I know that we can find the inverse of a function using solve e.g.
f(x):= a*x+b$
[invf]:solve(f=f(x),x);
However, I haven't been able to work out how to use this in the definition of a new function. I tried a few things, like:
invF(f):=rhs(invf);
but the f in rhs(invf) isn't recognized as the argument of the function invF(f). Is there any way to define an inverse, or would one simply work with this expression and use something like subst to evaluate it?
The right-hand side of := isn't evaluated (or even simplified), so anything like rhs(invf) in the function body doesn't actually compute the result until the function is called. What's needed here is to ensure that the result is pasted into the function body at the time the function is constructed.
There are at least two ways to do that. The more general way is to define the function with define instead of :=. Then the body is evaluated when the function is defined. E.g. define(invF(x), rhs(inv));.
Another way which only works at the top level (i.e., not inside any function) is to make use of the so-called quote-quote operator, '', which has the effect of interpolating the current value of something into the input, as if it were typed in. E.g. invF(x) := ''(rhs(inv));.
Quote-quote acts at the time the expression is parsed so that can be surprising if it appears in a function. E.g. y: 123; f(x) := block([y: 2*x], ''y); then f(4) returns 123, not 8. That said, I have found quote-quote to be convenient and helpful -- just have to confine it to top-level stuff.
The following modification of my earlier attempt seems to work:
f(x):=a*x+b;
[inv]:solve(f=f(x),x);
invF(x):=subst(x,f,rhs(inv));
Not sure why the use of subst changes the way the function definition is handled, but it does seem to work. Hope it might be helpful to others!
Related
To be to the point; I've done Lua for awhile, but never quite got the terminology down to specifics, so I've been Googling for hours and haven't come up with a definitive answer.
Related to OOP in Lua, the terminology used include:
Object
Class
Function
Method
Table
The question is, when are these properly used? Such as in the example below:
addon = { }
function addon:test_func( )
return 'hi'
end
Q: From my understanding with Lua and OOP, addon is a table, however, I've read that it can be an object as well -- but when it is technically an object? After a function is created within that table?
Q: test_func is a function, however, I've read that it becomes a "Method" when it's placed within a table (class).
Q: The entire line addon:test_func( ), I know the colon is an operator, but what is the term for the entire line set of text? A class itself?
Finally, for this example code:
function addon:test_func( id, name )
end
Q: What is id and name, because I've seen some people identify them as arguments, but then other areas classify them as parameters, so I've stuck with parameters.
So in short, what is the proper terminology for each of these, and when do they become what they are?
Thanks
From my understanding with Lua and OOP, addon is a table, however, I've read that it can be an object as well -- but when it is technically an object? After a function is created within that table?
Object is not a well-defined term. I've seen it defined (in C) as any value whatsoever. In Lua, I would consider it synonymous with a table. You could also define it as an instance of a class.
test_func is a function, however, I've read that it becomes a "Method" when it's placed within a table (class).
You're basically right. A method is any function that is intended to be called with the colon notation. Metamethods are also methods, because, like regular methods, they define the behavior of tables.
The entire line addon:test_func( ), I know the colon is an operator, but what is the term for the entire line set of text? A class itself?
There's no name for that particular piece of code. It's just part of a method definition.
Also, I wouldn't call the colon an operator. An operator would be the plus in x + y where x and y both mean something by themselves. In addon:test_func(), test_func only has meaning inside the table addon, and it's only valid to use the colon when calling or defining methods. The colon is actually a form of syntactic sugar where the real operator is the indexing operator: []. Assuming that you're calling the method, the expansion would be: addon['test_func'](addon).
What is id and name, because I've seen some people identify them as arguments, but then other areas classify them as parameters, so I've stuck with parameters.
They're parameters. Parameters are the names that you declare in the function signature. Arguments are the values that you pass to a function.
