%% -*- LaTeX -*- \documentclass[11pt]{typeclassopedia} %include lhs2TeX.fmt %options ghci \usepackage{amsmath} \usepackage{amsthm} \usepackage{paralist} \usepackage{prettyref} \usepackage{stmaryrd} \usepackage{graphicx} \usepackage{answers} \newcommand{\pref}[1]{\prettyref{#1}} \newrefformat{lst}{Listing~\ref{#1}} \newrefformat{sec}{page~\pageref{#1}} \newrefformat{ex}{Exercise~\ref{#1}} \newcommand{\term}[1]{\textbf{#1}} \renewcommand{\emph}[1]{\textit{#1}} \newcommand{\etal}{\textit{et al.}} \newenvironment{exercise}{\begin{ex}}{\end{ex}} \Newassociation{solution}{Solution}{sols} \title{The Typeclassopedia} \subtitle{Second Edition} \author{Brent Yorgey\email{byorgey@@cis.upenn.edu}} \begin{document} \Opensolutionfile{sols} \begin{introduction} The standard Haskell libraries feature a number of type classes with algebraic or category-theoretic underpinnings. Becoming a fluent Haskell hacker requires intimate familiarity with them all, yet acquiring this familiarity often involves combing through a mountain of tutorials, blog posts, mailing list archives, and IRC logs. The goal of this document is to serve as a starting point for the student of Haskell wishing to gain a firm grasp of its standard type classes. The essentials of each type class are introduced, with examples, commentary, and extensive references for further reading. \end{introduction} \section{Introduction} \label{sec:introduction} Have you ever had any of the following thoughts? \begin{itemize} \item \emph{What the heck is a monoid, and how is it different from a mon\emph{a}d?} \item \emph{I finally figured out how to use Parsec with do-notation, and someone told me I should use something called |Applicative| instead. Um, what?} \item \emph{Someone in the \#haskell IRC channel used |(***)|, and when I asked lambdabot to tell me its type, it printed out scary gobbledygook that didn't even fit on one line! Then someone used |fmap fmap fmap| and my brain exploded.} \item \emph{When I asked how to do something I thought was really complicated, people started typing things like |zip.ap fmap.(id &&& wtf)| and the scary thing is that they worked! Anyway, I think those people must actually be robots because there's no way anyone could come up with that in two seconds off the top of their head.} \end{itemize} If you have, look no further! You, too, can write and understand concise, elegant, idiomatic Haskell code with the best of them. There are two keys to an expert Haskell hacker's wisdom: \begin{inparaenum} \item Understand the types. \item Gain a deep intuition for each type class and its relationship to other type classes, backed up by familiarity with many examples. \end{inparaenum} It's impossible to overstate the importance of the first; the patient student of type signatures will uncover many profound secrets. Conversely, anyone ignorant of the types in their code is doomed to eternal uncertainty. ``Hmm, it doesn't compile\dots maybe I'll stick in an |fmap| here\dots nope, let's see\dots maybe I need another |(.)| somewhere? \dots um \dots '' The second key---gaining deep intuition, backed by examples---is also important, but much more difficult to attain. A primary goal of this document is to set you on the road to gaining such intuition. However--- \begin{center} \emph{There is no royal road to Haskell.} \\ ---Euclid\footnote{Well, he probably would have said it if he knew Haskell.} \end{center} This document can only be a starting point, since good intuition comes from hard work, not from learning the right metaphor~\cite{monad-tutorial-fallacy}. Anyone who reads and understands all of it will still have an arduous journey ahead---but sometimes a good starting point makes a big difference. It should be noted that this is not a Haskell tutorial; it is assumed that the reader is already familiar with the basics of Haskell, including the standard |Prelude|, the type system, data types, and type classes. \pref{fig:type-classes} shows the type classes we will be discussing and their interrelationships. Solid arrows point from the general to the specific; that is, if there is an arrow from |Foo| to |Bar| it means that every |Bar| is (or should be, or can be made into) a |Foo|. Dotted arrows indicate some other sort of relationship. The solid double arrow indicates that |Monad| and |ArrowApply| are equivalent. |Pointed| and |Comonad| are greyed out since they are not actually (yet) in the standard Haskell libraries (they are in the category-extras library~\cite{category-extras}). \begin{figure}[htp] \centering \includegraphics[scale=0.65]{dependencies.pdf} \vspace{1em} \caption{Relationships among standard Haskell type classes} \label{fig:type-classes} \end{figure} One more note before we begin. I've seen ``type class'' written as one word, ``typeclass'', but let's settle this once and for all: the correct spelling uses two words (the title of this document notwithstanding), as evidenced by, for example, the Haskell 98 Revised Report~\cite{haskell-report}, early papers on type classes~\cite{type-classes, Jones97typeclasses}, and Hudak {\it et al.