Perhaps is in older versions of julia, mine is Version 0.5.0-dev+2007 2016-01-05 18:26 UTC), but if you define:
julia> [1 +2,1]
ERROR: syntax: unexpected comma in matrix expression
in eval at ./boot.jl:265
Yo have to either define [1+2,1] or [1 + 2, 1]. Is this on purpose?
This is because Julia is parsing the +2 as the number "positive 2" and not treating the + as the addition operator. Since vectors of the form [a b,c] are invalid, you get the error.
Related
I typed the following in Julia's REPL:
julia> 6÷2(1+2)
1
julia> 6÷2*(1+2)
9
Why are the different results output?
Presh Talwalkar says 9 is correct in the movie
6÷2(1+2) = ? Mathematician Explains The Correct Answer - YouTube
YouTube notwithstanding, there is no correct answer. Which answer you get depends on what precedence convention you use to interpret the problem. Many of these viral "riddles" that go around periodically are contentious precisely because they are intentionally ambiguous. Not a math puzzle really, it's just a parsing problem. It's no deeper than someone saying a sentence with two interpretations. What do you do in that case in real life? You just ask which one they meant. This is no different. For this very reason, the ÷ symbol isn't often used in real mathematical notation—fraction notation is used instead, which clearly disambiguates this as either:
6
- (1 + 2) = 9
2
or as
6
--------- = 1
2 (1 + 2)
Regarding Julia specifically, this precedence behavior is documented here:
https://docs.julialang.org/en/v1/manual/integers-and-floating-point-numbers/#man-numeric-literal-coefficients
Specifically:
The precedence of numeric literal coefficients is slightly lower than that of unary operators such as negation. So -2x is parsed as (-2) * x and √2x is parsed as (√2) * x. However, numeric literal coefficients parse similarly to unary operators when combined with exponentiation. For example 2^3x is parsed as 2^(3x), and 2x^3 is parsed as 2*(x^3).
and the note:
The precedence of numeric literal coefficients used for implicit multiplication is higher than other binary operators such as multiplication (*), and division (/, \, and //). This means, for example, that 1 / 2im equals -0.5im and 6 // 2(2 + 1) equals 1 // 1.
I saw in someone code that they were using the + operator as if it was a function by doing +(1,2,3). Is it possible to use operators as functions in Julia?
In addition, I also saw things like A ⊗ B, where the behaviour of ⊗ was customizable. How can I do such a thing, and is there a list of symbols I can use in this way?
Yes, you indeed can use operators as functions in Julia.
From the Julia Docs:
In Julia, most operators are just functions with support for special syntax. (The exceptions are operators with special evaluation semantics like && and ||. These operators cannot be functions since Short-Circuit Evaluation requires that their operands are not evaluated before evaluation of the operator.) Accordingly, you can also apply them using parenthesized argument lists, just as you would any other function:
julia> 1 + 3 + 5
9
julia> +(1,3,5)
9
julia> 1 * 3 * 5
15
julia> *(1,3,5)
15
julia> h = *;
julia> h(1,3,5)
15
In addition, Julia allows you to define your own meaning to operators, and makes quite a few symbols available for the purpose. You can find the list of available symbols here:|
https://github.com/JuliaLang/julia/blob/master/src/julia-parser.scm
and define them as such:
⊗(a, b) = a * 3 - b # or some other arbitrary thing
a ⊗ b == a * 3 - b # true
When would I use vector-of to create a vector, instead of the vector function. Is the guideline to use vector most of the time and only for performance reason switch to vector-of?
I could not find good info on when to use vector-of.
vector-of is used for creating a vector of a single primitive type, :int, :long, :float, :double, :byte, :short, :char, or :boolean. It doesn't allow other types as it stores the values unboxed internally. So, if your vector need to include other types than those primitive types, you cannot use vector-of. But if you are sure that the vector will have data of a single primitive type, you can use vector-of for better performance.
user=> (vector-of :int 1 2 3 4 5)
[1 2 3 4 5]
user=> (vector-of :double 1.0 2.0)
[1.0 2.0]
user=> (vector-of :string "hello" "world")
Execution error (IllegalArgumentException) at user/eval5 (REPL:1).
Unrecognized type :string
As you can see, you should specify primitive type as an argument.
vector can be used to create a vector of any type.
user=> (vector 1 2.0 "hello")
[1 2.0 "hello"]
You can put any type when you use vector.
Also, there's another function vec, which is used for creating a new vector containing the contents of coll.
user=> (vec '(1 2 3 4 5))
[1 2 3 4 5]
Usually, you can get the basic information of a function/macro from the repl, like the following.
user=> (doc vector-of)
-------------------------
clojure.core/vector-of
([t] [t & elements])
Creates a new vector of a single primitive type t, where t is one
of :int :long :float :double :byte :short :char or :boolean. The
resulting vector complies with the interface of vectors in general,
but stores the values unboxed internally.