I'm learning Elixir and wonder why it has two types of function definitions:
functions defined in a module with def, called using myfunction(param1, param2)
anonymous functions defined with fn, called using myfn.(param1, param2)
Only the second kind of function seems to be a first-class object and can be passed as a parameter to other functions. A function defined in a module needs to be wrapped in a fn. There's some syntactic sugar which looks like otherfunction(&myfunction(&1, &2)) in order to make that easy, but why is it necessary in the first place? Why can't we just do otherfunction(myfunction))? Is it only to allow calling module functions without parenthesis like in Ruby? It seems to have inherited this characteristic from Erlang which also has module functions and funs, so does it actually comes from how the Erlang VM works internally?
It there any benefit having two types of functions and converting from one type to another in order to pass them to other functions? Is there a benefit having two different notations to call functions?
Just to clarify the naming, they are both functions. One is a named function and the other is an anonymous one. But you are right, they work somewhat differently and I am going to illustrate why they work like that.
Let's start with the second, fn. fn is a closure, similar to a lambda in Ruby. We can create it as follows:
x = 1
fun = fn y -> x + y end
fun.(2) #=> 3
A function can have multiple clauses too:
x = 1
fun = fn
y when y < 0 -> x - y
y -> x + y
end
fun.(2) #=> 3
fun.(-2) #=> 3
Now, let's try something different. Let's try to define different clauses expecting a different number of arguments:
fn
x, y -> x + y
x -> x
end
** (SyntaxError) cannot mix clauses with different arities in function definition
Oh no! We get an error! We cannot mix clauses that expect a different number of arguments. A function always has a fixed arity.
Now, let's talk about the named functions:
def hello(x, y) do
x + y
end
As expected, they have a name and they can also receive some arguments. However, they are not closures:
x = 1
def hello(y) do
x + y
end
This code will fail to compile because every time you see a def, you get an empty variable scope. That is an important difference between them. I particularly like the fact that each named function starts with a clean slate and you don't get the variables of different scopes all mixed up together. You have a clear boundary.
We could retrieve the named hello function above as an anonymous function. You mentioned it yourself:
other_function(&hello(&1))
And then you asked, why I cannot simply pass it as hello as in other languages? That's because functions in Elixir are identified by name and arity. So a function that expects two arguments is a different function than one that expects three, even if they had the same name. So if we simply passed hello, we would have no idea which hello you actually meant. The one with two, three or four arguments? This is exactly the same reason why we can't create an anonymous function with clauses with different arities.
Since Elixir v0.10.1, we have a syntax to capture named functions:
&hello/1
That will capture the local named function hello with arity 1. Throughout the language and its documentation, it is very common to identify functions in this hello/1 syntax.
This is also why Elixir uses a dot for calling anonymous functions. Since you can't simply pass hello around as a function, instead you need to explicitly capture it, there is a natural distinction between named and anonymous functions and a distinct syntax for calling each makes everything a bit more explicit (Lispers would be familiar with this due to the Lisp 1 vs. Lisp 2 discussion).
Overall, those are the reasons why we have two functions and why they behave differently.
I don't know how useful this will be to anyone else, but the way I finally wrapped my head around the concept was to realize that elixir functions aren't Functions.
Everything in elixir is an expression. So
MyModule.my_function(foo)
is not a function but the expression returned by executing the code in my_function. There is actually only one way to get a "Function" that you can pass around as an argument and that is to use the anonymous function notation.
It is tempting to refer to the fn or & notation as a function pointer, but it is actually much more. It's a closure of the surrounding environment.
If you ask yourself:
Do I need an execution environment or a data value in this spot?
And if you need execution use fn, then most of the difficulties become much
clearer.
I may be wrong since nobody mentioned it, but I was also under the impression that the reason for this is also the ruby heritage of being able to call functions without brackets.
Arity is obviously involved but lets put it aside for a while and use functions without arguments. In a language like javascript where brackets are mandatory, it is easy to make the difference between passing a function as an argument and calling the function. You call it only when you use the brackets.
my_function // argument
(function() {}) // argument
my_function() // function is called
(function() {})() // function is called
As you can see, naming it or not does not make a big difference. But elixir and ruby allow you to call functions without the brackets. This is a design choice which I personally like but it has this side effect you cannot use just the name without the brackets because it could mean you want to call the function. This is what the & is for. If you leave arity appart for a second, prepending your function name with & means that you explicitly want to use this function as an argument, not what this function returns.
