Oooh, scary! wren ng thornton wrnthorn@indiana.edu Cognitive - - PowerPoint PPT Presentation

Monads

Oooh, scary!

wren ng thornton

wrnthorn@indiana.edu

Cognitive Science & Computational Linguistics Indiana University Bloomington

wren ng thornton Probability Smoothing for NLP – p. 1/7

Outline of the talk

• Categories
• Functors
• Monads

wren ng thornton Probability Smoothing for NLP – p. 2/7

Categories

• Objects (e.g., types)
• Int, Bool, Int×Int, . . .
• Morphisms (e.g., functions)
• f :Char → Int, . . .
• Composition
• ∀ f :A → B. ∀g:B → C. ∃ g◦ f :A → C
• Identity morphisms
• ∀A. ∃idA : A → A such that

∀ f :A → B. f ◦idA = f ∀ f :B → A. idA ◦f = f

wren ng thornton Probability Smoothing for NLP – p. 3/7

Functors

• A map F :C → D from category C to category D
• Let’s assume C = D
• F takes C-objects to D-objects
• data List A = Nil | Cons A (List A)
• data Maybe A = Nothing | Just A
• data Tree A = Tip A | Bin (Tree A) (Tree A)
• F takes C-morphisms to D-morphisms
• mapList :(A → B) → (List A → List B)
• mapMaybe :(A → B) → (Maybe A → Maybe B)
• mapTree :(A → B) → (Tree A → Tree B)
• And follows some laws
• ∀A.

F (idA) = id(F A)

• ∀ f. ∀g.

(F f)◦(F g) = F (f ◦g)

wren ng thornton Probability Smoothing for NLP – p. 4/7

Monads

• A functor T with some extra stuff
• A “unit” natural transformation (aka pure, return, point,. . . )
• η :Id ˙

→T . . . What?!

• ∀A. ηA :A → T A

. . . not quite

• η :∀A. A → T A
• And a “join” natural transformation (or “bind” (>

> =) instead)

• µ :T ◦T ˙

→ T

• concat:∀A. List (List A) → List A
• Following some laws
• µA ◦ηA = idA
• µA ◦T ηA = idA
• µA ◦T µA = µT A ◦ µA

wren ng thornton Probability Smoothing for NLP – p. 5/7

Monads

wren ng thornton Probability Smoothing for NLP – p. 6/7

∼fin.

wren ng thornton Probability Smoothing for NLP – p. 7/7