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In speech and writing, I keep wanting to refer to the data inside a monad, but I don't know what to call it.
For example, in Scala, the argument to the function passed to flatMap gets bound to…er…that thing inside the monad. In:
List(1, 2, 3).flatMap(x => List(x, x))
x gets bound to that thing I don't have a word for.
Complicating things a bit, the argument passed to the Kleisli arrow doesn't necessarily get bound to all the data inside the monad. With List, Set, Stream, and lots of other monads, flatMap calls the Kleisli arrow many times, binding x to a different piece of the data inside the monad each time. Or maybe not even to "data", so long as the monad laws are followed. Whatever it is, it's wrapped inside the monad, and flatMap passes it to you without the wrapper, perhaps one piece at a time. I just want to know what to call the relevant inside-the-monad stuff that x refers to, at least in part, so I can stop mit all this fumbly language.
Is there a standard or conventional term for this thing/data/value/stuff/whatever-it-is?
If not, how about "the candy"?
Trying to say "x gets bound to" is setting you up for failure. Let me explain, and guide you towards a better way of expressing yourself when talking about these sorts of things.
Suppose we have:
someList.flatMap(x => some_expression)
If we know that someList has type List[Int], then we can safely say that inside of some_expression, x is bound to a value of type Int. Notice the caveat, "inside of some_expression". This is because, given someList = List(1,2,3), x will take on the values of each of them: 1, 2, and 3, in turn.
Consider a more generalized example:
someMonadicValue.flatMap(x => some_expression)
If we know nothing about someMonadicValue, then we don't know much about how some_expression is going to be invoked. It may be run once, or three times (as in the above example), or lazily, or asynchronously, or it may be scheduled for once someMonadicValue is finished (e.g. futures), or it may never be used (e.g. empty list, None). The Monad interface does not include reasoning about when or how someExpression will be used. So all you can say about what x will be is confined to the context of some_expression, whenever and however some_expression happens to be evaluated.
So back to the example.
someMonadicValue.flatMap(x => some_expression)
You are trying to say "x is the ??? of someMonadicValue." And you are looking for the word that accurately replaces ???. Well I'm here to tell you that you're doing it wrong. If you want to speak about x, then either do it
Within the context of some_expression. In this case, use the bolded phrase I gave you above: "inside of some_expression, x is bound to a value of type Foo." Or, alternatively, you can speak about x...
With additional knowledge about which monad you're dealing with.
In case #2, for example, for someList.flatMap(x => some_expression), you could say that "x is each element of someList." For someFuture.flatMap(x => some_expression), you could say that "x is the successful future value of someFuture, if it indeed ever completes and succeeds."
You see, that's the beauty of Monads. That ??? that you are trying to describe, is the thing which the Monad interface abstracts over. Now do you see why it's so difficult to give ??? a name? It's because it has a different name and a different meaning for each particular monad. And that's the point of having the Monad abstraction: to unify these differing concepts under the same computational interface.
Disclaimer: I'm definitely not an expert in functional programming terminology and I expect that the following will not be an answer to your question from your point of view. To me the problem rather is: If choosing a term requires expert knowledge, so does understanding.
Choosing an appropriate term largely depends on:
your desired level of linguistic correctness, and
your audience, and the corresponding connotations of certain terms.
Regarding the linguistic correctness the question is whether you properly want to refer to the values/data that are bound to x, or whether you can live with a certain (incorrect) abstraction. In terms of the audience, I would mainly differentiate between an audience with a solid background in functional programming and an audience coming from other programming paradigms. In case of the former, choosing the term is probably not entirely crucial, since the concept itself is familiar, and many terms would lead to the right association. The discussion in the comments already contains some very good suggestions for this case. However, the discussion also shows that you need a certain background in functional programming to see the rationale behind some terms.