}'s history of Haskell~\cite{lazy-with-class}. We now begin with the simplest type class of all: |Functor|. \section{Functor} \label{sec:functor} The |Functor| class~\cite{functor-docs} is the most basic and ubiquitous type class in the Haskell libraries. A simple intuition is that a |Functor| represents a ``container'' of some sort, along with the ability to apply a function uniformly to every element in the container. For example, a list is a container of elements, and we can apply a function to every element of a list, using |map|. As another example, a binary tree is also a container of elements, and it's not hard to come up with a way to recursively apply a function to every element in a tree. Another intuition is that a |Functor| represents some sort of ``computational context''. This intuition is generally more useful, but is more difficult to explain, precisely because it is so general. Some examples later should help to clarify the |Functor|-as-context point of view. In the end, however, a |Functor| is simply what it is defined to be; doubtless there are many examples of |Functor| instances that don't exactly fit either of the above intuitions. The wise student will focus their attention on definitions and examples, without leaning too heavily on any particular metaphor. Intuition will come, in time, on its own. \subsection{Definition} \label{sec:functor-definition} The type class declaration for |Functor| is shown in \pref{lst:functor}. |Functor| is exported by the |Prelude|, so no special imports are needed to use it. \begin{listing}[htp] \begin{code} class Functor f where fmap :: (a -> b) -> f a -> f b \end{code} \caption{The |Functor| type class} \label{lst:functor} \end{listing} First, the |f a| and |f b| in the type signature for |fmap| tell us that |f| isn't just a type; it is a \term{type constructor} which takes another type as a parameter. (A more precise way to say this is that the \term{kind} of |f| must be |* -> *|.) For example, |Maybe| is such a type constructor: |Maybe| is not a type in and of itself, but requires another type as a parameter, like |Maybe Integer|. So it would not make sense to say |instance Functor Integer|, but it could make sense to say |instance Functor Maybe|. Now look at the type of |fmap|: it takes any function from |a| to |b|, and a value of type |f a|, and outputs a value of type |f b|. From the container point of view, the intention is that |fmap| applies a function to each element of a container, without altering the structure of the container. From the context point of view, the intention is that |fmap| applies a function to a value without altering its context. Let's look at a few specific examples. \subsection{Instances} \label{sec:functor-examples} As noted before, the list constructor |[]| is a functor;\footnote{Recall that |[]| has two meanings in Haskell: it can either stand for the empty list, or, as here, it can represent the list type constructor (pronounced ``list-of''). In other words, the type |[a]| (list-of-|a|) can also be written |([] a)|.} we can use the standard list function |map| to apply a function to each element of a list.\footnote{You might ask why we need a separate |map| function. Why not just do away with the current list-only |map| function, and rename |fmap| to |map| instead? Well, that's a good question. The usual argument is that someone just learning Haskell, when using |map| incorrectly, would much rather see an error about lists than about |Functor|s.} The |Maybe| type constructor is also a functor, representing a container which might hold a single element. The function |fmap g| has no effect on |Nothing| (there are no elements to which |g| can be applied), and simply applies |g| to the single element inside a |Just|. Alternatively, under the context interpretation, the list functor represents a context of nondeterministic choice; that is, a list can be thought of as representing a single value which is nondeterministically chosen from among several possibilities (the elements of the list). Likewise, the |Maybe| functor represents a context with possible failure. These instances are shown in \pref{lst:functor-instances}. As an aside, in idiomatic Haskell code you will often see the letter |f| used to stand for both an arbitrary |Functor| and an arbitrary function. In this document, |f| represents only |Functor|s, and |g| or |h| always represent functions, but you should be aware of the potential confusion. In practice, what |f| stands for should always be clear from the context, by noting whether it is part of a type or part of the code. \begin{listing}[tp] \begin{code} instance Functor [] where fmap _ [] = [] fmap g (x:xs) = g x : fmap g xs -- or we could just say fmap = map instance Functor Maybe where fmap _ Nothing = Nothing fmap g (Just a) = Just (g a) \end{code} \caption{Two simple |Functor| instances} \label{lst:functor-instances} \end{listing} There are other |Functor| instances in the standard libraries; here are a few:\footnote{Note that some of these instances are not exported by the |Prelude|; to access them, you can import |Control.Monad.Instances|.