Monday, September 1, 2014

Peeping into Julia Dictionaries and using them in problem solving


Julia dictionaries are powerful mapping data structures.we here deal with two problems and using dictionaries.



1)Finding cube root of a no without using inbuilt cbrt() function i.e purely written in Julia

Here we see how to use a Julia dictionary and also map function


Solution:
function cubethRoot(n)
           d = factor(n)
           k,v=keys(d),values(d)
           fact = int([int(m) for m in v][1]/3)
           mykeys=[int(m) for m in k]
           res = map(x->x^fact,mykeys)
    prod(res)
end
@time cubeRoot(512)

2)Factorise from beginning to a given no

factor() is a julia inbuilt which factorises a no and returns a dictionary


Solution:
function showfactor(n)
           for i in 1:n
               d=factor(i)
               println("factors for $i")
               for k in d
                   print("$(k[1])^$(k[2])\t")
               end
               println('\n')
           end
       end
println("Enter a no to which factors to be printed")
m = int(rstrip(readline(STDIN)))
@time showfactor(m)



@time macro is optional to show efficiency of Julia function.

Tuesday, May 20, 2014

99 problems in Julia (11-20)

In this article i am proudly presenting solutions to 11-20 problems from classical 99 Haskell problem set.


 Problem 11

Modify the result of problem 10 in such a way that if an element has no duplicates it is simply copied into the result list. Only elements with duplicates are transferred as (N E) lists.
Example:
Haskell Example:
 
* (encode-modified '(a a a a b c c a a d e e e e))
((4 A) B (2 C) (2 A) D (4 E))

Solution:
julia>function modifiedEncoding(x)
        lest=[]
        for c in unique(x)
                if counter(x,c)>1
                        lest=[lest...,(counter(x,c),c)]
                else
                        lest=[lest...,c]
                end
        end
   lest

end

julia>function counter(l,el)
        temp=0
        for i=1:endof(l)
                if el==l[i]
                        temp+=1
                end
        end
     temp
end

Problem 12

(**) Decode a run-length encoded list.
Given a run-length code list generated as specified in problem 11. Construct its uncompressed version.
Example in Haskell:
P12> decodeModified 
       [Multiple 4 'a',Single 'b',Multiple 2 'c',
        Multiple 2 'a',Single 'd',Multiple 4 'e']

Solution:
julia>function de_encode(x)
        lest=[]
        for c in x
                if typeof(c)==(Int32,Int32)
                        for i=1:c[1]
                                lest=[lest...,c[2]]
                        end
                else
                        lest=[lest...,c]
                end
        end
    lest
end

Problem 13

(**) Run-length encoding of a list (direct solution).
Implement the so-called run-length encoding data compression method directly. I.e. don't explicitly create the sublists containing the duplicates, as in problem 9, but only count them. As in problem P11, simplify the result list by replacing the singleton lists (1 X) by X.
Example:
* (encode-direct '(a a a a b c c a a d e e e e))
((4 A) B (2 C) (2 A) D (4 E))

Solution:
Same as problem 11.We used shortcut solution already in our early 11th problem.

Problem 14

(*) Duplicate the elements of a list.
Example:
* (dupli '(a b c c d))
(A A B B C C C C D D)

Solution:
julia>function duplicate(x)
        lest=[]
        for c in x
                lest=[lest...,c,c]
        end
    lest
end

Problem 15

(**) Replicate the elements of a list a given number of times.
Example:
* (repli '(a b c) 3)
(A A A B B B C C C) 
 
Solution: 
julia>function replicate(x,k)
        lest=[]
        for c in x
                for i=1:k
                        lest=[lest...,c]
                end
        end
    lest
end

 Problem 16

(**) Drop every N'th element from a list.
Example:
* (drop '(a b c d e f g h i k) 3) (A B D E G H K)
 Solution:
julia>function dropN(x,n)
        lest=[]
        for i=1:endof(x)
                if i%n!=0
                        lest=[lest...,x[i]]
                end
        end
    lest
end

