Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
Principal type-schemes for functional programs
Search
Phil Freeman
June 28, 2017
Programming
0
330
Principal type-schemes for functional programs
Phil Freeman
June 28, 2017
Tweet
Share
More Decks by Phil Freeman
See All by Phil Freeman
The Future Is Comonadic!
paf31
14
4.6k
Incremental Programming in PureScript
paf31
3
980
An Overview of the PureScript Type System
paf31
5
1.9k
Fun with Profunctors
paf31
3
1.2k
Intro to psc-package
paf31
0
150
Stack Safety for Free
paf31
0
310
Other Decks in Programming
See All in Programming
Reactの歴史を振り返る
tutinoko
1
170
TypeScriptでDXを上げろ! Hono編
yusukebe
4
920
なぜあなたのオブザーバビリティ導入は頓挫するのか
ryota_hnk
5
560
MCPで実現できる、Webサービス利用体験について
syumai
7
2.3k
AIのメモリー
watany
12
1.2k
新世界の理解
koriym
0
130
Advanced Micro Frontends: Multi Version/ Framework Scenarios
manfredsteyer
PRO
0
140
#QiitaBash TDDで(自分の)開発がどう変わったか
ryosukedtomita
1
350
書き捨てではなく継続開発可能なコードをAIコーディングエージェントで書くために意識していること
shuyakinjo
0
180
実践 Dev Containers × Claude Code
touyu
1
120
抽象化という思考のツール - 理解と活用 - / Abstraction-as-a-Tool-for-Thinking
shin1x1
1
920
構文解析器入門
ydah
7
2k
Featured
See All Featured
Writing Fast Ruby
sferik
628
62k
Chrome DevTools: State of the Union 2024 - Debugging React & Beyond
addyosmani
7
790
Intergalactic Javascript Robots from Outer Space
tanoku
272
27k
Stop Working from a Prison Cell
hatefulcrawdad
271
21k
VelocityConf: Rendering Performance Case Studies
addyosmani
332
24k
How to train your dragon (web standard)
notwaldorf
96
6.1k
No one is an island. Learnings from fostering a developers community.
thoeni
21
3.4k
Java REST API Framework Comparison - PWX 2021
mraible
32
8.8k
Sharpening the Axe: The Primacy of Toolmaking
bcantrill
44
2.4k
What's in a price? How to price your products and services
michaelherold
246
12k
GraphQLの誤解/rethinking-graphql
sonatard
71
11k
Building an army of robots
kneath
306
45k
Transcript
Principal type-schemes for functional programs Luis Damas and Robin Milner
(POPL `82)
Agenda • Slides • Code
ML • Meta Language for LCF • Type inference •
Influence on Haskell, Rust, F#, OCaml, ... • “Sweet spot” in type system design
ML letrec f xs = if null xs then nil
else snoc (f (tl xs)) (hd xs) What type does this function have? null : ∀ ( list → bool) snoc : ∀ ( list → → list) hd, tl : ∀ ( list → ) nil : ∀ ( list)
ML What about: let s x y z = x
z (y z) ?
Type Inference f : ∀ ( list → list) •
Given f, how can we infer this type? • What does it even mean for a value to have a type? • How can we be sure we have the most general type?
Lambda Calculus Expressions e: • Identifiers: , , … •
Applications: e e’ • Abstractions: . e • Let bindings: let = e in e’
Lambda Calculus For example: . . . . let =
. . in
Types Monotypes : • Variables: • Primitives: • Functions: →
Type Schemes Type schemes : • Monomorphic: • Polymorphic: ∀
. Type schemes are types with identifiers bound by ∀ at the front.
Substitutions Mappings from variables to types • Can instantiate type
schemes using substitutions • Gives a simple subtyping relation on type schemes
Semantics Construct a semantic domain (CPO) V containing • Primitives
• Functions • An error element and a semantic function : e → (Id → V) → V
Semantics Identify types with subsets of V Define the judgment
A ╞ e : when (∀ ( : ’) ∈ A. ∈ ’) ⇒ e ∈
Declarative System Variable rule:
Declarative System Application rule:
Declarative System Abstraction rule:
Declarative System Let rule:
Declarative System Instantiation rule:
Declarative System Generalization rule:
Soundness If A e : then A ╞ e :
“Static behavior determines dynamic behavior”
Example Prove: . : ∀ . ( → → )
→ →
Algorithm W • The inference rules do not translate easily
into an algorithm (why not?) • Introduce w : Exp → Env → (Env, )
Algorithm W • W attempts to build a substitution, bottom-up
• W can fail with an error if there is no valid typing • Intuition: ◦ Collect constraints ◦ Then solve constraints • Reality: W is the fusion of these two steps • See the code!
Unification • Unification gives local information about types • We
assemble a global solution from local information
Unification Example: ( → ) ~ (( → ) →
) ~ ( → ) ~ ~ ( → )
Occurs Check Prevents inference of infinite types w( . ,
nil) = error! Can’t unify ~ if occurs in the body of . E.g. ~ → ~ ((… → ) → ) →
Soundness If w(A, e) = (S, ) then A e
: “Algorithm W constructs typing judgments”
Completeness If A e : then w(A, e) constructs a
typing judgment for e which generalises the above. “Algorithm W constructs principal types”
Further Reading More type systems • System F, F⍵ •
Rank-N types • Type Classes • Dependent Types • Refinement Types Other approaches • Constraints • Bidirectional typechecking • SMT See TAPL & ATAPL!
Acknowledgments DHM axioms reproduced from Wikipedia under the CC-3.0 Attribution/ShareAlike
license