} \begin{itemize} \item |Either e| is an instance of |Functor|; |Either e a| represents a container which can contain either a value of type |a|, or a value of type |e| (often representing some sort of error condition). It is similar to |Maybe| in that it represents possible failure, but it can carry some extra information about the failure as well. \item |((,) e)| represents a container which holds an ``annotation'' of type |e| along with the actual value it holds. (It might be clearer to write it as |(e,)|, by analogy with an operator section like |(1+)|, but that syntax is not allowed. However, you can certainly \emph{think} of it as |(e,)|.) \item |((->) e)| (which can be thought of as |(e ->)|; see above), the type of functions which take a value of type |e| as a parameter, is a |Functor|. As a container, |(e -> a)| represents a (possibly infinite) set of values of |a|, indexed by values of |e|. Alternatively, and more usefully, |(->) e| can be thought of as a context in which a value of type |e| is available to be consulted in a read-only fashion. This is also why |((->) e)| is sometimes referred to as the \term{reader monad}; more on this later. \item |IO| is a |Functor|; a value of type |IO a| represents a computation producing a value of type |a| which may have I/O effects. If |m| computes the value |x| while producing some I/O effects, then |fmap g m| will compute the value |g x| while producing the same I/O effects. \item Many standard types from the containers library~\cite{containers-library} (such as |Tree|, |Map|, |Sequence|, and |Stream|) are instances of |Functor|. A notable exception is |Set|, which cannot be made a |Functor| in Haskell (although it is certainly a mathematical functor) since it requires an |Ord| constraint on its elements, whereas |fmap| must be applicable to \emph{any} types |a| and |b|. However, |Set| (and other similarly restricted data types) can be made an instance of a suitable generalization of |Functor|~\cite{oleg-set-functor}. Adding \term{associated constraints} to Haskell as proposed by Orchard and Schrijvers~\cite{type-constraints-unleashed} represents another possible solution. \end{itemize} \begin{exercise} Implement |Functor| instances for |Either e| and % |((->) e)|. \begin{solution} \begin{spec} instance Functor (Either e) where fmap _ (Left e) = Left e fmap g (Right a) = Right (g a) instance Functor ((->) e) where fmap = (.) \end{spec} \end{solution} \end{exercise} \begin{exercise} Implement |Functor| instances for |((,) e)| and for |Pair|, defined as \begin{code} data Pair a = Pair a a \end{code} Explain their similarities and differences. \begin{solution} \begin{spec} instance Functor ((,) e) where fmap g (e,a) = (e, g a) instance Functor Pair where fmap g (Pair x y) = Pair (g x) (g y) \end{spec} |((,) e)| is the type of values with an ``annotation'' of type |e|; the two elements of the pair are not equal citizens. |fmap g| only applies |g| the value, not the annotation (indeed, in view of the types, this is the only thing it could do). |Pair|, on the other hand, stores a pair of values, so |fmap g| applies |g| to both (again, there is no other choice). \end{solution} \end{exercise} \begin{exercise} Give an example of a type which cannot be made an instance of |Functor| (without using |undefined|). \begin{solution} A good source of examples are types containing the type parameter in a \emph{negative} position (to the left of an odd number of arrows). For example, |(a -> Int)| cannot be given a |Functor| instance. \end{solution} \end{exercise} \subsection{Laws} As far as the Haskell language itself is concerned, the only requirement to be a |Functor| is an implementation of |fmap| with the proper type. Any sensible |Functor| instance, however, will also satisfy the \term{functor laws}, which are part of the definition of a mathematical functor. There are two, shown in \pref{lst:functor-laws}; together, these laws ensure that |fmap g| does not change the \emph{structure} of a container, only the elements. Equivalently, and more simply, they ensure that |fmap g| changes a value without altering its context.