I am learning Haskell and trying to understand the Monoid typeclass.
At the moment, I am reading the haskellbook and it says the following about the pattern (monoid):
One of the finer points of the Haskell community has been its
propensity for recognizing abstract patterns in code which have
well-defined, lawful representations in mathematics.
What does the author mean by abstract patterns?
Abstract in this sense is the opposite of concrete. This is probably one of the key things to understand about Haskell.
What is a concrete thing? Well, most values in Haskell are concrete. For example 'a' :: Char. The letter 'a' is a Char value, and it's a concrete value. Char is a concrete type. But in 1 :: Num a => a, the number 1 is actually a value of any type, so long as that type has the set of functions that the Num typeclass sets out as mandatory. This is an abstract value! We can have abstract values, abstract types, and therefore abstract functions. When the program is compiled, the Haskell compiler will pick a particular concrete value that supports all of our requirements.
Haskell, at its core, has a very simple, small but incredibly flexible language. It's very similar to an expression of maths, actually. This makes it very powerful. Why? because most things that would be built in language constructs in other languages are not directly built into Haskell, but defined in terms of this simple core.
One of the core pieces is the function, which, it turns out, most of computation is expressible in terms of. Because so much of Haskell is just defined in terms of this small simple core, it means we can extend it to almost anywhere we can imagine.
Typeclasses are probably the best example of this. Monoid, and Num are examples of typeclasses. These are constructs that allow programmers to use an abstraction like a function across a great many types but only having to define it once. Typeclasses let us use the same function names across a whole range of types if we can define those functions for those types. Why is that important or useful? Well, if we can recognise a pattern across, for example, all numbers, and we have a mechanism for talking about all numbers in the language itself, then we can write functions that work with all numbers at once. This is an abstract pattern. You'll notice some Haskellers are quite interested in a branch of mathematics called Category Theory. This branch is pretty much the mathematical definition of abstract patterns. Contrast this ability to encode such things with the inability of other languages, where in other languages the patterns the community notice are often far less rigorous and have to be manually written out, and without any respect for its mathematical nature. The beauty of following the mathematics is the extremely large body of stuff we get for free by aligning our language closer with mathematics.
This is a good explanation of these basics including typeclasses in a book that I helped author: http://www.happylearnhaskelltutorial.com/1/output_other_things.html
Because functions are written in a very general way (because Haskell puts hardly any limits on our ability to express things generally), we can write functions that use types which express such things as "any type, so long as it's a Monoid". These are called type constraints, as above.
Generally abstractions are very useful because we can, for example, write on single function to operate on an entire range of types which means we can often find functions that do exactly what we want on our types if we just make them instances of specific typeclasses. The Ord typeclass is a great example of this. Making a type we define ourselves an instance of Ord gives us a whole bunch of sorting and comparing functions for free.
This is, in my opinion, one of the most exciting parts about Haskell, because while most other languages also allow you to be very general, they mostly take an extreme dip in how expressive you can be with that generality, so therefore also are less powerful. (This is because they are less precise in what they talk about, because their types are less well "defined").
This is how we're able to reason about the "possible values" of a function, and it's not limited to Haskell. The more information we encode at the type level, the more toward the specificity end of the spectrum of expressivity we veer. For example, to take a classic case, the function const :: a -> b -> a. This function requires that a and b can be of absolutely any type at all, including the same type if we like. From that, because the second parameter can be a different type than the first, we can work out that it really only has one possible functionality. It can't return an Int, unless we give it an Int as its first value, because that's not any type, right? So therefore we know the only value it can return is the first value! The functionality is defined right there in the type! If that's not mindblowing, then I don't know what is. :)
As we move to dependent types (that is, a type system where types are first class, which means also that ordinary values can be encoded in the type system), we can get closer and closer to having the type system specify specifically what the constraints of possible functionality are. However, the kicker is, it doesn't necessarily speak about the implementation of the functionality unless we want it to, because we're in control of how abstract it is, but while maintaining expressivity and much precision. That's pretty fascinating, and amazingly powerful.