For an audience without a background in functional programming, I would rather sacrifice linguistic correctness in favor of comprehensibility. In such a situation I often just refer to it as "underlying type", just to avoid any confusion that I would probably create by trying to refer to the "thing(s) in the monad" itself. Obviously, it is literally wrong to say "x is bound to the underlying type". However, it is more important to me that my audience understands a concept at all. Since most programmers are familiar with containers and their underlying types, I'm aiming for the (flawed) association "underlying type" => "the thing(s) that are in a container" => "the thing(s) inside a monad", which often seems to work.
TL;DR: There always is a trade-off between correctness and accessibility. And when it comes to functional programming, it is sometimes helpful to shift the bias towards the latter.
flatMap does not call the Kleisli arrow many times. And "that thing" is not "inside" the monad.
flatMap lifts a Kleisli arrow to the monad. You could see this as the construction of an arrow M[A] => M[B] between types (A, B) lifted to the monad (M[A], M[B]), given a Kleisli arrow A => M[B].
So x in x => f(x) is the value being lifted.
What data?
data Monady a = Monady
Values of monads are values of monads, their wrapped type may be entirely a fiction. Which is to say that talking about it as if it exists may cause you pain.
What you want to talk about are the continuations like Monad m => a -> m b as they are guaranteed to exist. The funny stuff occurs in how (>>=) uses those continuations.
'Item' seems good? 'Item parameter' if you need to be more specific.
Either the source data can contain multiple items, or the operation can call it multiple times. Element would tend to be more specific for the first case, Value is singular as to the source & not sensible for list usages, but Item covers all cases correctly.
Disclaimer: I know more about understandable English than about FP.
Take a step back here.
A monad isn't just the functor m that wraps a value a. A monad is the stack of endofunctors (i.e., the compositions of m's), together with the join operator. That's where the famous quip -- that a monad is a monoid in the category of endofunctors, what's the problem? -- comes from.
(The whole story is that the quip means the composition of m's is another m, as witnessed by join)
A thing with the type (m a) is typically called a monad action. You can call a the result of the action.
I'm trying to learn scala and I'm unable to grasp this concept. Why does making an object immutable help prevent side-effects in functions. Can anyone explain like I'm five?
Interesting question, a bit difficult to answer.
Functional programming is very much about using mathematics to reason about programs. To do so, one needs a formalism that describe the programs and how one can make proofs about properties they might have.
There are many models of computation that provide such formalisms, such as lambda calculus and turing machines. And there's a certain degree of equivalency between them (see this question, for a discussion).
In a very real sense, programs with mutability and some other side effects have a direct mapping to functional program. Consider this example:
a = 0
b = 1
a = a + b
Here are two ways of mapping it to functional program. First one, a and b are part of a "state", and each line is a function from a state into a new state:
state1 = (a = 0, b = ?)
state2 = (a = state1.a, b = 1)
state3 = (a = state2.a + state2.b, b = state2.b)
Here's another, where each variable is associated with a time:
(a, t0) = 0
(b, t1) = 1
(a, t2) = (a, t0) + (b, t1)
So, given the above, why not use mutability?
Well, here's the interesting thing about math: the less powerful the formalism is, the easier it is to make proofs with it. Or, to put it in other words, it's too hard to reason about programs that have mutability.
As a consequence, there's very little advance regarding concepts in programming with mutability. The famous Design Patterns were not arrived at through study, nor do they have any mathematical backing. Instead, they are the result of years and years of trial and error, and some of them have since proved to be misguided. Who knows about the other dozens "design patterns" seen everywhere?
Meanwhile, Haskell programmers came up with Functors, Monads, Co-monads, Zippers, Applicatives, Lenses... dozens of concepts with mathematical backing and, most importantly, actual patterns of how code is composed to make up programs. Things you can use to reason about your program, increase reusability and improve correctness. Take a look at the Typeclassopedia for examples.
It's no wonder people not familiar with functional programming get a bit scared with this stuff... by comparison, the rest of the programming world is still working with a few decades-old concepts. The very idea of new concepts is alien.
