basic.md

Array algebra stripped down

Let's reduce the operations on rank-1 arrays (lists) down to very basic functions.

Two views on lists:

• A list is a function from indices to values
• A list is a free monoid

The first view gives ↕n, i⊑x and x⊏y, which correspond to the identity function, application and composition, and also gives f¨x and x f¨ y.

The second view gives ⟨⟩ and x∾y, ⋈x and f´x which are identity and multiplication of the monoid and unit and counit of the adjunction.

Extra: f⍟n x is directly related to the universal property of natural numbers.

Using this, let's implement some other functions:

• ≠x is +´1¨x
• n⥊x is n {𝕩⊏˜(≠𝕩)|↕𝕨} x
• ∾x is ∾´x
• x⋈y is ∾○⋈
• n↑x (for n≥0) is n {(d∾𝕩) ⊏˜ (≠𝕩)⊸≥⊸× 1+↕𝕨} x, where d is the default value of x
• n↓x (for n≥0) is {𝕩 ⊏˜ 𝕨+↕𝕨-˜≠𝕩}
• ⌽x is (↕∘≠-˜1-˜≠)⊸⊏ x and n⌽x is n {𝕩⊏˜(≠𝕩)|𝕨+↕≠𝕩} x
• m/l is m (∾⥊⟜⋈¨) x
• x⊐y is x {(⊑/∾≠)¨<˘𝕩≡⌜𝕨} y (table could be replaces with eaches)
• ∊x is {∨˝∧⟜(¬·»∨`)˘≡⌜˜𝕩} x

K world

K does not have that many primitives. Basic are:

• !n, x.i and x@y, also f'x
• (), x,y, ,x and f/x

Deriving the implementation of other verbs and adverbs is left to the reader as an exercise.