Now the anonymous function is bit different in that it is mainly used as an argument. Again this is a design choice but the rational behind it is that it is mainly used by iterators kind of functions which take functions as arguments. So obviously you don't need to use & because they are already considered arguments by default. It is their purpose.
Now the last problem is that sometimes you have to call them in your code, because they are not always used with an iterator kind of function, or you might be coding an iterator yourself. For the little story, since ruby is object oriented, the main way to do it was to use the call method on the object. That way, you could keep the non-mandatory brackets behaviour consistent.
my_lambda.call
my_lambda.call()
my_lambda_with_arguments.call :h2g2, 42
my_lambda_with_arguments.call(:h2g2, 42)
Now somebody came up with a shortcut which basically looks like a method with no name.
my_lambda.()
my_lambda_with_arguments.(:h2g2, 42)
Again, this is a design choice. Now elixir is not object oriented and therefore call not use the first form for sure. I can't speak for José but it looks like the second form was used in elixir because it still looks like a function call with an extra character. It's close enough to a function call.
I did not think about all the pros and cons, but it looks like in both languages you could get away with just the brackets as long as you make brackets mandatory for anonymous functions. It seems like it is:
Mandatory brackets VS Slightly different notation
In both cases you make an exception because you make both behave differently. Since there is a difference, you might as well make it obvious and go for the different notation. The mandatory brackets would look natural in most cases but very confusing when things don't go as planned.
Here you go. Now this might not be the best explanation in the world because I simplified most of the details. Also most of it are design choices and I tried to give a reason for them without judging them. I love elixir, I love ruby, I like the function calls without brackets, but like you, I find the consequences quite misguiding once in a while.
And in elixir, it is just this extra dot, whereas in ruby you have blocks on top of this. Blocks are amazing and I am surprised how much you can do with just blocks, but they only work when you need just one anonymous function which is the last argument. Then since you should be able to deal with other scenarios, here comes the whole method/lambda/proc/block confusion.
Anyway... this is out of scope.
I've never understood why explanations of this are so complicated.
It's really just an exceptionally small distinction combined with the realities of Ruby-style "function execution without parens".
Compare:
def fun1(x, y) do
x + y
end
To:
fun2 = fn
x, y -> x + y
end
While both of these are just identifiers...
fun1 is an identifier that describes a named function defined with def.
fun2 is an identifier that describes a variable (that happens to contain a reference to function).
Consider what that means when you see fun1 or fun2 in some other expression? When evaluating that expression, do you call the referenced function or do you just reference a value out of memory?
There's no good way to know at compile time. Ruby has the luxury of introspecting the variable namespace to find out if a variable binding has shadowed a function at some point in time. Elixir, being compiled, can't really do this. That's what the dot-notation does, it tells Elixir that it should contain a function reference and that it should be called.
And this is really hard. Imagine that there wasn't a dot notation. Consider this code:
val = 5
if :rand.uniform < 0.5 do
val = fn -> 5 end
end
IO.puts val # Does this work?
IO.puts val.() # Or maybe this?
Given the above code, I think it's pretty clear why you have to give Elixir the hint. Imagine if every variable de-reference had to check for a function? Alternatively, imagine what heroics would be necessary to always infer that variable dereference was using a function?
There's an excellent blog post about this behavior: link
Two types of functions
If a module contains this:
fac(0) when N > 0 -> 1;
fac(N) -> N* fac(N-1).
You can’t just cut and paste this into the shell and get the same
result.
It’s because there is a bug in Erlang. Modules in Erlang are sequences
of FORMS. The Erlang shell evaluates a sequence of
EXPRESSIONS. In Erlang FORMS are not EXPRESSIONS.
double(X) -> 2*X. in an Erlang module is a FORM
Double = fun(X) -> 2*X end. in the shell is an EXPRESSION
The two are not the same. This bit of silliness has been Erlang
forever but we didn’t notice it and we learned to live with it.