Problem 17

(*) Split a list into two parts; the length of the first part is given.
Do not use any predefined predicates.
Example:
* (split '(a b c d e f g h i k) 3)
( (A B C) (D E F G H I K))
 
Solution: 
julia>split(x,n)=x[1:n],x[n+1,end]

Problem 18

(**) Extract a slice from a list.
Given two indices, i and k, the slice is the list containing the elements between the i'th and k'th element of the original list (both limits included). Start counting the elements with 1.
Example:
* (slice '(a b c d e f g h i k) 3 7)
   (C D E F G)

Solution:
julia>slice(x,i,j)=x[i:j]

Problem 19

(**) Rotate a list N places to the left.
Hint: Use the predefined functions length and (++).
Examples:
* (rotate '(a b c d e f g h) 3)
(D E F G H A B C)

* (rotate '(a b c d e f g h) -2)
(G H A B C D E F)
 
Solution: 
 julia>rotLeft(x,n)=[x[n+1:end],x[1:n]]

Problem 20

(*) Remove the K'th element from a list.
Example in Prolog:
?- remove_at(X,[a,b,c,d],2,R).
X = b
R = [a,c,d]
 
Solution: 
julia>removeAt(x,n)=delete!(x,n)

This delete!() method will go to jail because it changes original list which 
is anti philosophy of functional programming.Here is functional version.

julia>function removeAt(x,n)
        lest=[]
        for i=1:endof(x)
                if i!=n
                        lest=[lest...,x[i]]
                end
        end
   lest
end

Thanks for your patience.see you next time.



Monday, May 19, 2014

99 problems in Julia programming Language (1-10)

Prolog and Haskell got 99 problems to understand basic logic constructs .This is an attempt to create Julia versions of solutions to those problems.Here first 10 problems are solved by me.Remaining will be looked in further articles.If you like to contribute to my work mail me at narenarya@live.com.

Problem 1:

Q) Find the last element of a list
Ex: julia>myLast([1,2,3,4,5])
5
Solution:
julia>myLast(l)= l[end]

Prblem 2:

Q) Find last but one element from list
Ex: julia>myButOne([1,2,3,4,5])
4
Solution:
julia>myButOne(l)=l[end-1]

Problem 3:

Q) Find the kth element of list
Ex: julia>kthElement([12,3,4,16,43,9],4)
16
since 16 is 4th element in the list.
Solution: 
julia>kthElement(l,k)=l[k]
Note:expression is so simple because Julia indexes from 1 not 0.

Problem 4:
Q)Find the no of elements in list
Ex:julia>ElementNo([1,2,3,4])
4
Solution:
julia>ElementNo(l)=length(l)

Problem 5:
Q)Reverse a list
Ex:julia>myReversedList([1,2,3,4])
[4,3,2,1]
Solution:
julia>myReversedList(l)=reverse(l)

Problem 6:
Q)Find out whether a list is a palindrome. A palindrome can be read forward or backward; e.g. (x a m a x).
Ex:julia>Palindrome([1,2,3,2,1])
true
Solution:
julia>Palindrome(l)= return l==reverse(l)

Problem 7:
Q)Flatten a nested list structure.
Ex:julia>myFlatten([1,[2,3,],16,[45,76,84],0])
1
2
3
16
45
76
84
0
Flattening means removing recursive lists i.e lists inside lists.So [1,[2,3]] list is broken into [1,2,3]
Solution:
julia>myFlatten(l)=l
This simple,yes it is utterly simple Julia automatically flats your nested structure.

