Julia: calculate an inner product using boolean algebra - julia

I've got two boolean vectors a = [1,1,1] and b = [0,1,1] where obviously 1 stands for true and 0 for false.
I want to calculate their inner product using Boolean algebra. The result I want is therefore
1*0 + 1*1 + 1*1 = 0 + 1 + 1 = 0
because addition plays the role of exclusive or (XOR).
I know that the product part can be done like this
a = [true, true, true] # could also use ones(Bool, 3)
b = [false, true, true]
bitwise_prod = a .& b
but I don't know how to do the sum. Any ideas?

I've actually found a good way now. First of all I don't need to use Boolean variables
a = [1, 1, 1] # or ones(Int, 3)
b = [0, 1, 1]
I can then use reduce with the xor function.
reduce(xor, a .& b)
Note that I tried using the bitwise xor operator $ that is found in the documentation (inside an anonymous function) but this operator is deprecated and Julia 0.6.2 suggests the xor function instead. I do think having the function name makes it very neat.

Related

Abstract typing and multiple dispatch for functions in julia

I want to have objects interact with specific interactions depending on their type.
Example problem: I have four particles, two are type A, and 2 are type B. when type A's interact I want to use the function
function interaction(parm1, parm2)
return parm1 + parm2
end
when type B's interact I want to use the function
function interaction(parm1, parm2)
return parm1 * parm2
end
when type A interacts with type B I want to use function
function interaction(parm1, parm2)
return parm1 - parm2
end
These functions are purposefully over simple.
I want to calculate a simple summation that depends on pairwise interactions:
struct part
parm::Float64
end
# part I need help with:
# initialize a list of length 4, where the entries are `struct part`, and the abstract types
# are `typeA` for the first two and `typeB` for the second two. The values for the parm can be
# -1.0,3, 4, 1.5 respectively
energy = 0.0
for i in range(length(particles)-1)
for j = i+1:length(particles)
energy += interaction(particles[i].parm, particles[j].parm)
end
end
println(energy)
assuming the use of parameters being particle[1].parm = -1, particle[2].parm = 3, particle[3].parm = 4, particle[4].parm = 1.5, energy should account for the interactions of
(1,2) = -1 + 3 = 2
(1,3) = -1 - 4 = -5
(1,4) = -1 - 1.5 = -2.5
(2,3) = 3 - 4 = -1
(2,4) = 3 - 1.5 = 1.5
(3,4) = 4 * 1.5 = 6
energy = 1
Doing this with if statements is almost trivial but not extensible. I am after a clean, tidy Julia approach...
You can do this (I use the simplest form of the implementation as in this case it is enough and it is explicit what happens I hope):
struct A
parm::Float64
end
struct B
parm::Float64
end
interaction(p1::A, p2::A) = p1.parm + p2.parm
interaction(p1::B, p2::B) = p1.parm * p2.parm
interaction(p1::A, p2::B) = p1.parm - p2.parm
interaction(p1::B, p2::A) = p1.parm - p2.parm # I added this rule, but you can leave it out and get MethodError if such case happens
function total_energy(particles)
energy = 0.0
for i in 1:length(particles)-1
for j = i+1:length(particles)
energy += interaction(particles[i], particles[j])
end
end
return energy
end
particles = Union{A, B}[A(-1), A(3), B(4), B(1.5)] # Union makes sure things are compiled to be fast
total_energy(particles)
I have no idea how to do this in your language, but what you need is an analogue to what we call the strategy pattern in object-oriented programming. A strategy is a pluggable, reusable algorithm. In Java I’d make an interface like:
interface Interaction<A, B>
{
double interact(A a, B b)
}
Then implement this three times and reuse those parts wherever you need things to interact. Another method can take an Interaction and use it without knowing how it’s implemented. I think this is the effect you’re after. Sorry I don’t know how to translate into your dialect.

