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Algorithm W infers data types for a pure lambda calculus with let polymorphism. Write $W(\Gamma ,e)=(S,\tau )$, where $S$ is a type substitution and $\tau$ the inferred data type. Composition $S_2{S}_1$ means applying $S_1$ first. The following presentation uses type schemes explicitly.^[ref]

For a program identifier, take a fresh instance of its type scheme in the typing context. For $\lambda x.e$, introduce fresh $\alpha$, infer $(S,\tau )=W(\Gamma [x:\alpha ],e)$, and return $(S,S\alpha \to \tau )$.

For an application $e_1{e}_2$, compute $(S_1,{\tau }_1)=W(\Gamma ,e_1)$ and $(S_2,{\tau }_2)=W(S_1\Gamma ,e_2)$. With fresh $\beta$, let $U$ be the most general unifier of $S_2{\tau }_1$ and ${\tau }_2\to \beta$. Return $(U{S}_2{S}_1,U\beta )$. An incompatible base data type or failed occurs check rejects the expression.

For let x = e1 in e2, first infer $(S_1,{\tau }_1)$. Generalize exactly those type variables of ${\tau }_1$ not free in $S_1\Gamma$, producing a type scheme $\sigma$. Infer $(S_2,{\tau }_2)$ for $e_2$ under $S_1\Gamma [x:\sigma ]$ and return $(S_2{S}_1,{\tau }_2)$.

For example, let id = \x -> x in let a = id 3 in id True has a Boolean data type result. The identity function receives type scheme $\forall \alpha .\alpha \to \alpha$, then separate uses instantiate it at integer and Boolean data types. A parameter to a lambda abstraction instead has one shared monomorphic type; treating that parameter like a let-bound type scheme would accept programs outside this type system.

A Boolean data type has two truth values. A binary probabilistic query returns such a value; rejection or failure must be represented separately if false outcomes are valid observations of the query.