Problem 8:
Q) Eliminate consecutive duplicates of list elements.
Ex:julia>deleteConsDuplicates([1,1,1,2,2,3,3,3,3,4,4,5])
1
2
3
4
5
Solution:
julia> function deleteConsDuplicates(l)
            myset=Set()
            for i=1:endof(l)
               push!(myset,l[i])
            end
           sort([x for x in myset])
        end
we can simply cheat using below oneliner
julia>deleteConsDuplicates(l)=unique(l)

Problem 9:
Q)Pack consecutive duplicates of list elements into sublists. If a list contains repeated elements they should be placed in separate sublists.
Ex: julia>packDuplicates([1,1,1,2,2,3,3,3,4,4,4,4,4])
[(1,1,1),(2,2),(3,3,3),(4,4,4,4)]
Solution:
julia>function packDuplicates(x)
               lest=[]
               for c in deleteConsDuplicates(x)
                     tup=()
                     for i=1:counter(x,c)
                             tup=tuple(tup...,(c))
                     end
                    lest=[lest,tup]
              end
              lest
       end
julia> function counter(l,x)
              temp=0
              for i=1:endof(l)
                   if l[i]==x
                   temp=temp+1
                   end
             end
             temp
        end

Problem 10:
Q)Run-length encoding of a list. Use the result of problem P09 to implement the so-called run-length encoding data compression method. Consecutive duplicates of elements are encoded as lists (N E) where N is the number of duplicates of the element E.
Ex: julia>encodeList([a,a,a,b,b,c,c,c,d,d,d,d])
[(3,a),(2,b),(3,c),(4,d)]
Solution:
 julia> function encodeList(x)
               finlist=[]
               for c in deleteConsDuplicates(x)
                       finlist=[finlist,(counter(x,c),c)]
               end
              finlist
         end

We cleverly used comprehensions for list and tuple to generate new lists,tuples respectively.From now each week new 10 problems will be solved until 99 project is completed,so please be updated with learnjulia.

















Sunday, May 18, 2014

Meet Julia for 10 minutes


Julia is a new homoiconic functional language focused on technical computing. While having the full power of homoiconic macros, first-class functions, and low-level control, Julia is as easy to learn and use as Python.
This is based on the current development version of Julia, as of October 18th, 2013.
# Single line comments start with a number symbol.
#= Multiline comments can be written
   by putting '#=' before the text  and '=#' 
   after the text. They can also be nested.
=#

####################################################
## 1. Primitive Datatypes and Operators
####################################################

# Everything in Julia is a expression.

# There are several basic types of numbers.
3 # => 3 (Int64)
3.2 # => 3.2 (Float64)
2 + 1im # => 2 + 1im (Complex{Int64})
2//3 # => 2//3 (Rational{Int64})

# All of the normal infix operators are available.
1 + 1 # => 2
8 - 1 # => 7
10 * 2 # => 20
35 / 5 # => 7.0
5 / 2 # => 2.5 # dividing an Int by an Int always results in a Float
div(5, 2) # => 2 # for a truncated result, use div
5 \ 35 # => 7.0
2 ^ 2 # => 4 # power, not bitwise xor
12 % 10 # => 2

# Enforce precedence with parentheses
(1 + 3) * 2 # => 8

# Bitwise Operators
~2 # => -3   # bitwise not
3 & 5 # => 1 # bitwise and
2 | 4 # => 6 # bitwise or
2 $ 4 # => 6 # bitwise xor
2 >>> 1 # => 1 # logical shift right
2 >> 1  # => 1 # arithmetic shift right
2 << 1  # => 4 # logical/arithmetic shift left

# You can use the bits function to see the binary representation of a number.
bits(12345)
# => "0000000000000000000000000000000000000000000000000011000000111001"
bits(12345.0)
# => "0100000011001000000111001000000000000000000000000000000000000000"

# Boolean values are primitives
true
false

# Boolean operators
!true # => false
!false # => true
1 == 1 # => true
2 == 1 # => false
1 != 1 # => false
2 != 1 # => true
1 < 10 # => true
1 > 10 # => false
2 <= 2 # => true
2 >= 2 # => true
# Comparisons can be chained
1 < 2 < 3 # => true
2 < 3 < 2 # => false

# Strings are created with "
"This is a string."

# Character literals are written with '
'a'

# A string can be indexed like an array of characters
"This is a string"[1] # => 'T' # Julia indexes from 1
# However, this is will not work well for UTF8 strings,
# so iterating over strings is recommended (map, for loops, etc).

