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Advanced Topic 44 of 50 Languages & Compilers

Hindley-Milner Inference

Classic algorithm for polymorphic type inference.

How it works

Hindley-Milner is a type inference algorithm for the polymorphic lambda calculus that infers the most general, principal type of an expression without annotations. It assigns fresh type variables, generates equality constraints from the structure of the code and solves them by unification. Let-polymorphism generalizes the types of let-bound definitions so functions like identity can be used at several types. Algorithm W and algorithm J are standard formulations.

Step-by-step walkthrough

  1. Assign unknown type variables to each expression.
  2. Collect constraints from how values are used, such as x + 1 means x is a number.
  3. Unification solves the constraints.
  4. The most general type comes out, like a to a for identity.
  5. ML, Haskell and OCaml use it.

Where it is used

It is the core of type inference in ML, OCaml, Haskell and F#, and influenced Rust, Elm and TypeScript's inference.

Common pitfall

It does not support all features directly: higher-rank types, subtyping, overloading and type classes need extensions or annotations. Error messages can point far from the real mistake because unification fails where constraints meet. The monomorphism restriction and value restriction limit polymorphism to preserve soundness with mutation. Add annotations to guide errors and document intent.

Where it fits in the course

Hindley-Milner Inference is topic 44 of 50 in Languages & Compilers, rated Advanced. It follows Ownership and Borrowing. The next topic is Dependent Types.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Languages & Compilers quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Hindley-Milner Inference?
Classic algorithm for polymorphic type inference. It belongs to Languages & Compilers and is an advanced-level topic.
Where do people use Hindley-Milner Inference?
It is the core of type inference in ML, OCaml, Haskell and F#, and influenced Rust, Elm and TypeScript's inference.
What is the most common mistake with Hindley-Milner Inference?
It does not support all features directly: higher-rank types, subtyping, overloading and type classes need extensions or annotations.
What should I learn before this topic?
Start with Ownership and Borrowing, the topic just before it in Languages & Compilers.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Languages & Compilers quiz.
Is Hindley-Milner Inference hard to learn?
It is an Advanced topic. Expect to read the explanation more than once and to connect it with earlier topics. Working through a small example by hand helps most.
How long does it take to learn Hindley-Milner Inference?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Hindley-Milner Inference?
Ownership and Borrowing, Dependent Types, Pattern Matching are closely connected. The full list is in the related topics section of this page.
What should I study after Hindley-Milner Inference?
Continue with Dependent Types, the next topic in Languages & Compilers.

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