Much math can be expressed in the language that underpins Haskell, the lambda calculus.
Related
I've been doing some light reading on functional programming concepts and ideas. So far, so good, I've read about three main concepts: algebraic structures, type classes, and algebraic data types. I have a fairly good understanding of what algebraic data types are. I think sum types and product types are fairly straightforward. For example, I can imagine creating an algebraic data type like a Card type which is a product type consisting of two enum types, Suit (with four values and symbols) and Rank (with 13 values and symbols).
However, I'm still hung up on trying to understand precisely what algebraic structures and type classes are. I just have a surface-level picture in my head but can't quite completely wrap my head around, for instance, the different types of algebraic structures like functors, monoids, monads, etc. How exactly are these different? How can they be used in a programming setting? How are type classes different from regular classes? Can anyone at least point me in the direction of a good book on abstract algebra and functional programming? Someone recommended I learn Haskell but do I really need to learn Haskell in order to understand functional programming?
"algebraic structure" is a concept that goes well beyond programming, it belongs to mathematics.
Imagine the unfathomably deep sea of all possible mathematical objects. Numbers of every stripe (the naturals, the reals, p-adic numbers...) are there, but also things like sequences of letters, graphs, trees, symmetries of geometrical figures, and all well-defined transformations and mappings between them. And much else.
We can try to "throw a net" into this sea and retain only some of those entities, by specifying conditions. Like "collections of things, for which there is an operation that combines two of those things into a third thing of the same type, and for which the operation is associative". We can give those conditions their own name, like, say, "semigroup". (Because we are talking about highly abstract stuff, choosing a descriptive name is difficult.)
That leaves out many inhabitants of the mathematical "sea", but the description still fits a lot of them! Many collections of things are semigroups. The natural numbers with the multiplication operation for example, but also non-empty lists of letters with concatenation, or the symmetries of a square with composition.
You can expand your description with extra conditions. Like "a semigroup, and there's also an element such that combining it with any other element gives the other element, unchanged". That restricts the number of mathematical entities that fit the description, because you are demanding more of them. Some valid semigroups will lack that "neutral element". But a lot of mathematical entities will still satisfy the expanded description. If you aren't careful, you can declare conditions so restrictive that no possible mathematical entity can actually fit them! At other times, you can be so precise that only one entity fits them.
Working purely with these descriptions of mathematical entities, using only the general properties we require of them, we can obtain unexpected results about them, non-obvious at first sight, results that will apply to all entities which fit the description. Think of these discoveries as the mathematical equivalent of "code reuse". For example, if we know that some collection of things is a semigroup, then we can calculate exponentials using binary exponentiation instead of tediously combining a thing with itself n times. But that only works because of the associative property of the semigroup operation.
You’ve asked quite a few questions here, but I can try to answer them as best I can:
… different types of algebraic structures like functors, monoids, monads, etc. How exactly are these different? How can they be used in a programming setting?
This is a very common question when learning Haskell. I won’t write yet another answer here — and a complete answer is fairly long anyway — but a simple Google search gives some very good answers: e.g. I can recommend 1 2 3
How are type classes different from regular classes?
(By ‘regular classes’ I assume you mean classes as found in OOP.)
This is another common question. Basically, the two have almost nothing in common except the name. A class in OOP is a combination of fields and methods. Classes are used by creating instances of that class; each instance can store data in its fields, and manipulate that data using its methods.
By contrast, a type class is simply a collection of functions (often also called methods, though there’s pretty much no connection). You can declare an instance of a type class for a data type (again, no connection) by redefining each method of the class for that type, after which you may use the methods with that type. For instance, the Eq class looks like this:
class Eq a where
(==) :: a -> a -> Bool
(/=) :: a -> a -> Bool
And you can define an instance of that class for, say, Bool, by implementing each function:
instance Eq Bool where
True == True = True
False == False = True
_ == _ = False
p /= q = not (p == q)
Can anyone at least point me in the direction of a good book on abstract algebra and functional programming?