# $ can be used for string interpolation:
"2 + 2 = $(2 + 2)" # => "2 + 2 = 4"
# You can put any Julia expression inside the parenthesis.

# Another way to format strings is the printf macro.
@printf "%d is less than %f" 4.5 5.3 # 5 is less than 5.300000

# Printing is easy
println("I'm Julia. Nice to meet you!")

####################################################
## 2. Variables and Collections
####################################################

# You don't declare variables before assigning to them.
some_var = 5 # => 5
some_var # => 5

# Accessing a previously unassigned variable is an error
try
    some_other_var # => ERROR: some_other_var not defined
catch e
    println(e)
end

# Variable names start with a letter.
# After that, you can use letters, digits, underscores, and exclamation points.
SomeOtherVar123! = 6 # => 6

# You can also use unicode characters
 = 8 # => 8
# These are especially handy for mathematical notation
2 * π # => 6.283185307179586

# A note on naming conventions in Julia:
#
# * Names of variables are in lower case, with word separation indicated by
#   underscores ('\_').
#
# * Names of Types begin with a capital letter and word separation is shown
#   with CamelCase instead of underscores.
#
# * Names of functions and macros are in lower case, without underscores.
#
# * Functions that modify their inputs have names that end in !. These
#   functions are sometimes called mutating functions or in-place functions.

# Arrays store a sequence of values indexed by integers 1 through n:
a = Int64[] # => 0-element Int64 Array

# 1-dimensional array literals can be written with comma-separated values.
b = [4, 5, 6] # => 3-element Int64 Array: [4, 5, 6]
b[1] # => 4
b[end] # => 6

# 2-dimentional arrays use space-separated values and semicolon-separated rows.
matrix = [1 2; 3 4] # => 2x2 Int64 Array: [1 2; 3 4]

# Add stuff to the end of a list with push! and append!
push!(a,1)     # => [1]
push!(a,2)     # => [1,2]
push!(a,4)     # => [1,2,4]
push!(a,3)     # => [1,2,4,3]
append!(a,b) # => [1,2,4,3,4,5,6]

# Remove from the end with pop
pop!(b)        # => 6 and b is now [4,5]

# Let's put it back
push!(b,6)   # b is now [4,5,6] again.

a[1] # => 1 # remember that Julia indexes from 1, not 0!

# end is a shorthand for the last index. It can be used in any
# indexing expression
a[end] # => 6

# we also have shift and unshift
shift!(a) # => 1 and a is now [2,4,3,4,5,6]
unshift!(a,7) # => [7,2,4,3,4,5,6]

# Function names that end in exclamations points indicate that they modify
# their argument.
arr = [5,4,6] # => 3-element Int64 Array: [5,4,6]
sort(arr) # => [4,5,6]; arr is still [5,4,6]
sort!(arr) # => [4,5,6]; arr is now [4,5,6]

# Looking out of bounds is a BoundsError
try
    a[0] # => ERROR: BoundsError() in getindex at array.jl:270
    a[end+1] # => ERROR: BoundsError() in getindex at array.jl:270
catch e
    println(e)
end

# Errors list the line and file they came from, even if it's in the standard
# library. If you built Julia from source, you can look in the folder base
# inside the julia folder to find these files.

# You can initialize arrays from ranges
a = [1:5] # => 5-element Int64 Array: [1,2,3,4,5]

# You can look at ranges with slice syntax.
a[1:3] # => [1, 2, 3]
a[2:end] # => [2, 3, 4, 5]

# Remove elements from an array by index with splice!
arr = [3,4,5]
splice!(arr,2) # => 4 ; arr is now [3,5]

# Concatenate lists with append!
b = [1,2,3]
append!(a,b) # Now a is [1, 2, 3, 4, 5, 1, 2, 3]

# Check for existence in a list with in
in(1, a) # => true

# Examine the length with length
length(a) # => 8

# Tuples are immutable.
tup = (1, 2, 3) # => (1,2,3) # an (Int64,Int64,Int64) tuple.
tup[1] # => 1
try:
    tup[1] = 3 # => ERROR: no method setindex!((Int64,Int64,Int64),Int64,Int64)
catch e
    println(e)
end

