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What is that and how it is applied in functional programming languages.

Currying, High Order Functions, Anonymous Functions

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- 1. Lambda Calculus 1
- 2. Origin First observed in the late 1890s Formalized in the 1930s Developed in order to study mathematical properties. Lambda calculus is a conceptually simple universal model of computation 2
- 3. Motivation The lambda calculus can be called the smallest universal programming language of the world 3
- 4. What is this? The name derives from the Greek letter lambda (λ) used to denote binding a variable in a function Single transformation rule -> variable substitution Single function definition schema Any computable function can be expressed and evaluated using this formalism Functional Programming essentially implements this calculus The λ-calculus provides a simple semantics for computation, enabling properties of computation to be studied formally 4
- 5. Lambda Terms a variable is itself a valid lambda term if t is a lambda term, and x is a variable, then ( λx.t) is a lambda term (called a lambda abstraction); if t and s are lambda terms, then (ts) is a lambda term (called an application). Thus a lambda term is valid if and only if it can be obtained by repeated application of these three rules 5
- 6. lambda abstraction λ x.t X is the input T is the expression λ x.x+2 == f(x) = x +2 6
- 7. Lambda Property - 1 Lambda: sqsum(x, y) = x*x + y*y (x, y) ↦ x*x + y*y 7
- 8. Lambda Property – 1 -Equivalent In computer programming, an anonymous function (also function constant, function literal, or lambda function) is a function (or a subroutine) defined, and possibly called, without being bound to an identifier. 8
- 9. Lambda Property-2 In lambda calculus, functions are taken to be 'first class values', so functions may be used as the inputs, or be returned as outputs from other functions. 9
- 10. Lambda Property-2 - Equivalent In mathematics and comp uter science, a higher- order function (also functional form, functional or funct or) is a function that does at least one of the following: take one or more functions as an input output a function 10
- 11. Lambda Property 3 1 - (x, y) ↦ x*x + y*y 1.1 – (5,2) == 5*5 + 2*2 = 29 2 - ((x ↦ (y ↦ x*x + y*y)) 2.2 - = (y ↦ 5*5 + y*y)(5) 2.2 - = 5*5 + 2*2 = 29 11
- 12. Lambda Property 3 - Equivalent In mathematics and com puter science, currying is the technique of transforming a function that takes multiple arguments (or a tuple of arguments) in such a way that it can be called as a chain of functions, each with a single argument 12
- 13. Much more α-conversion: changing bound variables (alpha); β-reduction: applying functions to their arguments (beta); η-conversion: which captures a notion of extensionality (eta). Recursion Parallelism and concurrency 13
- 14. That is all 14 http://www.linkedin.com/in/diegomendonca

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