\footnote{Technically, these laws make |f| and |fmap| together an endofunctor on \term{Hask}, the category of Haskell types (ignoring $\bot$, which is a party pooper).~\cite{wikibook-ct}} \begin{listing}[htp] \begin{code} fmap id = id fmap (g . h) = fmap g . fmap h \end{code} \caption{The |Functor| laws} \label{lst:functor-laws} \end{listing} The first law says that mapping the identity function over every item in a container has no effect. The second says that mapping a composition of two functions over every item in a container is the same as first mapping one function, and then mapping the other. As an example, the code shown in \pref{lst:functor-law-violation} is a ``valid'' instance of |Functor| (it typechecks), but it violates the functor laws. Do you see why? \begin{listing}[htp] \begin{code} instance Functor [] where fmap _ [] = [] fmap g (x:xs) = g x : g x : fmap g xs \end{code} \caption{A lawless |Functor| instance} \label{lst:functor-law-violation} \end{listing} Any Haskeller worth their salt would reject the code in \pref{lst:functor-law-violation} as a gruesome abomination. Unlike some other type classes we will encounter, a given type has at most one valid instance of |Functor|. This can be proven~\cite{roconnor-functor-unique} via the \term{free theorem}~\cite{wadler89free} for the type of |fmap|. In fact, the same argument shows that any |Functor| instance satisfying the first law (|fmap id = id|) will automatically satisfy the second law as well. Practically, this means that only the first law needs to be checked (usually by a very straightforward induction) to ensure that a |Functor| instance is valid. \begin{exercise} Although it is not possible for a |Functor| instance to satisfy the first |Functor| law but not the second, the reverse is possible. Give an example of a (bogus) |Functor| instance which satisfies the second law but not the first. \begin{solution} Here is one possible solution: \begin{spec} newtype Foo a = Foo [a] instance Functor Foo where fmap _ _ = Foo [] \end{spec} \end{solution} \end{exercise} \subsection{Intuition} \label{sec:functor-intuition} There are two fundamental ways to think about |fmap|. The first has already been mentioned: it takes two parameters, a function and a container, and applies the function ``inside'' the container, producing a new container. Alternately, we can think of |fmap| as applying a function to a value in a context (without altering the context). Just like all other Haskell functions of ``more than one parameter'', however, |fmap| is actually \term{curried}: it does not really take two parameters, but takes a single parameter and returns a function. For emphasis, we can write |fmap|'s type with extra parentheses: |fmap :: (a -> b) -> (f a -> f b)|. Written in this form, it is apparent that |fmap| transforms a ``normal'' function (|g :: a -> b|) into one which operates over %% XXX make sure this doesn't hang into the margin in the final version containers/contexts (|fmap g :: f a -> f b|). This transformation is often referred to as a \term{lift}; |fmap| ``lifts'' a function from the ``normal world'' into the ``|f| world''. \subsection{Further reading} \label{sec:functor-reading} A good starting point for reading about the category theory behind the concept of a functor is the excellent Haskell wikibook page on category theory~\cite{wikibook-ct}. %% XXX reorganize to introduce Apply class first, instead of Pointed? %% see http://hackage.haskell.org/package/functor-apply. %% Talk to copumpkin and edwardk. \section{Apply*} \label{sec:apply} The |Apply| type class\dots XXX write me. Much of the discussion from Applicative should be moved here. % \section{Pointed*} % \label{sec:pointed} % The |Pointed| type class represents \term{pointed functors}. It is not % actually a type class in the standard libraries (hence the % asterisk).\footnote{It is, however, a type class in the % category-extras library~\cite{category-extras}.} But it % \emph{could} be, and it's useful in understanding a few other % type classes, notably |Applicative| and |Monad|, so let's pretend for % a minute. % Given a |Functor|, the |Pointed| class represents the additional ability to put % a value into a ``default context''. Often, this corresponds to creating % a container with exactly one element, but it is more general than % that. The type class declaration for |Pointed| is shown in % \pref{lst:pointed}. % \begin{listing}[htp] % \begin{code} % class Functor f => Pointed f where % pure :: a -> f a -- aka singleton, return, unit, point % \end{code} % \caption{The |Pointed| type class} \label{lst:pointed} % \end{listing} % Most of the standard |Functor| instances could also be instances of % |Pointed|---for example, the |Maybe| instance of |Pointed| is |pure = % Just|; there are many possible implementations for lists, the most % natural of which is |pure x = [x]|; for |((->) e)| it is\dots well, % I'll let you work it out. (Just follow the types!) % One example of a |Functor| which is not |Pointed| is |((,) e)|. If % you try implementing |pure :: a -> (e,a)| you will quickly see why: % since the type |e| is completely arbitrary, there is no way to % generate a value of type |e| out of thin air! However, as we will % see, |((,) e)| can be made |Pointed| if we place an additional % restriction on |e| which allows us to generate a default value of type % |e| (the most common solution is to make |e| an instance of |Monoid|). % The |Pointed| class has only one law, shown in % \pref{lst:pointed-law}.