# Many list functions also work on tuples
length(tup) # => 3
tup[1:2] # => (1,2)
in(2, tup) # => true

# You can unpack tuples into variables
a, b, c = (1, 2, 3) # => (1,2,3)  # a is now 1, b is now 2 and c is now 3

# Tuples are created even if you leave out the parentheses
d, e, f = 4, 5, 6 # => (4,5,6)

# A 1-element tuple is distinct from the value it contains
(1,) == 1 # => false
(1) == 1 # => true

# Look how easy it is to swap two values
e, d = d, e  # => (5,4) # d is now 5 and e is now 4


# Dictionaries store mappings
empty_dict = Dict() # => Dict{Any,Any}()

# You can create a dictionary using a literal
filled_dict = ["one"=> 1, "two"=> 2, "three"=> 3]
# => Dict{ASCIIString,Int64}

# Look up values with []
filled_dict["one"] # => 1

# Get all keys
keys(filled_dict)
# => KeyIterator{Dict{ASCIIString,Int64}}(["three"=>3,"one"=>1,"two"=>2])
# Note - dictionary keys are not sorted or in the order you inserted them.

# Get all values
values(filled_dict)
# => ValueIterator{Dict{ASCIIString,Int64}}(["three"=>3,"one"=>1,"two"=>2])
# Note - Same as above regarding key ordering.

# Check for existence of keys in a dictionary with in, haskey
in(("one", 1), filled_dict) # => true
in(("two", 3), filled_dict) # => false
haskey(filled_dict, "one") # => true
haskey(filled_dict, 1) # => false

# Trying to look up a non-existant key will raise an error
try
    filled_dict["four"] # => ERROR: key not found: four in getindex at dict.jl:489
catch e
    println(e)
end

# Use the get method to avoid that error by providing a default value
# get(dictionary,key,default_value)
get(filled_dict,"one",4) # => 1
get(filled_dict,"four",4) # => 4

# Use Sets to represent collections of unordered, unique values
empty_set = Set() # => Set{Any}()
# Initialize a set with values
filled_set = Set(1,2,2,3,4) # => Set{Int64}(1,2,3,4)

# Add more values to a set
push!(filled_set,5) # => Set{Int64}(5,4,2,3,1)

# Check if the values are in the set
in(2, filled_set) # => true
in(10, filled_set) # => false

# There are functions for set intersection, union, and difference.
other_set = Set(3, 4, 5, 6) # => Set{Int64}(6,4,5,3)
intersect(filled_set, other_set) # => Set{Int64}(3,4,5)
union(filled_set, other_set) # => Set{Int64}(1,2,3,4,5,6)
setdiff(Set(1,2,3,4),Set(2,3,5)) # => Set{Int64}(1,4)


####################################################
## 3. Control Flow
####################################################

# Let's make a variable
some_var = 5

# Here is an if statement. Indentation is not meaningful in Julia.
if some_var > 10
    println("some_var is totally bigger than 10.")
elseif some_var < 10    # This elseif clause is optional.
    println("some_var is smaller than 10.")
else                    # The else clause is optional too.
    println("some_var is indeed 10.")
end
# => prints "some var is smaller than 10"


# For loops iterate over iterables.
# Iterable types include Range, Array, Set, Dict, and String.
for animal=["dog", "cat", "mouse"]
    println("$animal is a mammal")
    # You can use $ to interpolate variables or expression into strings
end
# prints:
#    dog is a mammal
#    cat is a mammal
#    mouse is a mammal

# You can use 'in' instead of '='.
for animal in ["dog", "cat", "mouse"]
    println("$animal is a mammal")
end
# prints:
#    dog is a mammal
#    cat is a mammal
#    mouse is a mammal

for a in ["dog"=>"mammal","cat"=>"mammal","mouse"=>"mammal"]
    println("$(a[1]) is a $(a[2])")
end
# prints:
#    dog is a mammal
#    cat is a mammal
#    mouse is a mammal

for (k,v) in ["dog"=>"mammal","cat"=>"mammal","mouse"=>"mammal"]
    println("$k is a $v")
end
# prints:
#    dog is a mammal
#    cat is a mammal
#    mouse is a mammal