Dot in calling fn
iex> f = fn(x) -> 2 * x end
#Function<erl_eval.6.17052888>
iex> f.(10)
20
In school I learned to call functions by writing f(10) not f.(10) -
this is “really” a function with a name like Shell.f(10) (it’s a
function defined in the shell) The shell part is implicit so it should
just be called f(10).
If you leave it like this expect to spend the next twenty years of
your life explaining why.
Elixir has optional braces for functions, including functions with 0 arity. Let's see an example of why it makes a separate calling syntax important:
defmodule Insanity do
def dive(), do: fn() -> 1 end
end
Insanity.dive
# #Function<0.16121902/0 in Insanity.dive/0>
Insanity.dive()
# #Function<0.16121902/0 in Insanity.dive/0>
Insanity.dive.()
# 1
Insanity.dive().()
# 1
Without making a difference between 2 types of functions, we can't say what Insanity.dive means: getting a function itself, calling it, or also calling the resulting anonymous function.
fn -> syntax is for using anonymous functions. Doing var.() is just telling elixir that I want you to take that var with a func in it and run it instead of referring to the var as something just holding that function.
Elixir has a this common pattern where instead of having logic inside of a function to see how something should execute, we pattern match different functions based on what kind of input we have. I assume this is why we refer to things by arity in the function_name/1 sense.
It's kind of weird to get used to doing shorthand function definitions (func(&1), etc), but handy when you're trying to pipe or keep your code concise.
In elixir we use def for simply define a function like we do in other languages.
fn creates an anonymous function refer to this for more clarification
Only the second kind of function seems to be a first-class object and can be passed as a parameter to other functions. A function defined in a module needs to be wrapped in a fn. There's some syntactic sugar which looks like otherfunction(myfunction(&1, &2)) in order to make that easy, but why is it necessary in the first place? Why can't we just do otherfunction(myfunction))?
You can do otherfunction(&myfunction/2)
Since elixir can execute functions without the brackets (like myfunction), using otherfunction(myfunction)) it will try to execute myfunction/0.
So, you need to use the capture operator and specify the function, including arity, since you can have different functions with the same name. Thus, &myfunction/2.
Looking at the source in FSharp.Core and PowerPack, I see that a lot of higher-order functions that accept a function with two or more parameters use FSharpFunc.Adapt. For example:
let mapi f (arr: ResizeArray<_>) =
let f = FSharpFunc<_,_,_>.Adapt(f)
let len = length arr
let res = new ResizeArray<_>(len)
for i = 0 to len - 1 do
res.Add(f.Invoke(i, arr.[i]))
res
The documentation on FSharpFunc.Adapt is fairly thin. Is this a general best practice that we should be using any time we have a higher-order function with a similar signature? Only if the passed-in function is called multiple times? How much of an optimization is it? Should we be using Adapt everywhere we can, or only rarely?
Thanks for your time.
That's quite interesting! I don't have any official information (and I didn't see this documented anywhere), but here are some thoughts on how the Adapt function might work.
Functions like mapi take curried form of a function, which means that the type of the argument is compiled to something like FSharpFunc<int, FSharpFunc<T, R>>. However, many functions are actually compiled directly as functions of two arguments, so the actual value would typically be FSharpFunc<int, T, R> which inherits from FSharpFunc<int, FSharpFunc<T, R>>.
If you call this function (e.g. f 1 "a") the F# compiler generates something like this:
FSharpFunc<int, string>.InvokeFast<a>(f, 1, "a");
If you look at the InvokeFast function using Reflector, you'll see that it tests if the function is compiled as the optimized version (f :? FSharpFunc<int, T, R>). If yes, then it directly calls Invoke(1, "a") and if not then it needs to make two calls Invoke(1).Invoke("a").
This check is done each time you call a function passed as an argument (it is probably faster to do the check and then use the optimized call, because that's more common).
What the Adapt function does is that it converts any function to FSharpFunc<T1, T2, R> (if the function is not optimized, it creates a wrapper for it, but that's not the case most of the time). The calls to the adapted function will be faster, because they don't need to do the dynamic check every time (the check is done only once inside Adapt).