\footnote{For those interested in category % theory, this law states precisely that |pure| is a natural % transformation from the identity functor to |f|.} % \begin{listing}[htp] % \begin{code} % fmap g . pure = pure . g % \end{code} % \caption{The |Pointed| law} \label{lst:pointed-law} % \end{listing} % However, you need not worry about it: this law is actually a so-called % ``free theorem'' guaranteed by parametricity~\cite{wadler89free}; it's % impossible to write an instance of |Pointed| which does not satisfy % it.\footnote{\dots modulo $\bot$, |seq|, and assuming a lawful |Functor| % instance.} \section{Applicative} \label{sec:applicative} A somewhat newer addition to the pantheon of standard Haskell type classes, applicative functors~\cite{applicative-docs} represent an abstraction lying exactly in between |Functor| and |Monad|, first described by McBride and Paterson~\cite{mcbride08applicative}. The title of McBride and Paterson's classic paper, \emph{Applicative Programming with Effects}, gives a hint at the intended intuition behind the |Applicative| type class. It encapsulates certain sorts of ``effectful'' computations in a functionally pure way, and encourages an ``applicative'' programming style. Exactly what these things mean will be seen later. \subsection{Definition} \label{sec:applicative-definition} The |Applicative| class adds a single capability to |Pointed| functors. Recall that |Functor| allows us to lift a ``normal'' function to a function on computational contexts. But |fmap| doesn't allow us to apply a function which is itself in a context to a value in another context. |Applicative| gives us just such a tool. \pref{lst:applicative} shows the type class declaration for |Applicative|, which is defined in |Control.Applicative|. Note that every |Applicative| must also be a |Functor|. In fact, as we will see, |fmap| can be implemented using the |Applicative| methods, so every |Applicative| is a functor whether we like it or not; the |Functor| constraint forces us to be honest. \begin{listing}[htp] \begin{code} class Functor f => Applicative f where pure :: a -> f a (<*>) :: f (a -> b) -> f a -> f b \end{code} \caption{The |Applicative| type class} \label{lst:applicative} \end{listing} As always, it's crucial to understand the type signature of |(<*>)|. The best way of thinking about it comes from noting that the type of |(<*>)| is similar to the type of |($)|,\footnote{Recall that |($)| is just function application: |f $ x = f x|.} but with everything enclosed in an |f|. % $ In other words, |(<*>)| is just function application within a computational context. The type of |(<*>)| is also very similar to the type of |fmap|; the only difference is that the first parameter is |f (a -> b)|, a function in a context, instead of a ``normal'' function |(a -> b)|. Of course, |pure| looks rather familiar. If we actually had a |Pointed| type class, |Applicative| could instead be defined as shown in \pref{lst:applicative-pointed}. \begin{listing}[htp] \begin{code} class Pointed f => Applicative' f where (<*>) :: f (a -> b) -> f a -> f b \end{code} \caption{Alternate definition of |Applicative| using |Pointed|} \label{lst:applicative-pointed} \end{listing} \subsection{Laws} \label{sec:applicative-laws} There are several laws that |Applicative| instances should satisfy~\cite{applicative-docs, mcbride08applicative}, but only one is crucial to developing intuition, because it specifies how |Applicative| should relate to |Functor| (the other four mostly specify the exact sense in which |pure| deserves its name). This law is shown in \pref{lst:applicative-law-functor}. \begin{listing}[htp] \begin{code} fmap g x = pure g <*> x \end{code} \caption{Law relating |Applicative| to |Functor|} \label{lst:applicative-law-functor} \end{listing} The law says that mapping a pure function |g| over a context |x| is the same as first injecting |g| into a context with |pure|, and then applying it to |x| with |(<*>)|. In other words, we can decompose |fmap| into two more atomic operations: injection into a context, and application within a context. The |Control.Applicative| module also defines |(<$>)| %$ as a synonym for |fmap|, so the above law can also be expressed as % |g <$> x = pure g <*> x|. %$ \subsection{Instances} \label{sec:applicative-instances} Most of the standard types which are instances of |Functor| are also instances of |Applicative|. |Maybe| can easily be made an instance of |Applicative|; writing such an instance is left as an exercise for the reader. The list type constructor |[]| can actually be made an instance of |Applicative| in two ways; essentially, it comes down to whether we want to think of lists as ordered collections of elements, or as contexts representing multiple results of a nondeterministic computation~\cite{wadler85successes}. Let's first consider the collection point of view. Since there can only be one instance of a given type class for any particular type, one or both of the list instances of |Applicative| need to be defined for a |newtype| wrapper; as it happens, the nondeterministic computation instance is the default, and the collection instance is defined in terms of a |newtype| called |ZipList|. This instance is shown in \pref{lst:ziplist}. \begin{listing}[htp] \begin{code} newtype ZipList a = ZipList { getZipList :: [a] } instance Applicative ZipList where pure = undefined -- exercise (ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) gs xs) \end{code} \caption{|ZipList| instance of |Applicative|} \label{lst:ziplist} \end{listing} % $ To apply a list of functions to a list of inputs with |(<*>)|, we just match up the functions and inputs elementwise, and produce a list of the resulting outputs. In other words, we ``zip'' the lists together with function application, |($)|; hence the name |ZipList|. As an %$ exercise, determine the correct definition of |pure|---there is only one implementation that satisfies the law in \pref{lst:applicative-law-functor}. The other |Applicative| instance for lists, based on the nondeterministic computation point of view, is shown in \pref{lst:lst-applicative}. Instead of applying functions to inputs pairwise, we apply each function to all the inputs in turn, and collect all the results in a list. \begin{listing}[htp] \begin{code} instance Applicative [] where pure x = [x] gs <*> xs = [ g x | g <- gs, x <- xs ] \end{code} \caption{|[]| instance of |Applicative|} \label{lst:lst-applicative} \end{listing} Now we can write nondeterministic computations in a natural style. To add the numbers |3| and |4| deterministically, we can of course write |(+) 3 4|. But suppose instead of |3| we have a nondeterministic computation that might result in |2|, |3|, or~|4|; then we can write \begin{code} pure (+) <*> [2,3,4] <*> pure 4 \end{code} or, more idiomatically, \begin{code} (+) <$> [2,3,4] <*> pure 4. \end{code} %$ There are several other |Applicative| instances as well: \begin{itemize} \item |IO| is an instance of |Applicative|, and behaves exactly as you would think: when |g <$> m1 <*> m2 <*> m3| is executed, the effects from the |mi|'s happen in order from left to right. %$ \item |((,) a)| is an |Applicative|, as long as |a| is an instance of |Monoid| (\pref{sec:monoid}). The |a| values are accumulated in parallel with the computation. \item The |Applicative| module defines the |Const| type constructor; a value of type |Const a b| simply contains an |a|. This is an instance of |Applicative| for any |Monoid a|; this instance becomes especially useful in conjunction with things like |Foldable| (\pref{sec:foldable}). \item The |WrappedMonad| and |WrappedArrow| newtypes make any instances of |Monad| (\pref{sec:monad}) or |Arrow| (\pref{sec:arrow}) respectively into instances of |Applicative|; as we will see when we study those type classes, both are strictly more expressive than |Applicative|, in the sense that the |Applicative| methods can be implemented in terms of their methods. \end{itemize} \subsection{Intuition} \label{sec:applicative-intuition} McBride and Paterson's paper introduces the notation $\llbracket g \; x_1 \; x_2 \; \cdots \; x_n \rrbracket$ to denote function application in a computational context. If each $x_i$ has type $f \; t_i$ for some applicative functor $f$, and $g$ has type $t_1 \to t_2 \to \dots \to t_n \to t$, then the entire expression $\llbracket g \; x_1 \; \cdots \; x_n \rrbracket$ has type $f \; t$. You can think of this as applying a function to multiple ``effectful'' arguments. In this sense, the double bracket notation is a generalization of |fmap|, which allows us to apply a function to a single argument in a context. Why do we need |Applicative| to implement this generalization of |fmap|? Suppose we use |fmap| to apply |g| to the first parameter |x1|. Then we get something of type |f (t2 -> ... t)|, but now we are stuck: we can't apply this function-in-a-context to the next argument with |fmap|. However, this is precisely