# While loops loop while a condition is true
x = 0
while x < 4
    println(x)
    x += 1  # Shorthand for x = x + 1
end
# prints:
#   0
#   1
#   2
#   3

# Handle exceptions with a try/catch block
try
   error("help")
catch e
   println("caught it $e")
end
# => caught it ErrorException("help")


####################################################
## 4. Functions
####################################################

# The keyword 'function' creates new functions
#function name(arglist)
#  body...
#end
function add(x, y)
    println("x is $x and y is $y")

    # Functions return the value of their last statement
    x + y
end

add(5, 6) # => 11 after printing out "x is 5 and y is 6"

# You can define functions that take a variable number of
# positional arguments
function varargs(args...)
    return args
    # use the keyword return to return anywhere in the function
end
# => varargs (generic function with 1 method)

varargs(1,2,3) # => (1,2,3)

# The ... is called a splat.
# We just used it in a function definition.
# It can also be used in a fuction call,
# where it will splat an Array or Tuple's contents into the argument list.
Set([1,2,3])    # => Set{Array{Int64,1}}([1,2,3]) # produces a Set of Arrays
Set([1,2,3]...) # => Set{Int64}(1,2,3) # this is equivalent to Set(1,2,3)

x = (1,2,3)     # => (1,2,3)
Set(x)          # => Set{(Int64,Int64,Int64)}((1,2,3)) # a Set of Tuples
Set(x...)       # => Set{Int64}(2,3,1)


# You can define functions with optional positional arguments
function defaults(a,b,x=5,y=6)
    return "$a $b and $x $y"
end

defaults('h','g') # => "h g and 5 6"
defaults('h','g','j') # => "h g and j 6"
defaults('h','g','j','k') # => "h g and j k"
try
    defaults('h') # => ERROR: no method defaults(Char,)
    defaults() # => ERROR: no methods defaults()
catch e
    println(e)
end

# You can define functions that take keyword arguments
function keyword_args(;k1=4,name2="hello") # note the ;
    return ["k1"=>k1,"name2"=>name2]
end

keyword_args(name2="ness") # => ["name2"=>"ness","k1"=>4]
keyword_args(k1="mine") # => ["k1"=>"mine","name2"=>"hello"]
keyword_args() # => ["name2"=>"hello","k1"=>4]

# You can combine all kinds of arguments in the same function
function all_the_args(normal_arg, optional_positional_arg=2; keyword_arg="foo")
    println("normal arg: $normal_arg")
    println("optional arg: $optional_positional_arg")
    println("keyword arg: $keyword_arg")
end

all_the_args(1, 3, keyword_arg=4)
# prints:
#   normal arg: 1
#   optional arg: 3
#   keyword arg: 4

# Julia has first class functions
function create_adder(x)
    adder = function (y)
        return x + y
    end
    return adder
end

# This is "stabby lambda syntax" for creating anonymous functions
(x -> x > 2)(3) # => true

# This function is identical to create_adder implementation above.
function create_adder(x)
    y -> x + y
end

# You can also name the internal function, if you want
function create_adder(x)
    function adder(y)
        x + y
    end
    adder
end

add_10 = create_adder(10)
add_10(3) # => 13


# There are built-in higher order functions
map(add_10, [1,2,3]) # => [11, 12, 13]
filter(x -> x > 5, [3, 4, 5, 6, 7]) # => [6, 7]

# We can use list comprehensions for nicer maps
[add_10(i) for i=[1, 2, 3]] # => [11, 12, 13]
[add_10(i) for i in [1, 2, 3]] # => [11, 12, 13]

####################################################
## 5. Types
####################################################

# Julia has a type system.
# Every value has a type; variables do not have types themselves.
# You can use the `typeof` function to get the type of a value.
typeof(5) # => Int64

# Types are first-class values
typeof(Int64) # => DataType
typeof(DataType) # => DataType
# DataType is the type that represents types, including itself.