So, the summary is that Adapt could improve the performance if you're calling a function passed as an argument that takes more than 1 argument a large number of times. As with any optimizations, I wouldn't use this blindly, but it is an interesting thing to be aware of when tuning the performance!
(BTW: Thanks for a very interesting question, I didn't know the compiler does this :-))
Here's something I've seen in erlang code a few times, but it's a tough thing to google and I can only find this example (the first code block in link below):
http://www.process-one.net/en/wiki/ejabberd_HTTP_request_handlers/
In the "head" of function definition of process/2
process(_LocalPath = ["world"], _Request) ->
there is a pattern match on first parameter / argument;
Does this act similarly like a guard, so the following clause will be executed only if the first argument passed to process/2 is string "world", or is "world" some kind of a default argument? Or i completely misunderstood/ mis-guessed?
Yes, this is a pattern match. The clause will be executed if the first argument is a list with a single element, the element being the string "world".
You are correct: _LocalPath = ["world"] acts as a pattern "guard". If the first parameter to the function "process" isn't equal to ["world"], then the emulator proceeds to find a match down.
One thing to note: _LocalPath serves as "decorator" to enhance readability since the identifier starts with an underscore.
The = in a pattern is used for an alias, it basically allows you to have your cake and eat it. It both does a normal pattern match and binds a variable to the whole matched data. It is practical if you need the whole data as it saves you having to reconstruct it. You can use it anywhere in a pattern. It has nothing to do with guards.
Starting a variable with a _ as in _LocalPath is too tell the compiler not to complain if this variable is not used. Normally the compiler whines a bit if you bind variables and don't use them. Apart from this there is nothing special about variables whose names start with _, you can use them as you would any variable.
The only really special variable is _, the anonymous variable. It always matches and is never bound so you can use it as an anonymous place holder. Which is why it exists in the first place.
I personally very rarely use variables starting with _ and prefer to use just _. I also feel that cluttering up patterns with unnecessary things is a Bad Thing so I wouldn't use aliases for documentation like that. I would write:
%% process(LocalPath, Request) -> ... .
process(["world"], _) ->
or perhaps a type declaration if you prefer. Keeps the code shorter and more legible, I think.
Since the age of the dinosaurs, Turbo Pascal and nowadays Delphi have a Write() and WriteLn() procedure that quietly do some neat stuff.
The number of parameters is variable;
Each variable can be of all sorts of types; you can supply integers, doubles, strings, booleans, and mix them all up in any order;
You can provide additional parameters for each argument:
Write('Hello':10,'World!':7); // alignment parameters
It even shows up in a special way in the code-completion drowdown:
Write ([var F:File]; P1; [...,PN] )
WriteLn ([var F:File]; [ P1; [...,PN]] )
Now that I was typing this I've noticed that Write and WriteLn don't have the same brackets in the code completion dropdown. Therefore it looks like this was not automatically generated, but it was hard-coded by someone.
Anyway, am I able to write procedures like these myself, or is all of this some magic hardcoded compiler trickery?
Writeln is what we call a compiler "magic" function. If you look in System.pas, you won't find a Writeln that is declared anything like what you would expect. The compiler literally breaks it all down into individual calls to various special runtime library functions.
In short, there is no way to implement your own version that does all the same things as the built-in writeln without modifying the compiler.
As the Allen said you can't write your own function that does all the same things.
You can, however, write a textfile driver that does something custom and when use standard Write(ln) to write to your textfile driver. We did that in ye old DOS days :)
("Driver" in the context of the previous statement is just a piece of Pascal code that is hooked into the system by switching a pointer in the System unit IIRC. Been a long time since I last used this trick.)
As far as I know, the pascal standards don't include variable arguments.
Having said that, IIRC, GNU Pascal let's you say something like:
Procecdure Foo(a: Integer; b: Integer; ...);
Try searching in your compiler's language docs for "Variable Argument Lists" or "conformant arrays". Here's an example of the later: http://www.gnu-pascal.de/demos/conformantdemo.pas.