Optionally takes one or more elements to populate the vector.
Reference:
https://clojuredocs.org/clojure.core/vector-of
https://clojuredocs.org/clojure.core/vector
https://clojuredocs.org/clojure.core/vec
Nobody really ever uses vector-of. If you don't super care about performance, vector is fine, and if you do super care about performance you usually want a primitive array or some other java type. Honestly I would expect occasional weird snags when passing a vector-of to anything that expects an ordinary vector or sequence - maybe it works fine, but it's just such a rare thing to see that it wouldn't surprise me if it caused issues.
I was delighted to learn that Julia allows a beautifully succinct way to form inner products:
julia> x = [1;0]; y = [0;1];
julia> x'y
1-element Array{Int64,1}:
0
This alternative to dot(x,y) is nice, but it can lead to surprises:
julia> #printf "Inner product = %f\n" x'y
Inner product = ERROR: type: non-boolean (Array{Bool,1}) used in boolean context
julia> #printf "Inner product = %f\n" dot(x,y)
Inner product = 0.000000
So while i'd like to write x'y, it seems best to avoid it, since otherwise I need to be conscious of pitfalls related to scalars versus 1-by-1 matrices.
But I'm new to Julia, and probably I'm not thinking in the right way. Do others use this succinct alternative to dot, and if so, when is it safe to do so?
There is a conceptual problem here. When you do
julia> x = [1;0]; y = [0;1];
julia> x'y
0
That is actually turned into a matrix * vector product with dimensions of 2x1 and 1 respectively, resulting in a 1x1 matrix. Other languages, such as MATLAB, don't distinguish between a 1x1 matrix and a scalar quantity, but Julia does for a variety of reasons. It is thus never safe to use it as alternative to the "true" inner product function dot, which is defined to return a scalar output.
Now, if you aren't a fan of the dots, you can consider sum(x.*y) of sum(x'y). Also keep in mind that column and row vectors are different: in fact, there is no such thing as a row vector in Julia, more that there is a 1xN matrix. So you get things like
julia> x = [ 1 2 3 ]
1x3 Array{Int64,2}:
1 2 3
julia> y = [ 3 2 1]
1x3 Array{Int64,2}:
3 2 1
julia> dot(x,y)
ERROR: `dot` has no method matching dot(::Array{Int64,2}, ::Array{Int64,2})
You might have used a 2d row vector where a 1d column vector was required.
Note the difference between 1d column vector [1,2,3] and 2d row vector [1 2 3].
You can convert to a column vector with the vec() function.
The error message suggestion is dot(vec(x),vec(y), but sum(x.*y) also works in this case and is shorter.
julia> sum(x.*y)
10
julia> dot(vec(x),vec(y))
10
Now, you can write x⋅y instead of dot(x,y).
To write the ⋅ symbol, type \cdot followed by the TAB key.
If the first argument is complex, it is conjugated.
Now, dot() and ⋅ also work for matrices.
Since version 1.0, you need
using LinearAlgebra
before you use the dot product function or operator.
There are some special operators in Prolog, one of them is is, however, recently I came across the =:= operator and have no idea how it works.
Can someone explain what this operator does, and also where can I find a predefined list of such special operators and what they do?
I think the above answer deserves a few words of explanation here nevertheless.
A short note in advance: Arithmetic expressions in Prolog are just terms ("Everything is a term in Prolog"), which are not evaluated automatically. (If you have a Lisp background, think of quoted lists). So 3 + 4 is just the same as +(3,4), which does nothing on its own. It is the responsibility of individual predicates to evaluate those terms.
Several built-in predicates do implicit evaluation, among them the arithmetic comparsion operators like =:= and is. While =:= evaluates both arguments and compares the result, is accepts and evaluates only its right argument as an arithmetic expression.
The left argument has to be an atom, either a numeric constant (which is then compared to the result of the evaluation of the right operand), or a variable. If it is a bound variable, its value has to be numeric and is compared to the right operand as in the former case. If it is an unbound variable, the result of the evaluation of the right operand is bound to that variable. is is often used in this latter case, to bind variables.
To pick up on an example from the above linked Prolog Dictionary: To test if a number N is even, you could use both operators:
0 is N mod 2 % true if N is even
0 =:= N mod 2 % dito
But if you want to capture the result of the operation you can only use the first variant. If X is unbound, then:
X is N mod 2 % X will be 0 if N is even
X =:= N mod 2 % !will bomb with argument/instantiation error!
