# Polymorphic recursion

> recursive parametrically polymorphic function

**Wikidata**: [Q7226600](https://www.wikidata.org/wiki/Q7226600)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Polymorphic_recursion)  
**Source**: https://4ort.xyz/entity/polymorphic-recursion

## Summary
Polymorphic recursion refers to a recursive function that is parametrically polymorphic, meaning it can operate on multiple types without being explicitly defined for each one. This concept is foundational in type theory and programming language design, particularly in systems that support higher-order functions and type inference.

## Key Facts
- **Instance of**: Programming language (as per Wikidata)
- **Aliases**: Milner–Mycroft typability, Milner–Mycroft calculus
- **Wikipedia title**: Polymorphic recursion
- **Wikipedia languages**: English (en)
- **Wikidata description**: Recursive parametrically polymorphic function
- **Microsoft Academic ID (discontinued)**: 2776834425
- **Sitelink count**: 1 (as of the provided data)

## FAQs
### Q: What is the difference between polymorphic recursion and regular recursion?
A: Polymorphic recursion involves functions that can operate on multiple types, while regular recursion typically operates on a single type or data structure.

### Q: Which programming languages support polymorphic recursion?
A: Languages like Haskell, ML, and Scala are known for supporting polymorphic recursion, though specific implementations may vary.

### Q: What is the Milner–Mycroft calculus?
A: The Milner–Mycroft calculus is a type system that formalizes polymorphic recursion, particularly in the context of higher-order functions.

### Q: How does polymorphic recursion relate to type inference?
A: Polymorphic recursion often relies on type inference to determine the appropriate type instantiations during function application.

### Q: Is polymorphic recursion used in functional programming?
A: Yes, polymorphic recursion is a common feature in functional programming languages, enabling more flexible and reusable code.

## Why It Matters
Polymorphic recursion is significant in programming language theory and practice because it enables the creation of generic, reusable functions that can adapt to different data types. This concept is particularly valuable in functional programming, where higher-order functions and type abstraction are central. By allowing functions to operate on multiple types without explicit type annotations, polymorphic recursion simplifies code development and enhances modularity. It also plays a crucial role in type inference systems, where the compiler automatically deduces the appropriate type instantiations. Historically, the Milner–Mycroft calculus formalized this idea, providing a theoretical foundation for polymorphic recursion in languages like ML and Haskell. Overall, polymorphic recursion is a key feature that enhances the expressiveness and flexibility of programming languages, making it an essential concept for both theorists and practitioners.

## Notable For
- Being a foundational concept in type theory and functional programming.
- Enabling the creation of generic, reusable functions across multiple types.
- Serving as the basis for the Milner–Mycroft calculus, a formal type system.
- Supporting higher-order functions and type inference in modern programming languages.
- Facilitating modular and flexible code design in functional programming paradigms.

## Body
### Definition and Characteristics
Polymorphic recursion is a recursive function that is parametrically polymorphic, meaning it can operate on multiple types without being explicitly defined for each one. This concept is central to type theory and programming language design, particularly in systems that support higher-order functions and type inference.

### Historical Context
The Milner–Mycroft calculus formalized polymorphic recursion, providing a theoretical framework for its implementation in programming languages. This calculus is named after Robin Milner and Alan Mycroft, who contributed to its development.

### Applications
Polymorphic recursion is widely used in functional programming languages such as Haskell, ML, and Scala. These languages leverage polymorphic recursion to create flexible and reusable functions that can adapt to different data types.

### Type Inference
Polymorphic recursion often relies on type inference to determine the appropriate type instantiations during function application. This allows programmers to write more concise and generic code without explicitly specifying types.

### Impact on Programming
By enabling the creation of generic functions, polymorphic recursion enhances the modularity and reusability of code. It is particularly valuable in functional programming, where higher-order functions and type abstraction are fundamental.