As the prev poster said, writeln() is magic. I think the problem has to do with how the stack is assembled in a pascal function, but it's been a real long time since I've thought about where things were on the stack :)
However, unless you're writing the "writeln" function (which is already written), you probably don't need to implement a procedure with a variable arguments. Try iteration or recursion instead :)
It is magic compiler behaviour rather than regular procedure. And no, there is no way to write such subroutines (unfortunately!). Code generation resolves count of actual parameters and their types and translates to appropriate RTL calls (eg. Str()) at compile time. This opposes frequently suggested array of const (single variant array formal parameter, actually) which leads to doing the same at runtime. I'm finding later approach clumsy, it impairs code readability somewhat, and Bugland (Borland/Inprise/Codegear/Embarcadero/name it) broke Code Insight for variant open array constructors (yes, i do care, i use OutputDebugString(PChar(Format('...', [...])))) and code completion does not work properly (or at all) there.
So, closest possible way to simulate magic behaviour is to declare lot of overloaded subroutines (really lot of them, one per specific formal parameter type in the specific position). One could call this a kludge too, but this is the only way to get flexibility of variable parameter list and can be hidden in the separate module.
PS: i left out format specifiers aside intentionally, because syntax doesn't allow to semicolons use where Str(), Write() and Writeln() are accepting them.
Yes, you can do it in Delphi and friends (e.g. free pascal, Kylix, etc.) but not in more "standard" pascals. Look up variant open array parameters, which are used with a syntax something like this:
procedure MyProc(args : array of const);
(it's been a few years and I don't have manuals hand, so check the details before proceeding). This gives you an open array of TVarData (or something like that) that you can extract RTTI from.
One note though: I don't think you'll be able to match the x:y formatting syntax (that is special), and will probably have to go with a slightly more verbose wrapper.
Most is already said, but I like to add a few things.
First you can use the Format function. It is great to convert almost any kind of variable to string and control its size. Although it has its flaws:
myvar := 1;
while myvar<10000 do begin
Memo.Lines.Add(Format('(%3d)', [myVar]));
myvar := myvar * 10;
end;
Produces:
( 1)
( 10)
(100)
(1000)
So the size is the minimal size (just like the :x:y construction).
To get a minimal amount of variable arguments, you can work with default parameters and overloaded functions:
procedure WriteSome(const A1: string; const A2: string = ''; const A3: string = '');
or
procedure WriteSome(const A1: string); overload;
procedure WriteSome(const A1: Integer); overload;
You cannot write your own write/writeln in old Pascal. They are generated by the compiler, formatting, justification, etc. That's why some programmers like C language, even the flexible standard functions e.g. printf, scanf, can be implemented by any competent programmers.
You can even create an identical printf function for C if you are inclined to create something more performant than the one implemented by the C vendor. There's no magic trickery in them, your code just need to "walk" the variable arguments.
P.S.
But as MarkusQ have pointed out, some variants of Pascal(Free Pascal, Kylix, etc) can facilitate variable arguments. I last tinker with Pascal, since DOS days, Turbo Pascal 7.
Writeln is not "array of const" based, but decomposed by the compiler into various calls that convert the arguments to string and then call the primitive writestring. The "LN" is just a function that writes the lineending as a string. (OS dependant). The procedure variables (function pointers) for the primitives are part of the file type (Textrec/filerec), which is why they can be customized. (e.g. AssignCrt in TP)
If {$I+} mode is on, after each element, a call to the iocheck function is made.
The GPC construct made above is afaik the boundless C open array. FPC (and afaik Delphi too) support this too, but with different syntax.
procedure somehting (a:array of const);cdecl;
will be converted to be ABI compatible to C, printf style. This means that the relevant function (somehting in this case) can't get the number of arguments, but must rely on formatstring parsing. So this is something different from array of const, which is safe.
Although not a direct answer to you question, I would like to add the following comment:
I have recently rewritten some code using Writeln(...) syntax into using a StringList, filling the 'lines' with Format(...) and just plain IntToStr(...), FloatToStr(...) functions and the like.
The main reason for this change was speed improvement. Using a StringList and SaveFileTo is much, much more quicker than the WriteLn, Write combination.
If you are writing a program which creates a lot of text files (I was working on a web site creation program), this makes a lot of difference.