.

⊢ Right is {⍵}
⊣ Left is {⍵}⁖{⍺}
⍤ Atop is ⦅F⋄G⦆
⍥ Over is ⦅F⍤⊣⋄G⋄F⍤⊢⦆
⍨ Constant is ⦅n⦆
⍨ Commute and ⍨ Duplicate are ⦅⊢⋄F⋄⊣⦆
⊸ Left Hook is ⦅F⍤⊣⋄G⋄⊢⦆
⟜ Right Hook is ⦅⊣⋄F⋄G⍤⊢⦆
⇾ Left Fork is ⦅F⋄G⋄⊢⦆
⇽ Right Fork is ⦅⊣⋄F⋄G⦆
⸚ Mirror is ⦅G⍨⋄F⋄G⦆
«» Fork is ⦅F⋄G⋄H⦆
∘ After is ⦅F⋄G⦆⁖⦅⊣⋄F⋄G⍤⊢⦆
⍛ Before is ⦅G⋄F⦆⁖⦅F⍤⊣⋄G⋄⊢⦆

Promote, Demote, Rerank

∧ Promote is {⍵⍴⍨1⍪⍴⍵}.

∨ Demote is

{ 0=ϼ⍵: ■⍵
⋄ 1=ϼ⍵: ■⊃⍵
⋄ ⍵⍴⍨(×⍆2↑⍴⍵)⍪2↓⍴⍵ }

ϼ Rerank is

{ ⍺=ϼ⍵: ■⍵
⋄ ⍺>ϼ⍵: ■⍺ ∇ ∧⍵
⋄ ⍺ ∇ ∨⍵ }

Set operations

∪ Unique is ≠⊸⌿
∪ Union is {⍺⍪⍵⌿⍨~⍵∊⍺}
∩ Intersection is {⍺⌿⍨⍺∊⍵}
~ Difference is {⍺⌿⍨~⍺∊⍵}
§ Symmetric Difference is ∪«~»∩

Comparisons

All comparisons can be reconstructed from just = Equal To and ≤ Less Than or Equal To.

≠ Not Equal To is ~⍤=
< Less Than is ≤«∧»≠
> Greater Than is ~⍤≤
≥ Greater Than or Equal To is ~⍤<

Similarly, TAO comparisons can be reconstructed from ≡ Identical and ⊴ Precedes Or Identical.

≢ Not Identical is ~⍤≡
⊲ Precedes is ⊴«∧»≢
⊳ Succeeds is ~⍤⊴
⊴ Precedes Or Identical is ~⍤⊲

But TAO comparisons can be reconstructed from grading:

≡ Identical is ⦅⍋⋄=ᑒ⋄⍒⦆⍤⍮
⊴ Precedes Or Identical is ⦅⍋⋄≤ᑒ⋄⍒⦆⍤⍮

Min/max are also easy:

⇂ Minimal is {⍺⊴⍵:■⍺⋄⍵}
↾ Maximal is {⍺⊳⍵:■⍺⋄⍵}
⌊ Minimum is ⇂⌓
⌈ Maximum is ↾⌓

Floor and Round

⌊ Floor is "complex floor":

{ RealFloor←1∘|⊸-
⋄ x←1|ℜ⍵
⋄ y←1|ℑ⍵
⋄ p←(ℜ⍵)⊕⍥RealFloorℑ⍵
⋄ 1>x+y: ■p
⋄ x≥y: ■p+1
⋄ p+0ᴊ1 }

⸠ Round is componentwise round: ⦅ℜ⋄⊕⍥(⌊⍤+∘0.5)⋄ℑ⦆

Reduce

Reductions are special cases of folds where the initial value is the first/last cell of the array.

⍆ Reduce is _{(⊃↓⍵)⍶⍶⍆(1↓↓⍵)}
⍅ Reduce Back is _{(⊇↓⍵)⍶⍶⍆(¯1↓⍵)}

Remainder, GCD and LCM

| Remainder is { ⍺=0: ■⍵ ⋄ ⍵-⍺×⌊⍵÷⍺ }⌓
∨ Greatest Common Divisor is { 0=⍺|⍵: ⍺ ⋄ ⍵ ∇⍨ ⍺|⍵ }⌓
∧ Least Common Multiple is ×«÷»∨

Inner Product and Table

∙ Inner Product is F⍤(G◠◡)⍤1‿∞
⊞ Table is F◠⍤0‿∞

Tally and Depth

≢ Tally is ⦅⊃⋄0⋄⍪⍨⋄⍴⦆

≡ Depth is

{ (0≡ϼ⍵)∧⍵≡⊂⍵: ■0
⋄ 0≡ϼ⍵: ■1+∇⊃⍵
⋄ 0∊⍴⍵:■1
⋄ 1+⌈⍆∇ᑒ¨,⍵ }

Repeat and Until

Monadic ⍣ Repeat is _{ 0=⍵⍵: ■⍵ ⋄ ⍶⍶ _∇_ (⍵⍵-1) ⍶⍶ ⍵ }_; dyadic is an application of ∘ Bind.

Similarly, monadic ⍣ Until is _{(⍶⍶ ⍵) ⍶⍶ _{ ⍺ ⍹⍹ ⍵: ■⍺ ⋄ (⍶⍶ ⍺) ⍶⍶ _∇_ ⍹⍹ ⍺ }_ ⍹⍹ ⍵}_; dyadic as above.

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