what |(<*>)| allows us to do. This suggests the proper translation of the idealized notation $\llbracket g \; x_1 \; x_2 \; \cdots \; x_n \rrbracket$ into Haskell, namely \begin{code} g <$> x1 <*> x2 <*> ... <*> xn, \end{code} %$ recalling that |Control.Applicative| defines |(<$>)| as a convenient %$ infix shorthand for |fmap|. This is what is meant by an ``applicative style''---effectful computations can still be described in terms of function application; the only difference is that we have to use the special operator |(<*>)| for application instead of simple juxtaposition. \subsection{Further reading} \label{sec:applicative-further-reading} There are many other useful combinators in the standard libraries implemented in terms of |pure| and |(<*>)|: for example, |(*>)|, |(<*)|, |(<**>)|, |(<$)|, and so on~\cite{applicative-docs}. %$ Judicious use of such secondary combinators can often make code using |Applicative|s much easier to read. McBride and Paterson's original paper~\cite{mcbride08applicative} is a treasure-trove of information and examples, as well as some perspectives on the connection between |Applicative| and category theory. Beginners will find it difficult to make it through the entire paper, but it is extremely well-motivated---even beginners will be able to glean something from reading as far as they are able. Conal Elliott has been one of the biggest proponents of |Applicative|. For example, the Pan library for functional images~\cite{Elliott03:FOP} and the reactive library for functional reactive programming (FRP)~\cite{Elliott08:FRP} make key use of it; his blog also contains many examples of |Applicative| in action~\cite{conal-blog-applicative}. Building on the work of McBride and Paterson, Elliott also built the TypeCompose library~\cite{TypeCompose}, which embodies the observation (among others) that |Applicative| types are closed under composition; therefore, |Applicative| instances can often be automatically derived for complex types built out of simpler ones. Although the Parsec parsing library~\cite{parsec, leijen01parsec} was originally designed for use as a monad, in its most common use cases an |Applicative| instance can be used to great effect; Bryan O'Sullivan's blog post is a good starting point~\cite{bos08applicative}. If the extra power provided by |Monad| isn't needed, it's usually a good idea to use |Applicative| instead. A couple other nice examples of |Applicative| in action include the ConfigFile and HSQL libraries~\cite{done09config-hsql} and the formlets library~\cite{CLWY08idiomsguide}. \section{Monad} \label{sec:monad} It's a safe bet that if you're reading this, you've heard of monads---although it's quite possible you've never heard of |Applicative| before, or |Arrow|, or even |Monoid|. Why are monads such a big deal in Haskell? There are several reasons. \begin{itemize} \item Haskell does, in fact, single out monads for special attention by making them the framework in which to construct I/O operations. \item Haskell also singles out monads for special attention by providing a special syntactic sugar for monadic expressions: the |do|-notation. \item |Monad| has been around longer than other abstract models of computation such as |Applicative| or |Arrow|. \item The more monad tutorials there are, the harder people think monads must be, and the more new monad tutorials are written by people who think they finally ``get'' monads~\cite{monad-tutorial-fallacy}. \end{itemize} I will let you judge for yourself whether these are good reasons. In the end, despite all the hoopla, |Monad| is just another type class. Let's take a look at its definition. \subsection{Definition} \label{sec:monad-definition} The type class declaration for |Monad|~\cite{monad-docs} is shown in \pref{lst:monad}. The |Monad| type class is exported by the |Prelude|, along with a few standard instances. However, many utility functions are found in |Control.Monad|, and there are also several instances (such as |((->) e)|) defined in |Control.Monad.Instances|. \begin{listing}[htp] \begin{code} class Monad m where return :: a -> m a (>>=) :: m a -> (a -> m b) -> m b (>>) :: m a -> m b -> m b m >> n = m >>= \_ -> n fail :: String -> m a \end{code} \caption{The |Monad| type class} \label{lst:monad} \end{listing} Let's examine the methods in the |Monad| class one by one. The type of |return| should look familiar; it's the same as |pure|. Indeed, |return| \emph{is} |pure|, but with an unfortunate name. (Unfortunate, since someone coming from an imperative programming background might think that |return| is like the C or Java keyword of the same name, when in fact the similarities are minimal.) From a mathematical point of view, every monad is an applicative functor, but for historical reasons (\ie because |Monad| has