I must admit that I can’t help with this (and it’s off-topic for Stack Overflow anyway).
Someone recommended I learn Haskell but do I really need to learn Haskell in order to understand functional programming?
No, you don’t — you can learn functional programming from any functional language, including Lisp (particularly Scheme dialects), OCaml, F#, Elm, Scala etc. Haskell happens to be a particularly ‘pure’ functional programming language, and I would recommend it as well, but if you just want to learn and understand functional programming then any one of those will do.
In the Haskell community, we are slowly adding features of dependent types. Dependent types is an advanced typing feature by which types can depend on values. Some languages like Agda and Idris already have them. It appears to be a very advanced feature requiring an advanced type system, until you realize that python has had dependent types has had the dynamic typing version of dependent types, which may or may not be actual dependent types, from the beginning.
For most any program in a functional programming language, there is a way to reperesent it as an untyped lambda calculus term, no matter how advanced the typing. That's because typing only eliminates programs, not enable new ones.
Strong Typing wins us safety. How classes of errors that happened at run time can no longer happen at run time. This safety is rather nice. Besides this safety though, what does strong typing give you?
Are there an additional benefits of a strong type system besides safety?
(Note that I'm not saying that strong typing is worthless. Safety is a huge benefit in and of itself. I'm just wondering if there are additional benefits.)
First, we need to talk a bit about the history of the simply typed lambda calculus.
There are two historical developments of the simply typed lambda calculus.
When Alonzo Church described the lambda calculus the types were baked in as part of the meaning / operational behavior of the terms.
When Haskell Curry described the lambda calculus the types were annotations put on the terms.
So we have the lambda calculus a la Church and the lambda calculus a la Curry. See https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus#Intrinsic_vs._extrinsic_interpretations for more.
Ironically, the language Haskell, which is named after Curry is based on a lambda calculus a la Church!
What this means is the types aren't simply annotations that rule out bad programs for you. They can "do stuff" too. Such types don't erase without leaving residue.
This shows up in Haskell's notion of type classes, which are really why Haskell is a language a la Church.
In Haskell, when I make a function
sort :: Ord a => [a] -> [a]
We're passing an object or dictionary for Ord a as the first argument.
But you aren't forced to plumb that argument around yourself in the code, it is the job of the compiler to build that up and use it.
instance Ord Char
instance Ord Int
instance Ord a => Ord [a]
So if you go and use sort on a list of strings, which are themselves lists of chars, then this will build up the dictionary by passing the Ord Char instance through the instance for Ord a => Ord [a] to get Ord [Char], which is the same as Ord String, then you can sort a list of strings.
Calling sort above, is a lot less verbose than manually building a LexicographicComparator<List<Char>> by passing it an IComparator<Char> to its constructor and calling the function with an extra second argument, if I were to compare the complexity of calling such a sort function in Haskell to calling it in C# or Java.
This shows us that programming with types can be significantly less verbose, because mechanisms like implicits and typeclasses can infer a large part of the code for your program during type checking.
On a simpler basis, even the sizes of arguments can depend on types, unless you want to pay fairly massive costs for boxing everything in your language up so that it has a homogeneous representation.
This shows us that programming with types can be significantly more efficient, because it can use dedicated representations, rather than paying for boxed structures everywhere in your code. An int can't just be a machine integer, because it has to somehow look like everything else in the system. If you're willing to give up an order of magnitude or more worth of performance at runtime, then this may not matter to you.
Finally, once we have types "doing stuff" for us, it is often beneficial to consider the refactoring benefits that mere safety provides.
If I refactor the smaller set of code that remains, it'll rewrite all that type-class plumbing for me. It'll figure out the new ways it can rewrite the code to unbox more arguments. I'm not stuck elaborating all of this stuff by hand, I can leave these mundane tasks to the type-checker.