Rule of thumb: If you just need arithmetic comparison, use =:=. If you want to capture the result of an evaluation, use is.
?- 2+3 =:= 6-1.
true.
?- 2+3 is 6-1.
false.
Also please see docs http://www.swi-prolog.org/pldoc/man?predicate=is/2
Complementing the existing answers, I would like to state a few additional points:
An operator is an operator
First of all, the operator =:= is, as the name indicates, an operator. In Prolog, we can use the predicate current_op/3 to learn more about operators. For example:
?- current_op(Prec, Type, =:=).
Prec = 700,
Type = xfx.
This means that the operator =:= has precedence 700 and is of type xfx. This means that it is a binary infix operator.
This means that you can, if you want, write a term like =:=(X, Y) equivalently as X =:= Y. In both cases, the functor of the term is =:=, and the arity of the term is 2. You can use write_canonical/1 to verify this:
?- write_canonical(a =:= b).
=:=(a,b)
A predicate is not an operator
So far, so good! This has all been a purely syntactical feature. However, what you are actually asking about is the predicate (=:=)/2, whose name is =:= and which takes 2 arguments.
As others have already explained, the predicate (=:=)/2 denotes arithmetic equality of two arithmetic expressions. It is true iff its arguments evaluate to the same number.
For example, let us try the most general query, by which we ask for any solution whatsoever, using variables as arguments:
?- X =:= Y.
ERROR: Arguments are not sufficiently instantiated
Hence, this predicate is not a true relation, since we cannot use it for generating results! This is a quite severe drawback of this predicate, clashing with what you commonly call "declarative programming".
The predicate only works in the very specific situation that both arguments are fully instantiated. For example:
?- 1 + 2 =:= 3.
true.
We call such predicates moded because they can only be used in particular modes of usage. For the vast majority of beginners, moded predicates are a nightmare to use, because they require you to think about your programs procedurally, which is quite hard at first and remains hard also later. Also, moded predicates severely limit the generality of your programs, because you cannot use them on all directions in which you could use pure predicates.
Constraints are a more general alternative
Prolog also provides much more general arithmetic predicates in the form of arithmetic constraints.
For example, in the case of integers, try your Prolog system's CLP(FD) constraints. One of the most important CLP(FD) constraints denotes arithmetic equality and is called (#=)/2. In complete analogy to (=:=)/2, the operator (#=)/2 is also defined as an infix operator, and so you can write for example:
| ?- 1 + 2 #= 3.
yes
I am using GNU Prolog as one particular example, and many other Prolog systems also provide CLP(FD) implementations.
A major attraction of constraints is found in their generality. For example, in contrast to (=:=)/2, we get with the predicate (#=)/2:
| ?- X + 2 #= 3.
X = 1
| ?- 1 + Y #= 3.
Y = 2
And we can even ask the most general query:
| ?- X #= Y.
X = _#0(0..268435455)
Y = _#0(0..268435455)
Note how naturally these predicates blend into Prolog and act as relations between integer expressions that can be queried in all directions.
Depending on the domain of interest, my recommendition is to use CLP(FD), CLP(Q), CLP(B) etc. instead of using more low-level arithmetic predicates.
Also see clpfd, clpq and clpb for more information.
Coincidentally, the operator =:= is used by CLP(B) with a completely different meaning:
?- sat(A =:= B+1).
A = 1,
sat(B=:=B).
This shows that you must distinguish between operators and predicates. In the above case, the predicate sat/1 has interpreted the given expression as a propositional formula, and in this context, =:= denotes equality of Boolean expressions.
I found my own answer, http://www.cse.unsw.edu.au/~billw/prologdict.html
Its an ISO core standard predicate operator, which cannot be bootstrapped from unification (=)/2 or syntactic equality (==)/2. It is defined in section 8.7 Arithmetic Comparison. And it basically behaves as follows:
E =:= F :-
X is E,
Y is F,
arithmetic_compare(=, X, Y).
So both the left hand side (LHS) and right hand side (RHS) must be arithmetic expressions that are evaluted before they are compared. Arithmetic comparison can compare across numeric types. So we have:
GNU Prolog 1.4.5 (64 bits)
?- 0 = 0.0.
no
?- 0 == 0.0
no
?- 0 =:= 0.0.
yes
From Erlang I think it could be good to annotate that as syntax are mostly look alike to Prolog.
=:= expression is meaning of exactly equal.
such as in JavaScript you can use === to also see if the type of the variables are same.
Basically it's same logic but =:= is used in functional languages as Prolog, Erlang.
Not much information but hope it could help in some way.
=:= is a comparison operator.A1 =:= A2 succeeds if values of expressions A1 and A2 are equal.
A1 == A2 succeeds if terms A1 and A2 are identical;