Get a number from an array of digits

To split a number into digits in a given base, Julia has the digits() function:
julia> digits(36, base = 4)
3-element Array{Int64,1}:
0
1
2
What's the reverse operation? If you have an array of digits and the base, is there a built-in way to convert that to a number? I could print the array to a string and use parse(), but that sounds inefficient, and also wouldn't work for bases > 10.
The previous answers are correct, but there is also the matter of efficiency:
sum([x[k]*base^(k-1) for k=1:length(x)])
collects the numbers into an array before summing, which causes unnecessary allocations. Skip the brackets to get better performance:
sum(x[k]*base^(k-1) for k in 1:length(x))
This also allocates an array before summing: sum(d.*4 .^(0:(length(d)-1)))
If you really want good performance, though, write a loop and avoid repeated exponentiation:
function undigit(d; base=10)
s = zero(eltype(d))
mult = one(eltype(d))
for val in d
s += val * mult
mult *= base
end
return s
end
This has one extra unnecessary multiplication, you could try to figure out some way of skipping that. But the performance is 10-15x better than the other approaches in my tests, and has zero allocations.
Edit: There's actually a slight risk to the type handling above. If the input vector and base have different integer types, you can get a type instability. This code should behave better:
function undigits(d; base=10)
(s, b) = promote(zero(eltype(d)), base)
mult = one(s)
for val in d
s += val * mult
mult *= b
end
return s
end
The answer seems to be written directly within the documentation of digits:
help?> digits
search: digits digits! ndigits isdigit isxdigit disable_sigint
digits([T<:Integer], n::Integer; base::T = 10, pad::Integer = 1)
Return an array with element type T (default Int) of the digits of n in the given base,
optionally padded with zeros to a specified size. More significant digits are at higher
indices, such that n == sum([digits[k]*base^(k-1) for k=1:length(digits)]).
So for your case this will work:
julia> d = digits(36, base = 4);
julia> sum([d[k]*4^(k-1) for k=1:length(d)])
36
And the above code can be shortened with the dot operator:
julia> sum(d.*4 .^(0:(length(d)-1)))
36
Using foldr and muladd for maximum conciseness and efficiency
undigits(d; base = 10) = foldr((a, b) -> muladd(base, b, a), d, init=0)

Refer to struct fields without dot notation (in Julia)

In Julia I've defined a type and I need to write some functions that work with the fields of that type. Some of the functions contain complicated formulas and it gets messy to use the field access dot notation all over the place. So I end up putting the field values into local variables to improve readability. It works fine, but is there some clever way to avoid having to type out all the a=foo.a lines or to have Julia parse a as foo.a etc?
struct Foo
a::Real
b::Real
c::Real
end
# this gets hard to read
function bar(foo::Foo)
foo.a + foo.b + foo.c + foo.a*foo.b - foo.b*foo.c
end
# this is better
function bar(foo::Foo)
a = foo.a
b = foo.b
c = foo.c
a + b + c + a*b - b*c
end
# this would be great
function bar(foo::Foo)
something clever
a + b + c + a*b - b*c
end
Because Julia generally encourages the use of generalized interfaces to interact with fields rather than accessing the fields directly, a fairly natural way of accomplishing this would be unpacking via iteration. In Julia, objects can be "unpacked" into multiple variables by iteration:
julia> x, y = [1, 2, 3]
3-element Array{Int64,1}:
1
2
3
julia> x
1
julia> y
2
We can implement such an iteration protocol for a custom object, like Foo. In v0.7, this would look like:
Base.iterate(foo::Foo, state = 1) = state > 3 ? nothing : (getfield(foo, state), state + 1)
Note that 3 is hardcoded (based on the number of fields in Foo) and could be replaced with fieldcount(Foo). Now, you can simply "unpack" an instance of Foo as follows:
julia> a, b, c = Foo("one", 2.0, 3)
Foo("one", 2.0, 3)
julia> a
"one"
julia> b
2.0
julia> c
3
This could be the "something clever" at the beginning of your function. Additionally, as of v0.7, you can unpack the fields in the function argument itself:
function bar((a, b, c)::Foo)
a + b + c + a*b - b*c
end
Although this does require that you mention the field names again, it comes with two potential advantages:
In the case that your struct is refactored and the fields are renamed, all code accessing the fields will remain intact (as long as the field order doesn't change or the iterate implementation is changed to reflect the new object internals).
Longer field names can be abbreviated. (i.e. rather than using the full apples field name, you can opt to use a.)
If it's important that the field names not be repeated, you could define a macro to generate the required variables (a = foo.a; b = foo.b; c = foo.c); however, this would likely be more confusing for the readers of your code and lack the advantages listed above.
As of Julia 1.6, the macros in this package look relevant: https://github.com/mauro3/UnPack.jl.
The syntax would look like:
function bar(foo::Foo)
# something clever!
#unpack a, b, c = f
a + b + c + a*b - b*c
end
In Julia 1.7, it looks like this feature will be added with the syntax
function bar(foo::Foo)
# something clever!
(; a, b, c) = f
a + b + c + a*b - b*c
end
Here is the merged pull request: https://github.com/JuliaLang/julia/pull/39285

How do I refactor this function in ELM?