# Types are used for documentation, optimizations, and dispatch.
# They are not statically checked.

# Users can define types
# They are like records or structs in other languages.
# New types are defined used the `type` keyword.

# type Name
#   field::OptionalType
#   ...
# end
type Tiger
  taillength::Float64
  coatcolor # not including a type annotation is the same as `::Any`
end

# The default constructor's arguments are the properties
# of the type, in the order they are listed in the definition
tigger = Tiger(3.5,"orange") # => Tiger(3.5,"orange")

# The type doubles as the constructor function for values of that type
sherekhan = typeof(tigger)(5.6,"fire") # => Tiger(5.6,"fire")

# These struct-style types are called concrete types
# They can be instantiated, but cannot have subtypes.
# The other kind of types is abstract types.

# abstract Name
abstract Cat # just a name and point in the type hierarchy

# Abstract types cannot be instantiated, but can have subtypes.
# For example, Number is an abstract type
subtypes(Number) # => 6-element Array{Any,1}:
                 #     Complex{Float16}
                 #     Complex{Float32}
                 #     Complex{Float64}
                 #     Complex{T<:Real}
                 #     ImaginaryUnit
                 #     Real
subtypes(Cat) # => 0-element Array{Any,1}

# Every type has a super type; use the `super` function to get it.
typeof(5) # => Int64
super(Int64) # => Signed
super(Signed) # => Real
super(Real) # => Number
super(Number) # => Any
super(super(Signed)) # => Number
super(Any) # => Any
# All of these type, except for Int64, are abstract.

# <: is the subtyping operator
type Lion <: Cat # Lion is a subtype of Cat
  mane_color
  roar::String
end

# You can define more constructors for your type
# Just define a function of the same name as the type
# and call an existing constructor to get a value of the correct type
Lion(roar::String) = Lion("green",roar)
# This is an outer constructor because it's outside the type definition

type Panther <: Cat # Panther is also a subtype of Cat
  eye_color
  Panther() = new("green")
  # Panthers will only have this constructor, and no default constructor.
end
# Using inner constructors, like Panther does, gives you control
# over how values of the type can be created.
# When possible, you should use outer constructors rather than inner ones.

####################################################
## 6. Multiple-Dispatch
####################################################

# In Julia, all named functions are generic functions
# This means that they are built up from many small methods
# Each constructor for Lion is a method of the generic function Lion.

# For a non-constructor example, let's make a function meow:

# Definitions for Lion, Panther, Tiger
function meow(animal::Lion)
  animal.roar # access type properties using dot notation
end

function meow(animal::Panther)
  "grrr"
end

function meow(animal::Tiger)
  "rawwwr"
end

# Testing the meow function
meow(tigger) # => "rawwr"
meow(Lion("brown","ROAAR")) # => "ROAAR"
meow(Panther()) # => "grrr"

# Review the local type hierarchy
issubtype(Tiger,Cat) # => false
issubtype(Lion,Cat) # => true
issubtype(Panther,Cat) # => true

# Defining a function that takes Cats
function pet_cat(cat::Cat)
  println("The cat says $(meow(cat))")
end

pet_cat(Lion("42")) # => prints "The cat says 42"
try
    pet_cat(tigger) # => ERROR: no method pet_cat(Tiger,)
catch e
    println(e)
end

# In OO languages, single dispatch is common;
# this means that the method is picked based on the type of the first argument.
# In Julia, all of the argument types contribute to selecting the best method.

# Let's define a function with more arguments, so we can see the difference
function fight(t::Tiger,c::Cat)
  println("The $(t.coatcolor) tiger wins!")
end
# => fight (generic function with 1 method)

fight(tigger,Panther()) # => prints The orange tiger wins!
fight(tigger,Lion("ROAR")) # => prints The orange tiger wins!

# Let's change the behavior when the Cat is specifically a Lion
fight(t::Tiger,l::Lion) = println("The $(l.mane_color)-maned lion wins!")
# => fight (generic function with 2 methods)

fight(tigger,Panther()) # => prints The orange tiger wins!
fight(tigger,Lion("ROAR")) # => prints The green-maned lion wins!

