Download C311 First Class Objects

C311 First Class Objects
http://www.cs.iusb.edu/~danav/teach/c311/c311_2_firstclass.html
Dana Vrajitoru
C311 Programming Languages
First Class Objects
Classes of Objects
A classification of entities one can find in a program by what can be done with them.
First class: can be used without restriction. Example: integers in C++ or any other language.
Second class: cannot be assigned to a variable or returned by functions. Example: Classes in
Smalltalk.
Third class: cannot be passed as function parameters. Example: data types in C++.
First Class Objects
Expressible as an anonymous literal value
Storable in variables
Storable in data structures
Having an intrinsic identity (independent of any given name)
Comparable for equality with other entities
Passable as a parameter to a function
Returnable as the result of a function call
Constructable at runtime
First Class Functions
A programming language supports first class functions if it allows functions to be first class
objects.
Supported by all functional languages: Lisp, Scheme, ML, as well as many of the interpreted
languages: Python, Perl, JavaScript.
Creating a function at runtime means your program must contain a compiler.
In C++ only function references (pointers) are first class objects. Function Pointer Tutorials.
Higher-Order Functions
A higher-order function accepts functions as arguments and is able to return a function as its
result.
A higher-order language supports higher-order functions and allows functions to be constituents
of data structures.
Functions of order 0: both the domain and the range of the function are non-function data
(numbers, lists, arrays).
Functions of order k: Both the domain and the range can be functions of order k-1.
Higher order: where the order is at least 2.
Lambda Expressions
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C311 First Class Objects
http://www.cs.iusb.edu/~danav/teach/c311/c311_2_firstclass.html
Also called anonymous functions or methods.
They describe functions by directly describing their behavior.
Inspired from lambda calculus, a branch of mathematics that studies functions, recursion,
computability. Invented by A. Church in 1930.
Present in many languages: C#, Python, Perl.
Usually they mean the language allows first class functions.
Lambda Calculus
A mathematical tool to study functions, applications, recursion.
It also provides a meta-language for formal definition of functions.
Some notations:
lambda x. 2*x-1 describes a function f(x) = 2x-1
(lambda x. 2*x-1) 7 means the previous function applied to the constant 7 and is evaluated to
13.
lambda x. lambda y. x % y describes a function with two variables f(x, y) = x%y
Arithmetic and Lambda Calculus
Church numerals: defines the numerals as the n-th composition of a function with itself.
0: lambda f x. x
1: lambda f x. f x
2: lambda f x. f ( f x)
Arithmetic operations:
n+m: lambda f x. (n f) (m f x)
n*m: lambda f x. (n (mf x))
SUCC := lambda n f x. f (n f x)
PLUS := lambda m n f x. m f (n f x)
MULT := lambda m n f. m (n f)
Predicates in Lambda Calculus
TRUE := lambda x y. x
FALSE := lambda x y. y
AND := lambda p q. p q FALSE
OR := lambda p q. p TRUE q
NOT := lambda p. p FALSE TRUE
IFTHENELSE := lambda p x y. p x y
Computability
A function is called computable if there exists an algorithm to compute it.
Church-Turing thesis - states that any computable function can be computed by a Turing
machine with unlimited time/storage and the other way around.
A function can be defined as computable if and only if there exists a lambda expression f such
that for every pair of x, y in N, F(x) = y if and only if
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C311 First Class Objects
http://www.cs.iusb.edu/~danav/teach/c311/c311_2_firstclass.html
f x == y, where x and y are the Church numerals corresponding to x and y, respectively.
Lambda Expressions in Lisp
(lambda (arg-variables...)
[documentation-string]
[interactive-declaration]
body-forms...)
A lambda expression evaluates to itself.
(lambda (x y) (* x y))
((lambda (x y) (* x y)) 2 3)
6
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