I am trying to pick up functional programming and decided to start with Problem 1 on Project Euler: basically add all numbers less than 1000 divisible by 3 or 5 (link: a link).
This is the code that I have written. It outputs a list of factors of 3 or 5 (still need to figure out how to sum).
import Html exposing (text)
import Array
main =
text (
toString
[findSum_maxZ 3 5 1000]
)
findSum_maxZ x y max_z =
Array.filter isDivisible_x_or_y (Array.initialize max_z identity)
isDivisible_x_or_y x =
if x % 3 == 0 || x % 5 == 0 then True else False
My issue is that I reference 3 and 5 twice but I cannot call isDivisible with the additional parameters of the more abstract 'x' and'y'. My goal is to determine effective methods of removing these artificially mutable values so the end user only has to modify each input value once. Any advice?
I apologize if this question is dumb, there is not a lot of information on ELM available (especially compared to python, c, c++, java, etc which I have used) and I am still not fully comfortable with the functional programming jargon. Any and all help is appreciated.
The cool thing about ML languages is that you are pretty much free to build your own "dialect" to solve problems.
You can use currying to apply just the x and y arguments to your function, creating a new function where the supplied values are already set.
import Html exposing (text)
import Array
main = [findSum 3 5 1000]
|>toString
|>text
findSum x y maxZ =
let
isDivisibleByX = isDivisible x
isDivisibleByY = isDivisible y
in
Array.initialize maxZ identity
|>Array.filter isDivisibleByX
|>Array.filter isDivisibleByY
--as you can see, it is possible to use a list instead of creating
--new functions, it is up to you to check which abstraction works
--the best
isDivisible a b =
b % a == 0
You can also work with a single function, without resorting to currying:
import Html exposing (text)
import Array
main = [findSum 3 5 1000]
|>toString
|>text
findSum x y maxZ =
Array.initialize maxZ identity
|>Array.filter (\n-> isDivisible x n ) --or just (isDivisible x)
|>Array.filter (\n-> isDivisible y n)
isDivisible a b =
b % a == 0
If you want to filter the array with just one line, you can do this:
import Html exposing (text)
main = findSum 3 5 1000
|>toString
|>text
findSum x y maxZ =
let
divisibles = \n-> isDivisible x n && isDivisible y n
in
List.range 0 maxZ
|>List.filter divisibles
isDivisible a b =
b % a == 0
The most direct answer to your question is that you can have isDivisible_x_or_y take the two factors, and then use currying to pass the partially applied function to Array.filter.
That is, you can define isDivisible_x_or_y like this (I also removed the if True then True else False syntax and just return the expression directly):
isDivisible_x_or_y x y val =
val % x == 0 || val % y == 0
Currying is the ability to only supply some of the parameters to a function, and get back a function that takes the rest of the parameters. So, the type definition of isDivisible_x_or_y is Int -> Int -> Int -> Bool (that is, it takes in three Int values and returns a Bool). If we supply values for the x and y arguments (e.g. isDivisible_x_y 3 5), we now get a function with the type definition of Int -> Bool. This is the type expected by Array.filter.
You can see a working example at https://ellie-app.com/sdxWFL9ynka1
Another couple of notes:
List is much more common than Array in Elm. You would only use Array if you need to get items at specific indexes. Instead of Array.initialize, you can use List.range
Using the pipeline operator |> can often make your code a lot simpler to read. Instead of text (toString (getValue)), you have getValue |> toString |> text, which is now in the order that the operations occur, and doesn't have extra parenthesis. This whole program could be one simple pipeline (in a lot of scenarios putting everything into one pipeline can be excessive, though):
main =
List.range 0 max_z
|> List.filter (isDivisible_x_or_y 3 5)
|> toString
|> text
isDivisible_x_or_y x y val =
val % x == 0 || val % y == 0