But even when I do change the types, I can move arguments around fairly willy-nilly, comfortable that the compiler will very likely catch my errors. Types give you "free theorems" which are like unit tests for whole classes of such errors.
On the other hand, once I lock down an API in a language like Python I'm deathly afraid of changing it, because it'll silently break at runtime for all my downstream dependencies! This leads to baroque APIs that lean heavily on easily bit-rotted keyword-arguments, and the API of something that evolves over time rarely resembles what you'd build out of the box if you had it to do over again. Consequently, even the mere safety concern has long-term impact in API design once you ever want people to build on top of your work, rather than simply replace it when it gets too unwieldy.
That's because typing only eliminates programs, not enable new ones.
This is not a correct statement. Type-classes make it possible to generate parts of your program from type-level information.
Consider two expressions:
readMaybe "15" :: Maybe Integer
readMaybe "15" :: Maybe Bool
Here I'm using the readMaybe function from the Text.Read module. At term level those expressions are identical, only their type annotations are different. However, the results they produce at runtime differ (Just 15 in the first case, Nothing in the second case).
This is because the compiler generates code for you from the static type information you have. To be more precise, it selects a suitable type class instance and passes its dictionary to the polymorphic function (readMaybe in our case).
This example is simple, but there are way more complex use cases. Using the mtl library you can write computations that run in different computational contexts (aka Monads). The compiler will automatically insert a lot of code that manages the computational contexts. In a dynamically typed language, you would have no static information to make this possible.
As you can see, static typing not only cuts off incorrect programs but also writes correct ones for you.
You need "safety" when you already know what and how you want to write. It's a very small part of what types are useful for. The most important thing about types is that they make your reasoning structured. When someone writes in Python a + b he doesn't see a and b as some abstract variables — he sees them as some numbers. Types are already there in the internal language of humans, Python just doesn't have a type system to talk about them. The actual question in the "typed vs untyped (unityped) programming" dispute is "do we want to reflect our internal structured concepts in a safe and explicit or unsafe and implicit way?". Types don't introduce new concepts — it's untyped reasoning forgets the existing ones.
When someone looks at a tree (I mean a real green one) he doesn't see every single leaf on it, but he doesn't treat it as an abstract nameless object as well. "A tree" — is an approximation that is good enough for most cases and that's why we have Hindley-Milner type systems, but sometimes you want to talk about a specific tree and you do want to look at leaves. And that's what dependent types give you: the ability to zoom. "A tree without leaves", "a tree in the forest", "a tree of a particular form"... Dependently typed programming is just another step towards how humans think.
On a less abstract note, I have a type checker for a toy dependently typed language, where all typing rules are expressed as constructors of a data type. You don't need to dive into the type checking procedure to understand the rules of the system. That's the power of "zooming": you can introduce as complex invariants as you want, thus distinguishing essential parts from not important ones.
Another example of the power dependent types give you is various forms of reflection. Look e.g. at the Pierre-Évariste Dagand thesis, which proves that
generic programming is just programming
And of course types are hints, many functions and abstractions I defined I would define in a far more clumsy way in a weakly typed language, but types suggested better alternatives.
There is just no question "What to choose: simple types or dependent types?". Dependent types are always better and they of course subsume simple types. The question is "What to choose: no types or dependent types?", but that question doesn't stand for me.
Refactoring. By having a strong type system you can safely refactor code and have the compiler tell you whether what you are doing now even makes sense. The stronger the typing system, the more refactor errors are avoided. This of course means your code is a lot more maintainable.
pigworker once asked how to express that a type is infinitely differentiable. This question brought to mind the fact that in complex analysis, a function that is differentiable (on an open set) must be infinitely differentiable (on that set). Is there a way to talk about complex differentiation of datatypes? If so, does a similar theorem hold?
Not really an answer... but this rant is way too long for a comment.