# We don't need a Tiger in order to fight
fight(l::Lion,c::Cat) = println("The victorious cat says $(meow(c))")
# => fight (generic function with 3 methods)

fight(Lion("balooga!"),Panther()) # => prints The victorious cat says grrr
try
  fight(Panther(),Lion("RAWR")) # => ERROR: no method fight(Panther,Lion)
catch
end

# Also let the cat go first
fight(c::Cat,l::Lion) = println("The cat beats the Lion")
# => Warning: New definition
#    fight(Cat,Lion) at none:1
# is ambiguous with
#    fight(Lion,Cat) at none:2.
# Make sure
#    fight(Lion,Lion)
# is defined first.
#fight (generic function with 4 methods)

# This warning is because it's unclear which fight will be called in:
fight(Lion("RAR"),Lion("brown","rarrr")) # => prints The victorious cat says rarrr
# The result may be different in other versions of Julia

fight(l::Lion,l2::Lion) = println("The lions come to a tie")
fight(Lion("RAR"),Lion("brown","rarrr")) # => prints The lions come to a tie


# Under the hood
# You can take a look at the llvm  and the assembly code generated.

square_area(l) = l * l      # square_area (generic function with 1 method)

square_area(5) #25

# What happens when we feed square_area an integer?
code_native(square_area, (Int32,))  
    #       .section    __TEXT,__text,regular,pure_instructions
    #   Filename: none
    #   Source line: 1              # Prologue
    #       push    RBP
    #       mov RBP, RSP
    #   Source line: 1
    #       movsxd  RAX, EDI        # Fetch l from memory?
    #       imul    RAX, RAX        # Square l and store the result in RAX
    #       pop RBP                 # Restore old base pointer
    #       ret                     # Result will still be in RAX

code_native(square_area, (Float32,))
    #       .section    __TEXT,__text,regular,pure_instructions
    #   Filename: none
    #   Source line: 1
    #       push    RBP
    #       mov RBP, RSP
    #   Source line: 1
    #       vmulss  XMM0, XMM0, XMM0  # Scalar single precision multiply (AVX)
    #       pop RBP
    #       ret

code_native(square_area, (Float64,))
    #       .section    __TEXT,__text,regular,pure_instructions
    #   Filename: none
    #   Source line: 1
    #       push    RBP
    #       mov RBP, RSP
    #   Source line: 1
    #       vmulsd  XMM0, XMM0, XMM0 # Scalar double precision multiply (AVX)
    #       pop RBP
    #       ret
    #   
# Note that julia will use floating point instructions if any of the
# arguements are floats.
# Let's calculate the area of a circle 
circle_area(r) = pi * r * r     # circle_area (generic function with 1 method)
circle_area(5)                  # 78.53981633974483

code_native(circle_area, (Int32,))
    #       .section    __TEXT,__text,regular,pure_instructions
    #   Filename: none
    #   Source line: 1
    #       push    RBP
    #       mov RBP, RSP
    #   Source line: 1
    #       vcvtsi2sd   XMM0, XMM0, EDI          # Load integer (r) from memory
    #       movabs  RAX, 4593140240              # Load pi
    #       vmulsd  XMM1, XMM0, QWORD PTR [RAX]  # pi * r
    #       vmulsd  XMM0, XMM0, XMM1             # (pi * r) * r
    #       pop RBP
    #       ret
    #

code_native(circle_area, (Float64,))
    #       .section    __TEXT,__text,regular,pure_instructions
    #   Filename: none
    #   Source line: 1
    #       push    RBP
    #       mov RBP, RSP
    #       movabs  RAX, 4593140496
    #   Source line: 1
    #       vmulsd  XMM1, XMM0, QWORD PTR [RAX]
    #       vmulsd  XMM0, XMM1, XMM0
    #       pop RBP
    #       ret
    #   

Further Reading

You can get a lot more detail from The Julia Manual