List comprehensions and tuples in Julia

I am trying to do in Julia what this Python code does. (Find all pairs from the two lists whose combined value is above 7.)
#Python
def sum_is_large(a, b):
return a + b > 7
l1 = [1,2,3]
l2 = [4,5,6]
l3 = [(a,b) for a in l1 for b in l2 if sum_is_large(a, b)]
print(l3)
There is no if for list comprehensions in Julia. And if I use filter(), I'm not sure if I can pass two arguments. So my best suggestion is this:
#Julia
function sum_is_large(pair)
a, b = pair
return a + b > 7
end
l1 = [1,2,3]
l2 = [4,5,6]
l3 = filter(sum_is_large, [(i,j) for i in l1, j in l2])
print(l3)
I don't find this very appealing. So my question is, is there a better way in Julia?
Using the very popular package Iterators.jl, in Julia:
using Iterators # install using Pkg.add("Iterators")
filter(x->sum(x)>7,product(l1,l2))
is an iterator producing the pairs. So to get the same printout as the OP:
l3iter = filter(x->sum(x)>7,product(l1,l2))
for p in l3iter println(p); end
The iterator approach is potentially much more memory efficient. Ofcourse, one could just l3 = collect(l3iter) to get the pair vector.
#user2317519, just curious, is there an equivalent iterator form for python?
Guards (if) are now available in Julia v0.5 (currently in the release-candidate stage):
julia> v1 = [1, 2, 3];
julia> v2 = [4, 5, 6];
julia> v3 = [(a, b) for a in v1, b in v2 if a+b > 7]
3-element Array{Tuple{Int64,Int64},1}:
(3,5)
(2,6)
(3,6)
Note that generators are also now available:
julia> g = ( (a, b) for a in v1, b in v2 if a+b > 7 )
Base.Generator{Filter{##18#20,Base.Prod2{Array{Int64,1},Array{Int64,1}}},##17#19}(#17,Filter{##18#20,Base.Prod2{Array{Int64,1},Array{Int64,1}}}(#18,Base.Prod2{Array{Int64,1},Array{Int64,1}}([1,2,3],[4,5,6])))
Another option similar to the one of #DanGetz using also Iterators.jl:
function expensive_fun(a, b)
return (a + b)
end
Then, if the condition is also complicated, it can be defined as a function:
condition(x) = x > 7
And last, filter the results:
>>> using Iterators
>>> result = filter(condition, imap(expensive_fun, l1, l2))
result is an iterable that is only computed when needed (inexpensive) and can be collected collect(result) if required.
The one-line if the filter condition is simple enough would be:
>>> result = filter(x->(x > 7), imap(expensive_fun, l1, l2))
Note: imap works natively for arbitrary number of parameters.
Perhaps something like this:
julia> filter(pair -> pair[1] + pair[2] > 7, [(i, j) for i in l1, j in l2])
3-element Array{Tuple{Any,Any},1}:
(3,5)
(2,6)
(3,6)
although I'd agree it doesn't look like it ought to be the best way...
I'm surprised nobody mentions the ternary operator to implement the conditional:
julia> l3 = [sum_is_large((i,j)) ? (i,j) : nothing for i in l1, j in l2]
3x3 Array{Tuple,2}:
nothing nothing nothing
nothing nothing (2,6)
nothing (3,5) (3,6)
or even just a normal if block within a compound statement, i.e.
[ (if sum_is_large((x,y)); (x,y); end) for x in l1, y in l2 ]
which gives the same result.
I feel this result makes a lot more sense than filter(), because in julia the a in A, b in B construct is interpreted dimensionally, and therefore the output is in fact an "array comprehension" with appropriate dimensionality, which clearly in many cases would be advantageous and presumably the desired behaviour (whether we include a conditional or not).
Whereas filter will always return a vector. Obviously, if you really want a vector result you can always collect the result; or for a conditional list comprehension like the one here, you can simply remove nothing elements from the array by doing l3 = l3[l3 .!= nothing].
Presumably this is still clearer and no less efficient than the filter() approach.
You can use the #vcomp (vector comprehension) macro in VectorizedRoutines.jl to do Python-like comprehensions:
using VectorizedRoutines
Python.#vcomp Int[i^2 for i in 1:10] when i % 2 == 0 # Int[4, 16, 36, 64, 100]

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