A little bit of Lamba Calculus. What is that and how it is applied in functional programming languages. Currying, High Order Functions, Anonymous Functions

1. Lambda Calculus
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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
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3. Motivation
 The lambda calculus can be called the smallest
universal programming language of the world
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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
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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
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6. lambda abstraction
 λ x.t
 X is the input
 T is the expression
 λ x.x+2 == f(x) = x +2
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7. Lambda Property - 1
 Lambda:
 sqsum(x, y) = x*x + y*y
 (x, y) ↦ x*x + y*y
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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.
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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.
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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
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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
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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
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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
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14. That is all
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