computer science

Parametricity

Parametricity is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Parametricity rather than just read about it. In short: In programming language theory, parametricity is an abstract uniformity property enjoyed by parametrically polymorphic functions, which captures the intuition that all instances of a polymorphic function act the same way. Idea Consider this example, based on a set X and the type T(X) = [X → X] of functions from X to itself.

Key takeaways

Parametricity belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Parametricity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Parametricity from memory before moving on to harder problems.

Reference excerpt

In programming language theory, parametricity is an abstract uniformity property enjoyed by parametrically polymorphic functions, which captures the intuition that all instances of a polymorphic function act the same way.

Idea Consider this example, based on a set X and the type T(X) = [X → X] of functions from X to itself. The higher-order function twiceX : T(X) → T(X) given by twiceX(f) = f ∘ f, is intuitively independent of the set X. The family of all such functions twiceX, parametrized by sets X, is called a "parametrically polymorphic function". We simply write twice for the entire family of these functions and write its type as ∀ {\displaystyle \forall } X. T(X) → T(X). The individual functions twiceX are called the components or instances of the polymorphic function. Notice that all the component functions twiceX act "the same way" because they are given by the same rule. Other families of functions obtained by picking one arbitrary function from each T(X) → T(X) would not have such uniformity. They are called "ad hoc polymorphic functions". Parametricity is the abstract property enjoyed by the uniformly acting families such as twice, which distinguishes them from ad hoc families. With an adequate formalization of parametricity, it is possible to prove that the parametrically polymorphic functions of type ∀ {\displaystyle \forall } X. T(X) → T(X) are one-to-one with natural numbers. The function corresponding to the natural number n is given by the rule f ↦ {\displaystyle \mapsto } fn, i.e., the polymorphic Church numeral for n. In contrast, the collection of all ad hoc families would be too large to be a set.

History The parametricity theorem was originally stated by John C. Reynolds, who called it the abstraction theorem. In his paper "Theorems for free!", Philip Wadler described an application of parametricity to derive theorems about parametrically polymorphic functions based on their types.

Programming language implementation Parametricity is the basis for many program transformations implemented in compilers for the programming language Haskell. These transformations were traditionally thought to be correct in Haskell because of Haskell's non-strict semantics. Despite being a lazy evaluation programming language, Haskell does support certain primitive operations, such as the operator seq—that enable so-called selective strictness, allowing evaluation to be forced for certain expressions. In their paper "Free theorems in the presence of seq", Patricia Johann and Janis Voigtlaender showed that because of the presence of these operations, the general parametricity theorem does not hold for Haskell programs; thus, these transformations are unsound in general.

Dependent types

See also Parametric polymorphism Non-strict programming language

References

External links Wadler: Parametricity

Worked examples

Example 1 — a first encounter with Parametricity

Start with the simplest possible case. Write down what Parametricity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Parametricity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Parametricity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Parametricity

In research: Parametricity appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Parametricity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Parametricity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Polymorphism (computer science), Programming language topics, Type theory, so understanding it makes those chapters shorter.
In everyday life: Look for Parametricity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Parametricity in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Parametricity means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Parametricity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Parametricity in simple terms?

Why does Parametricity matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Parametricity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Parametricity.