I find it a bit misleading to think complex differentiability just implies infinite differentiability. It's in fact much stronger than that: if a function is complex differentiable, then its derivatives at any point determine the entire function. And because infinite differentiability gives you a full Taylor series, you have an analytic function which is equal to your function, i.e. is your function itself. So, in a sense complex differentiable functions are analytic... because they are.
From a (standard) calculus perspective, the key contrast between real diff'ability and complex diff'ability is that in the reals, there is only one direction in which you can take the limit of difference-quotients (f(x+δ) - f x)/δ. You merely require that the left limit equals the right limit. But because that's an equality after the limit, this has only an effect locally. (Topologically speaking, the constraint just compares two discrete values, so it doesn't really deal with continuity properties at all.)
OTOH, for complex differentiability we require that the limit of the difference quotient is the same if we approach x from any direction in the entire complex plane. That's an entire continuous degree of freedom constrained. You can then go on to perform topological tricks (Cauchy integrals are essentially that) to “spread” the constraint through the entire domain.
I consider this a bit problematic philosophically. Holomorphic functions aren't really functions at all, as in: they're not so much defined by the entirety of their result values across the domain, as by some way to write them with analytic formulas (i.e. possibly-infinite algebraic expressions / polynomials).
Most mathematicians and physicists apparently like this a lot – such expressions are just the way in which they generally write functions.
I don't, really, like it at all: to me, a function should be a function, something defined by individual values, like field strengths you can measure in space or results you can define in Haskell.
Anyway, I digress...
If we translate this issue from functions on numbers to functors on Haskell types, I suppose the upshot is that complex diff'ability means nothing else but: a type can be written as a (possibly infinite?) ADT polynomial. And how to get infinite differentiability for such ADTs was shown in the post you linked to.
Another spin... perhaps closer to an answer.
These “derivatives” of Haskell types aren't really derivatives in the calculus sense. As in, they aren't motivated by a concept of small-pertubation response analysis†. It so happens that you can mathematically proove, for a very specific class of functions – those defined by an algebraic expression – that the calculus-derivative can again be written in a simple algebraic way (given by the well-known differentiation rules). That means trivially that you can differentiate infinitely often.
The usefulness of this symbolic differentiation also motivates to think about it as a more abstract operation. And when you're differentiating Haskell types, it is mainly just this algebraic definition you're going after, not the original calculus one.
Which is fine... but once you're doing algebra rather than calculus, it's not very meaningful to distinguish “real” from “complex” – it's actually neither, because you're not handling values but symbolic representations of values. An untyped language, if you will (and indeed, Haskell's type language is still untyped, with everything having kind *).
†Be it with traditional convergent limits or NSA-infinitesimals.
I know what covariance and contravariance of types are. My question is why haven't I encountered discussion of these concepts yet in my study of Haskell (as opposed to, say, Scala)?
It seems there is a fundamental difference in the way Haskell views types as opposed to Scala or C#, and I'd like to articulate what that difference is.
Or maybe I'm wrong and I just haven't learned enough Haskell yet :-)
There are two main reasons:
Haskell lacks an inherent notion of subtyping, so in general variance is less relevant.
Contravariance mostly appears where mutability is involved, so most data types in Haskell would simply be covariant and there'd be little value to distinguishing that explicitly.
However, the concepts do apply--for instance, the lifting operation performed by fmap for Functor instances is actually covariant; the terms co-/contravariance are used in Category Theory to talk about functors. The contravariant package defines a type class for contravariant functors, and if you look at the instance list you'll see why I said it's much less common.
There are also places where the idea shows up implicitly, in how manual conversions work--the various numeric type classes define conversions to and from basic types like Integer and Rational, and the module Data.List contains generic versions of some standard functions. If you look at the types of these generic versions you'll see that Integral constraints (giving toInteger) are used on types in contravariant position, while Num constraints (giving fromInteger) are used for covariant position.
There are no "sub-types" in Haskell, so covariance and contravariance don't make any sense.
