Curried Functions - Computerphile
TLDRThe video script delves into the concept of curried functions in the context of lambda calculus, a foundational mathematical theory for functions. It starts by defining a function and its basic anatomy, using Haskell as an example. The script then introduces curried functions, explaining how they accept multiple inputs sequentially rather than as a set. It illustrates the utility of curried functions through practical examples, including the use of 'map' and the 'add' function. The explanation further breaks down the concept using lambda expressions to reveal the essence of curried functions. It concludes by relating the concept to real-world applications, like ATMs, which operate as curried functions, taking inputs step by step.
Takeaways
- ๐ Lambda calculus is a mathematical theory of functions that is simple yet powerful.
- ๐ A function is defined as taking an input, processing it, and then producing an output.
- ๐ Haskell is used as an example programming language to illustrate concepts, but the principles apply to any modern language supporting lambda calculus.
- ๐ข The script introduces an 'increment' function as an example, demonstrating how functions are defined and used.
- ๐บ๏ธ The concept of 'map' is explained as a higher-order function that applies another function across a list.
- ๐ค The idea of a function with multiple inputs is introduced, using an 'add' function as an example.
- ๐ฒ Curried functions are functions that take multiple inputs but process them one at a time, rather than as a single package.
- ๐ฆ Partial application of curried functions is shown, where a function can be applied with only some of its inputs, returning a new function waiting for the remaining inputs.
- ๐ก The script uses a cash machine as a real-world example of a curried function, illustrating how inputs are processed sequentially.
- ๐ The term 'currying' is named after Haskell Curry, a mathematician and logician, although the concept is attributed to Moses Schรถnfinkel.
- ๐ Functional programming languages manage details like memory management, making them popular for modern programming practices.
Q & A
What is a function in the context of the provided script?
-A function takes an input, processes it, and produces an output. It can be thought of as a little box or machine that performs these operations.
How is the increment function defined in Haskell?
-The increment function is defined as 'increment of x is x plus one', which adds one to a given number.
What is a higher-order function, and can you provide an example from the script?
-A higher-order function is a function that takes another function as one of its inputs. An example from the script is the 'map' function, which takes the 'increment' function and a list of numbers, applying 'increment' to each number in the list.
How is the 'add' function defined to take multiple inputs, and what is its output?
-The 'add' function is defined as taking two numbers, x and y, as inputs and returning their sum. For example, 'add 1 and 2' gives 3, and 'add 2 and 3' gives 5.
What distinguishes a curried function from a regular function with multiple inputs?
-A curried function takes its inputs one at a time rather than all at once. For example, instead of 'add of x y is x plus y', a curried function would be 'add of x and then a space and then a y is x plus y'.
What happens when a curried function is given only part of its required inputs?
-When a curried function is given only part of its inputs, it returns a new function that takes the remaining inputs. For example, 'add of one' results in a function that still expects a second parameter.
How can curried functions be useful in practical programming scenarios?
-Curried functions allow for partial application of arguments, enabling more modular and reusable code. For example, 'add one' can be used to increment all numbers in a list using the 'map' function.
What is the lambda notation, and how does it relate to curried functions?
-Lambda notation represents nameless functions. For example, 'add of x is lambda y, arrow x plus y' means that given x, the function returns another function that takes y and computes x plus y. This notation helps in understanding the currying process.
Who is credited with the concept of currying, and who originally studied it?
-The concept of currying is named after Haskell Curry, but it was originally studied by Moses Schoenfinkel. Despite the credit given to Curry, Schoenfinkel was the first to work on this idea.
Can you provide a real-world analogy to explain curried functions?
-A cash machine serves as a real-world analogy for curried functions. It takes inputs one at a time: card, pin number, and request. Each step returns a function expecting the next input, ultimately resulting in the output of cash.
Outlines
๐ Introduction to Functions and Currying
The video begins by revisiting the concept of lambda calculus, a foundational theory in computer science. The focus then shifts to 'curried functions,' which are functions that accept multiple arguments but process them one at a time. The presenter uses Haskell to illustrate the concept, starting with a simple increment function that adds one to a given number. This serves as a basic example of a function's structure and purpose. The video then explores the idea of higher-order functions, like 'map,' which apply another function to a list. The concept of currying is introduced by redefining the 'add' function to accept its arguments sequentially rather than as a pair, demonstrating how this allows for partial application of functions.
๐ Deep Dive into Currying and Practical Applications
This paragraph delves deeper into the mechanics of curried functions, explaining how they can be partially applied to a subset of inputs. The presenter uses the 'add' function as an example, showing how it can be applied incrementally. The explanation is further clarified by transforming the function definition into a lambda expression, which helps to understand that a curried function essentially returns another function waiting for the next input. The historical origins of currying are also briefly discussed, crediting Haskell Curry and Moses Schรถnfinkel for their contributions. The paragraph concludes with a real-world analogy, comparing a cash machine's operation to a curried function, where inputs are provided sequentially and the machine dispenses money as the final output.
๐ Conclusion on Currying and Its Relevance in Modern Programming
The final paragraph wraps up the discussion on curried functions by emphasizing their simplicity and utility. It highlights that functional programming languages, such as Haskell, manage many implementation details automatically, which might be more cumbersome in older languages. The presenter suggests that the concept of currying is not only relevant in computer science but also naturally occurs in everyday scenarios, reinforcing its practical value. The video concludes by reinforcing the idea that curried functions are a straightforward yet powerful concept in programming, particularly in languages that support functional programming paradigms.
Mindmap
Keywords
๐กLambda Calculus
๐กCurried Functions
๐กFunction
๐กIncrement Function
๐กMap
๐กAdd Function
๐กHigher Order Function
๐กPartial Application
๐กLambda Expression
๐กCash Machine
Highlights
Introduction to lambda calculus and its significance in understanding functions.
Explanation of the concept of curried functions in the context of functional programming.
Defining a function as a process that takes input, processes it, and gives an output.
Using Haskell to demonstrate the concept of functions with a simple increment example.
The anatomy of a function definition, including naming, input parameters, and output calculation.
Demonstration of applying a function to a list using the map higher-order function.
Concept of a function with more than one input parameter, exemplified by an add function.
Redefinition of the add function as a curried function, taking inputs one at a time.
Partial application of curried functions, allowing for the creation of new functions with fewer arguments.
Practical example of using a curried add function to increment a list without a custom increment function.
Lambda calculus trick to understand curried functions as a series of functions waiting for inputs.
Historical background on the naming of currying after Haskell Curry, a mathematician and logician.
Real-world analogy of a cash machine as a curried function, taking inputs step by step.
Definition of a cash machine as a curried function using lambda notation for clarity.
Conclusion emphasizing the simplicity and usefulness of curried functions in programming.
Discussion on the benefits of functional programming languages in managing implementation details.
Transcripts
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