Parametricityyppiin

Parametricityyppiin, often shortened to parametricity, is a fundamental concept in programming language theory, particularly within statically typed functional languages. It describes a property that holds for universally quantified types. In essence, parametricity states that a function with a polymorphic type should behave uniformly regardless of the type of the values it operates on. This means that a function like `id :: forall a. a -> a` (the identity function) cannot inspect the type of its argument. If it receives an integer, it must return an integer. If it receives a string, it must return a string. It cannot, for instance, behave differently based on whether the argument is a primitive type or a complex data structure.

The principle of parametricity, often referred to as the abstraction of types, implies that polymorphic functions can only use values of their arguments in ways that are independent of the argument's type. This severely restricts the possible implementations of such functions. For example, a function that takes a list of elements of any type and returns a boolean cannot, on its own, create new elements of that type or distinguish between different types. It can only rely on operations that are generic across all types, such as checking for emptiness or accessing elements.

This property is not just a theoretical curiosity; it has significant practical implications for program design and verification. It allows programmers to reason about the behavior of polymorphic functions with greater confidence, as their implementation is constrained by the type system. Furthermore, parametricity is a cornerstone of type inference systems and is crucial for proving properties about programs, contributing to the robustness and safety of software written in languages that leverage it.