In Scala, you have e.g. Option[+A] with the subclasses Some[+A] and None. You have to provide the covariance annotations + to say that an Option[Foo] is an Option[Bar] if Foo extends Bar. Because of the presence of sub-types, this is necessary.
In Haskell, there are no sub-types. The equivalent of Option in Haskell, called Maybe, has this definition:
data Maybe a = Nothing | Just a
The type variable a can only ever be one type, so no further information about it is necessary.
As mentioned, Haskell does not have subtypes. However, if you're looking at typeclasses it may not be clear how that works without subtyping.
Typeclasses specify predicates on types, not types themselves. So when a Typeclass has a superclass (e.g. Eq a => Ord a), that doesn't mean instances are subtypes, because only the predicates are inherited, not the types themselves.
Also, co-, contra-, and in- variance mean different things in different fields of math (see Wikipedia). For example the terms covariant and contravariant are used in functors (which in turn are used in Haskell), but the terms mean something completely different. The term invariant can be used in a lot of places.
Reading "Real world Haskell" i found some intresting question about data types:
This pattern matching and positional
data access make it look like you have
very tight coupling between data and
code that operates on it (try adding
something to Book, or worse change the
type of an existing part).
This is usually a very bad thing in
imperative (particularly OO)
languages... is it not seen as a
problem in Haskell?
source at RWH comments
And really, writing some Haskell programs I found that when I make small change to data type structure it affects almost all functions that use that data type. Maybe there are some good practices for data type usage. How can i minimize code coupling?
What you are describing is commonly known as the expression problem -- http://en.wikipedia.org/wiki/Expression_Problem.
There is a definite trade-off to be made, haskell code in general, and algebraic data types in particular, tends to fall into the hard to change the type but easy to add functions over the type. This optimizes for (up front) well-designed, complete, data types.
All that said, there are a number of things that you can do to reduce the coupling.
Define good library functions, by defining a complete set of combinators and higher order functions that are useful for interacting with your data type you will reduce coupling. It is often said that when ever you think of pattern matching there is a better solution using higher-order functions. If you look for these situations you will be in a better spot.
Expose your data structures as more abstract types. This means implementing all appropriate type classes. This will assist with defining a library functions as you will get a bunch for free with any of the type classes you implement, for examples look at operations over Functor or Monad.
Hide (as much as possible) any type constructors. Constructors expose implementation detail and will encourage coupling. Hint: this links in with defining a good api for interacting with your type, consumers of your type should rarely, if ever, have to use the type constructors.
The haskell community seems particularly good at this, if you look at many of the libraries on hackage you will find really good examples of implementing type classes and exposing good library functions.
In addition to what's been said:
One interesting approach is the "scrap your boilerplate" style of defining functions over data types, which makes use of generic functions (as opposed to explicit pattern matching) to define functions over the constructors of a data type. Looking at the "scrap your boilerplate" papers, you will see examples of functions which can cope with changes to the structure of a data type.
A second method, as Hibberd pointed out, is to use folds, maps, unfolds, and other recursion combinators, to define your functions. When you write functions using higher order functions, oftentimes small changes to the underlying data type can be dealt with in the instance declarations for Functor, Foldable, and so on.
First, I'd like to mention that in my view, there are two kinds of couplings:
One that makes your code cease to compile when you change one and forget to change the other
One that makes your code buggy when you change one and forget to change the other
While both are problematic, the former is significantly less of a headache, and that seems to be the one you're talking about.
I think the main problem you're mentioning is due to over-using positional arguments. Haskell almost forces you to have positional arguments in your ordinary functions, but you can avoid them in your type products (records).
Just use records instead of multiple anonymous fields inside data constructors, and then you can pattern-match any field you want out of it, by name.
bad (Blah _ y) = ...
good (Blah{y = y}) = ...
Avoid over-using tuples, especially those beyond 2-tuples, and liberally create records/newtypes around things to avoid positional meaning.