Unfortunately, all these patterns, all these concepts, only apply with the code they are working with does not contain mutability (or other side effects). If it does, then their properties cease to be valid, and you can't rely on them. You are back to guessing, testing and debugging.
In short, if a function mutates an object then it has side effects. Mutation is a side effect. This is just true by definition.
In truth, in a purely functional language it should not matter if an object is technically mutable or immutable, because the language will never "try" to mutate an object anyway. A pure functional language doesn't give you any way to perform side effects.
Scala is not a pure functional language, though, and it runs in the Java environment in which side effects are very popular. In this environment, using objects that are incapable of mutation encourages you to use a pure functional style because it makes a side-effect oriented style impossible. You are using data types to enforce purity because the language does not do it for you.
Now I will say a bunch of other stuff in the hope that it helps this make sense to you.
Fundamental to the concept of a variable in functional languages is referential transparency.
Referential transparency means that there is no difference between a value, and a reference to that value. In a language where this is true, it makes it much simpler to think about a program works, since you never have to stop and ask, is this a value, or a reference to a value? Anyone who's ever programmed in C recognizes that a great part of the challenge of learning that paradigm is knowing which is which at all times.
In order to have referential transparency, the value that a reference refers to can never change.
(Warning, I'm about to make an analogy.)
Think of it this way: in your cell phone, you have saved some phone numbers of other people's cell phones. You assume that whenever you call that phone number, you will reach the person you intend to talk to. If someone else wants to talk to your friend, you give them the phone number and they reach that same person.
If someone changes their cell phone number, this system breaks down. Suddenly, you need to get their new phone number if you want to reach them. Maybe you call the same number six months later and reach a different person. Calling the same number and reaching a different person is what happens when functions perform side effects: you have what seems to be the same thing, but you try to use it, it turns out it's different now. Even if you expected this, what about all the people you gave that number to, are you going to call them all up and tell them that the old number doesn't reach the same person anymore?
You counted on the phone number corresponding to that person, but it didn't really. The phone number system lacks referential transparency: the number isn't really ALWAYS the same as the person.
Functional languages avoid this problem. You can give out your phone number and people will always be able to reach you, for the rest of your life, and will never reach anybody else at that number.
However, in the Java platform, things can change. What you thought was one thing, might turn into another thing a minute later. If this is the case, how can you stop it?
Scala uses the power of types to prevent this, by making classes that have referential transparency. So, even though the language as a whole isn't referentially transparent, your code will be referentially transparent as long as you use immutable types.
Practically speaking, the advantages of coding with immutable types are:
Your code is simpler to read when the reader doesn't have to look out for surprising side effects.
If you use multiple threads, you don't have to worry about locking because shared objects can never change. When you have side effects, you have to really think through the code and figure out all the places where two threads might try to change the same object at the same time, and protect against the problems that this might cause.
Theoretically, at least, the compiler can optimize some code better if it uses only immutable types. I don't know if Java can do this effectively, though, since it allows side effects. This is a toss-up at best, anyway, because there are some problems that can be solved much more efficiently by using side effects.
I'm running with this 5 year old explanation:
class Account(var myMoney:List[Int] = List(10, 10, 1, 1, 1, 5)) {
def getBalance = println(myMoney.sum + " dollars available")
def myMoneyWithInterest = {
myMoney = myMoney.map(_ * 2)
println(myMoney.sum + " dollars will accru in 1 year")
}
}
Assume we are at an ATM and it is using this code to give us account information.
You do the following:
scala> val myAccount = new Account()
myAccount: Account = Account#7f4a6c40
scala> myAccount.getBalance
28 dollars available
scala> myAccount.myMoneyWithInterest
56 dollars will accru in 1 year
scala> myAccount.getBalance
56 dollars available
We mutated the account balance when we wanted to check our current balance plus a years worth of interest. Now we have an incorrect account balance. Bad news for the bank!
If we were using val instead of var to keep track of myMoney in the class definition, we would not have been able to mutate the dollars and raise our balance.
