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Algorithm W performs type inference by traversing the program's abstract syntax tree, introducing fresh type variables, combining type substitutions obtained by type unification, and generalizing eligible let bindings. It distinguishes fresh instances of a type scheme from the shared monomorphic types of parameters to a lambda abstraction.

The pure Hindley-Milner type system combines function types, type inference and let polymorphism. It solves constraints by type unification and generalizes eligible type variables at let bindings. Robin Milner's 1978 paper presents Algorithm W and proves its soundness for the paper's language; it also acknowledges Roger Hindley's earlier work on finding general types. ^[ref]

In the examples here, a simple type is a finite expression built from the base types int and bool using function types:For example, $\mathrm{int}\to \mathrm{bool}$ describes a function accepting an integer and producing a Boolean value. During type inference, the grammar is extended with type variables such as $\alpha$ for types not yet determined. A type scheme additionally binds such type variables; it is distinct from the individual simple type used at one call.

Type inference discovers types consistent with a program's structure and typing context. In our examples, it introduces fresh type variables, generates equations from typing judgments, and solves them by type unification. For $f3$, if $f$ initially has unknown type $\alpha$ and the result has fresh type $\beta$, lambda application requires $\alpha =\mathrm{int}\to \beta$.

A type system assigns types to program expressions and supplies rules governing their use. A type constrains permitted operations: in our teaching programming language, int admits integer addition, bool supplies true and false for conditions, and a function type specifies an argument type and a result type. Storage as bits alone does not explain these distinctions; the type system determines whether an expression can serve as an argument, a condition or an operand.

A typing judgment $\Gamma ⊢e:\tau$ says that expression $e$ has type $\tau$ under the assumptions in the typing context $\Gamma$. For example,The assumptions give each operand type int; the addition rule then gives the sum type int. Changing $y$ to type bool prevents this derivation. Our teaching programming language has distinct int and bool types and no implicit conversions between them.

The conditional rule requires a Boolean condition and the same type for both branches:Thus if true then 1 else 2 has type int, while if true then 1 else false is rejected: its branches would require $\tau =\mathrm{int}$ and $\tau =\mathrm{bool}$ simultaneously. Even a branch that will not execute is checked by this rule. This is a rule of this particular type system; a different programming language can choose different rules. ^[ref]

The useful feature is compositional reasoning: after checking the parts, a rule explains how to check the whole. The function type diagram shows this for a call. Type inference can discover the required types automatically, and type variables let one express relationships that recur across many concrete types. The result is both a practical early error detector and a small, readable derivation explaining why an expression fits its intended use.

The guarantee depends on the rules and execution model. Type safety can exclude specified failures such as adding a Boolean or calling an integer as a function. It does not by itself establish every desired runtime property or guarantee that execution terminates. A type system that permits unrestricted recursion can accept a computation that runs forever; exception handling also needs its own account of permitted outcomes.