Untyped

Untyped refers to systems that do not assign or enforce a static type to data or expressions. In programming language theory, an untyped system imposes no compile-time or run-time type discipline, allowing values of any kind to be used interchangeably in most contexts. This flexibility can simplify language design and early prototyping but makes certain classes of errors possible only at runtime, such as applying a non-function as a function.

A canonical example is the untyped lambda calculus, a formal system introduced by Alonzo Church that includes only variables, abstraction, and application. In the untyped lambda calculus, every expression can be applied to any argument, and there is no notion of a universal data type; the system is Turing complete and serves as a foundational model of computation. This contrasts with simply-typed or polymorphically typed lambda calculi, which enforce a type discipline that prevents certain kinds of errors but restricts expressiveness.

In practical programming, most languages employ some form of typing, whether statically, dynamically, or through gradual typing. The term untyped is typically reserved for theoretical formalisms or historical discussions, with real-world languages usually described as dynamically or statically typed, or as having type systems that can be evaluated at runtime. Nonetheless, untyped models remain of interest for reasoning about computation without the constraints of a type system and for illustrating the limits of type-free computation.

Critics of untyped systems cite difficulties in proving program correctness, absence of safety guarantees, and challenges for optimization lacking type information. Advocates emphasize simplicity, expressive power, and tight alignment with mathematical models of computation.

See also: untyped lambda calculus; dynamic typing; type system; simply typed lambda calculus.