When defining the class (in the REPL) with val:
error: reassignment to val
myMoney = myMoney.map(_ * 2
Scala is telling us that we wanted an immutable value but are trying to change it!
Thanks to Scala, we can switch to val, re-write our myMoneyWithInterest method and rest assured that our Account class will never alter the balance.
One important property of functional programming is: If I call the same function twice with the same arguments I'll get the same result. This makes reasoning about code much easier in many cases.
Now imagine a function returning the attribute content of some object. If that content can change the function might return different results on different calls with the same argument. => no more functional programming.
First a few definitions:
A side effect is a change in state -- also called a mutation.
An immutable object is an object which does not support mutation, (side effects).
A function which is passed mutable objects (either as parameters or in the global environment) may or may not produce side effects. This is up to the implementation.
However, it is impossible for a function which is passed only immutable objects (either as parameters or in the global environment) to produce side effects. Therefore, exclusive use of immutable objects will preclude the possibility of side effects.
Nate's answer is great, and here is some example.
In functional programming, there is an important feature that when you call a function with same argument, you always get same return value.
This is always true for immutable objects, because you can't modify them after create it:
class MyValue(val value: Int)
def plus(x: MyValue) = x.value + 10
val x = new MyValue(10)
val y = plus(x) // y is 20
val z = plus(x) // z is still 20, plus(x) will always yield 20
But if you have mutable objects, you can't guarantee that plus(x) will always return same value for same instance of MyValue.
class MyValue(var value: Int)
def plus(x: MyValue) = x.value + 10
val x = new MyValue(10)
val y = plus(x) // y is 20
x.value = 30
val z = plus(x) // z is 40, you can't for sure what value will plus(x) return because MyValue.value may be changed at any point.
Why do immutable objects enable functional programming?
They don't.
Take one definition of "function," or "prodecure," "routine" or "method," which I believe applies to many programming languages: "A section of code, typically named, accepting arguments and/or returning a value."
Take one definition of "functional programming:" "Programming using functions." The ability to program with functions is indepedent of whether state is modified.
For instance, Scheme is considered a functional programming language. It features tail calls, higher-order functions and aggregate operations using functions. It also has mutable objects. While mutability destroys some nice mathematical qualities, it does not necessarily prevent "functional programming."
I've read all the answers and they don't satisfy me, because they mostly talk about "immutability", and not about its relation to FP.
The main question is:
Why do immutable objects enable functional programming?
So I've searched a bit more and I have another answer, I believe the easy answer to this question is: "Because Functional Programming is basically defined on the basis of functions that are easy to reason about". Here's the definition of Functional Programming:
The process of building software by composing pure functions.
If a function is not pure -- which means receiving the same input, it's not guaranteed to always produce the same output (e.g., if the function relies on a global object, or date and time, or a random number to compute the output) -- then that function is unpredictable, that's it! Now exactly the same story goes about "immutability" as well, if objects are not immutable, a function with the same object as its input may have different results (aka side effects) each time used, and this will make it hard to reason about the program.
I first tried to put this in a comment, but it got longer than the limit, I'm by no means a pro so please take this answer with a grain of salt.
I understand different notions of functional programming by itself: side effects, immutability, pure functions, referential transparency. But I can't connect them together in my head. For example, I have the following questions:
What is the relation between ref. transparency and immutability. Does one implies the other?
Sometimes side effects and immutability is used interchangeably. Is it correct?
This question requires some especially nit-picky answers, since it's about defining common vocabulary.
First, a function is a kind of mathematical relation between a "domain" of inputs and a "range" (or codomain) of outputs. Every input produces an unambiguous output. For example, the integer addition function + accepts input in the domain Int x Int and produces outputs in the range Int.
object Ex0 {
def +(x: Int, y: Int): Int = x + y
}
Given any values for x and y, clearly + will always produce the same result. This is a function. If the compiler were extra clever, it could insert code to cache the results of this function for each pair of inputs, and perform a cache lookup as an optimization. That's clearly safe here.
