Simply typed lambda calculus

The simply typed lambda calculus (${\displaystyle \lambda ^{\to }}$), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor: ${\displaystyle \to }$ that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties.

The term simple type is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The former are still considered simple because the Church encodings of such structures can be done using only ${\displaystyle \to }$ and suitable type variables, while polymorphism and dependency cannot.

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