Functional programming, lambda calculus.

1. Functional
Programming
I’m loving it.

2. onad

3. • Algorithm
• David Hilbert (1928) — Entscheidungsproblem
• Halting Problem

4. Turing Machine

5. Alonzo Church (1903 – 1995)

6. Alonzo Church (1936) —
Lambda Calculus

7. Alonzo Church (1932) — λ-calculus
L, M, N ::= x
| (λx. N)
| (L M)

9. Rules
• Alpha equivalence
• Beta reduction
λx.x λy.y
(λx.x) y == y

10. λ-calculus
λx.xIdentity
λs.s sSelf Apply
x => x
s => s(s)

11. λ-calculus
(s => s(s))(x => x)
(x => x)(x => x)
(x => x)

12. λ-calculus Recursion
(s => s(s))(s => s(s))
(λs.s s)(λs.s s)

13. Church Encoding
λx. λy. x
λx. λy. y
True
False
λp. λx. λy. p x yIf
λp. λq. p q pAnd
λp. λq. p p qOr

14. Church Encoding
λf. λx. x
λf. λx. f x
0
1
2 λf. λx. f (f x)
λm. λn. λf. λx. m f (n f x)m + n

15. Church Encoding
λx. λy. λz. z x yPair
First λp. p (λx. λy. x)
Second λp. p (λx. λy. y)

16. Modus Ponens

17. Alonzo Church (1940) —
Typed λ-calculus

18. Howard 1980

19. Curry-Howard
Correspondence
• Proposition
• Proof
• Normalization
• Type
• Program
• Evaluation

20. Correspondence

21. Languages
• Lisp (McCarthy, 1960)
• Scheme
• Racket (1994)
• Clojure (Hickey, 2007)
• ML (Milner, 1979)
• Haskell (1987)
• O’Caml (Leroy, 1996)
• Erlang (Armstrong, 1996)
• Scala (Odersky, 2004)
• F# (Syme, 2006)

22. ML Example

23. Proof Assistants
• Automath (de Bruijn, 1970)
• Type Theory (Martin Löf, 1975)
• Coq (Huet and Coquand, 1988)
• Isabelle (Paulson, 1993)
• Epigram (McBride and McKinna, 2004)
• Agda (Norell, 2005)
• Idris

24. Functional Programming
• First-class Functions / Higher Order Functions
• Pure Functions / Referential Transparency
• Immutability
• Recursion

25. Benefits
• Guaranties!
• Modularity
• Composability
• Conciseness
• Concurrency / Distribution
• Simplicity
• Beauty

26. Real World

27. Thank You!
Questions?
anemish@gmail.com
http://fb.com/anemish
@atlanter
http://github.com/nau
http://gitter.im/lasca-lang/compiler

28. Resources
• "Propositions as Types" by Philip Wadler
https://www.youtube.com/watch?v=FiflFiZ6pPI&t=1430s
• Category Theory for the Working Hacker by Philip Wadler
https://www.youtube.com/watch?v=V10hzjgoklA
• The Value of Values with Rich Hickey
https://www.youtube.com/watch?v=-6BsiVyC1kM
• Why Functional Programming Matters by John Hughes
https://www.youtube.com/watch?v=XrNdvWqxBvA
• Robert C Martin - Functional Programming; What? Why? When?
https://www.youtube.com/watch?v=7Zlp9rKHGD4
• Persistent Data Structures. MIT 6.851 Advanced Data Structures
https://www.youtube.com/watch?v=T0yzrZL1py0
• Simon Peyton Jones - Haskell is useless
https://www.youtube.com/watch?v=iSmkqocn0oQ&t=199s
• Brian Beckman: Don't fear the Monad
https://www.youtube.com/watch?v=ZhuHCtR3xq8
• Erik Meijer: Functional Programming
https://www.youtube.com/watch?v=z0N1aZ6SnBk
• http://learnyouahaskell.com/
• Scala for the Impatient – http://www.horstmann.com/scala/