The problem is that in software, the term "function" has been somewhat abused: although functions accept arguments and return values as declared in their signature, they can also read and write to some external context. For example:
class Ex1 {
def +(x: Int): Int = x + Random.nextInt
}
We can't think of this as a mathematical function anymore, because for a given value of x, + can produce different results (depending on a random value, which doesn't appear anywhere in +'s signature). The result of + can not be safely cached as described above. So now we have a vocabulary problem, which we solve by saying that Ex0.+ is pure, and Ex1.+ isn't.
Okay, since we've now accepted some degree of impurity, we need to define what kind of impurity we're talking about! In this case, we've said the difference is that we can cache Ex0.+'s results associated with its inputs x and y, and that we can't cache Ex1.+'s results associated with its input x. The term we use to describe cacheability (or, more properly, substitutability of a function call with its output) is referential transparency.
All pure functions are referentially transparent, but some referentially transparent functions aren't pure. For example:
object Ex2 {
var lastResult: Int
def +(x: Int, y: Int): Int = {
lastResult = x + y
lastResult
}
}
Here we're not reading from any external context, and the value produced by Ex2.+ for any inputs x and y will always be cacheable, as in Ex0. This is referentially transparent, but it does have a side effect, which is to store the last value computed by the function. Somebody else can come along later and grab lastResult, which will give them some sneaky insight into what's been happening with Ex2.+!
A side note: you can also argue that Ex2.+ is not referentially transparent, because although caching is safe with respect to the function's result, the side effect is silently ignored in the case of a cache "hit." In other words, introducing a cache changes the program's meaning, if the side effect is important (hence Norman Ramsey's comment)! If you prefer this definition, then a function must be pure in order to be referentially transparent.
Now, one thing to observe here is that if we call Ex2.+ twice or more in a row with the same inputs, lastResult will not change. The side effect of invoking the method n times is equivalent to the side effect of invoking the method only once, and so we say that Ex2.+ is idempotent. We could change it:
object Ex3 {
var history: Seq[Int]
def +(x: Int, y: Int): Int = {
result = x + y
history = history :+ result
result
}
}
Now, each time we call Ex3.+, the history changes, so the function is no longer idempotent.
Okay, a recap so far: a pure function is one that neither reads from nor writes to any external context. It is both referentially transparent and side effect free. A function that reads from some external context is no longer referentially transparent, whereas a function that writes to some external context is no longer free of side effects. And finally, a function which when called multiple times with the same input has the same side effect as calling it only once, is called idempotent. Note that a function with no side effects, such as a pure function, is also idempotent!
So how does mutability and immutability play into all this? Well, look back at Ex2 and Ex3. They introduce mutable vars. The side effects of Ex2.+ and Ex3.+ is to mutate their respective vars! So mutability and side effects go hand-in-hand; a function that operates only on immutable data must be side effect free. It might still not be pure (that is, it might not be referentially transparent), but at least it will not produce side effects.
A logical follow-up question to this might be: "what are the benefits of a pure functional style?" The answer to that question is more involved ;)
"No" to the first - one implies the other, but not the converse, and a qualified "Yes" to the second.
"An expression is said to be referentially transparent if it can be replaced with its value without changing the behavior of a program".
Immutable input suggests that an expression (function) will always evaluate to the same value, and therefore be referentially transparent.
However, (mergeconflict has kindly correct me on this point) being referentially transparent does not necessarily require immutability.
By definition, a side-effect is an aspect of a function; meaning that when you call a function, it changes something.
Immutability is an aspect of data; it cannot be changed.
Calling a function on such does imply that there can be no side-effects. (in Scala, that's limited to "no changes to the immutable object(s)" - the developer has responsibilities and decisions).
While side-effect and immutability don't mean the same thing, they are closely related aspects of a function and the data the function is applied to.
Since Scala is not a pure functional programming language, one must be careful when considering the meaning of such statements as "immutable input" - the scope of the input to a function may include elements other than those passed as parameters. Similarly for considering side-effects.
It rather depends on the specific definitions you use (there can be disagreement, see for example Purity vs Referential transparency), but I think this is a reasonable interpretation:
Referential transparency and 'purity' are properties of functions/expressions. A function/expression may or may not have side-effects. Immutability, on the other hand, is a property of objects, not functions/expressions.
Referential transparency, side-effects and purity are closely related: 'pure' and 'referentially transparent' are equivalent, and those notions are equivalent to the absence of side-effects.
An immutable object may have methods which are not referentially transparent: those methods will not change the object itself (as that would make the object mutable), but may have other side-effects such as performing I/O or manipulating their (mutable) parameters.
This may seem really obvious to the FP cognoscenti here, but what is point free style in Scala good for? What would really sell me on the topic is an illustration that shows how point free style is significantly better in some dimension (e.g. performance, elegance, extensibility, maintainability) than code solving the same problem in non-point free style.
Quite simply, it's about being able to avoid specifying a name where none is needed, consider a trivial example:
List("a","b","c") foreach println
In this case, foreach is looking to accept String => Unit, a function that accepts a String and returns Unit (essentially, that there's no usable return and it works purely through side effect)
There's no need to bind a name here to each String instance that's passed to println. Arguably, it just makes the code more verbose to do so:
List("a","b","c") foreach {println(_)}
Or even
List("a","b","c") foreach {s => println(s)}
Personally, when I see code that isn't written in point-free style, I take it as an indicator that the bound name may be used twice, or that it has some significance in documenting the code. Likewise, I see point-free style as a sign that I can reason about the code more simply.
One appeal of point-free style in general is that without a bunch of "points" (values as opposed to functions) floating around, which must be repeated in several places to thread them through the computation, there are fewer opportunities to make a mistake, e.g. when typing a variable's name.
However, the advantages of point-free are quickly counterbalanced in Scala by its meagre ability to infer types, a fact which is exacerbated by point-free code because "points" serve as clues to the type inferencer. In Haskell, with its almost-complete type inferencing, this is usually not an issue.
I see no other advantage than "elegance": It's a little bit shorter, and may be more readable. It allows to reason about functions as entities, without going mentally a "level deeper" to function application, but of course you need getting used to it first.
I don't know any example where performance improves by using it (maybe it gets worse in cases where you end up with a function when a method would be sufficient).
Scala's point-free syntax is part of the magic Scala operators-which-are-really-functions. Even the most basic operators are functions:
For example:
val x = 1
val y = x + 1
...is the same as...
val x = 1
val y = x.+(1)
...but of course, the point-free style reads more naturally (the plus appears to be an operator).
I was checking a friend's code, and this pattern showed up quite a bit whenever he wrote functions that returned a boolean value:
def multiple_of_three(n):
if (n % 3) is 0:
return True
else:
return False
I maintain that it's simpler (and perhaps a bit faster) to write:
def multiple_of_three(n):
return (n % 3) is 0
Am I correct that the second implementation is faster? Also, is it any less readable or somehow frowned upon?
I very much doubt there is a compiler or interpreter still out there where there is a significant speed difference - most will generate exactly the same code in both situations. But your "direct return" method is, in my opinion, clearer and more maintainable.
I can't speak of the Python interpreter's exact behavior, but doing one way over another like that (in any language) simply for reasons of "faster" is misguided, and qualifies as "premature optimization". As Paul Tomblin stated in another answer, the difference in speed, if any, is quite negligible. Common practice does dictate, however, that the second form in this case is more readable. If an expression is implicitly boolean, then the if statement wrapper is frivolous.
See also http://en.wikipedia.org/wiki/Program_optimization#When_to_optimize
The second form is the preferred form. In my experience the first form is usually the sign of an inexperienced programmer (and this does not apply solely to Python - this crops up in most languages).