been around longer than |Applicative|), the |Monad| type class declaration does not (yet) require this. We can see that |(>>)| is a specialized version of |(>>=)|, with a default implementation given. It is only included in the type class declaration so that specific instances of |Monad| can override the default implementation of |(>>)| with a more efficient one, if desired. % Also, note that although |_ >> n = n| would be a % type-correct implementation of |(>>)|, it would not correspond to the % intended semantics: the intention is that |m >> n| ignores the % \emph{result} of |m|, but not its \emph{effects}. The |fail| function is an awful hack that has no place in the |Monad| class; more on this later. The only really interesting thing to look at---and what makes |Monad| strictly more powerful than |Applicative|---is |(>>=)|, which is often called \emph{bind}. An alternative definition of |Monad| could look like \pref{lst:monad-alt1}. \begin{listing}[htp] \begin{code} class Applicative m => Monad' m where (>>=) :: m a -> (a -> m b) -> m b \end{code} \caption{An alternative definition of |Monad|} \label{lst:monad-alt1} \end{listing} We could spend a while talking about the intuition behind |(>>=)|---and we will. But first, let's look at some examples. \subsection{Instances} \label{sec:monad-instances} Even if you don't understand the intuition behind the |Monad| class, you can still create instances of it by just seeing where the types lead you. You may be surprised to find that this actually gets you a long way towards understanding the intuition; at the very least, it will give you some concrete examples to play with as you read more about the |Monad| class in general. The first few examples are from the standard |Prelude|; the remaining examples are from the monad transformer library (mtl)~\cite{mtl}. \begin{itemize} \item The simplest possible instance of |Monad| is |Identity|~\cite{identity-docs}, which is described in Dan Piponi's highly recommended blog post on ``The Trivial Monad''~\cite{piponi-07-trivial-monad}. Despite being ``trivial'', it is a great introduction to the |Monad| type class, and contains some good exercises to get your brain working. \item The next simplest instance of |Monad| is |Maybe|. We already know how to write |return|/|pure| for |Maybe|. So how do we write |(>>=)|? Well, let's think about its type. Specializing for |Maybe|, we have \begin{code} (>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b. \end{code} If the first argument to |(>>=)| is |Just x|, then we have something of type |a| (namely, |x|), to which we can apply the second argument---resulting in a |Maybe b|, which is exactly what we wanted. What if the first argument to |(>>=)| is |Nothing|? In that case, we don't have anything to which we can apply the |a -> Maybe b| function, so there's only one thing we can do: yield |Nothing|. This instance is shown in \pref{lst:maybe-monad}. We can already get a bit of intuition as to what is going on here: if we build up a computation by chaining together a bunch of functions with |(>>=)|, as soon as any one of them fails, the entire computation will fail (because |Nothing >>= f| is |Nothing|, no matter what |f| is). The entire computation succeeds only if all the constituent functions individually succeed. So the |Maybe| monad models computations which may fail. \end{itemize} \begin{listing}[htp] \begin{code} instance Monad Maybe where return = Just (Just x) >>= g = g x Nothing >>= _ = Nothing \end{code} \caption{The |Maybe| instance of |Monad|} \label{lst:maybe-monad} \end{listing} \begin{itemize} \item The |Monad| instance for the list constructor |[]| is similar to its |Applicative| instance; see \pref{ex:list-monad}. \item Of course, the |IO| constructor is famously a |Monad|, but its implementation is somewhat magical, and may in fact differ from compiler to compiler. It is worth emphasizing that the |IO| monad is the \emph{only} monad which is magical. It allows us to build up, in an entirely pure way, values representing possibly effectful computations. The special value |main|, of type % |IO ()|, is taken by the runtime and actually executed, producing actual effects. Every other monad is functionally pure, and requires no special compiler support. We often speak of monadic values as ``effectful computations'', but this is because some monads allow us to write code \emph{as if} it has side effects, when in fact the monad is hiding the plumbing which allows these apparent side effects to be implemented in a functionally pure way. \item As mentioned earlier, |((->) e)| is known as the \emph{reader monad}, since it describes computations in which a value of type |e| is available as a read-only environment; see \pref{ex:reader-monad}. The |Control.Monad.Reader| module~\cite{reader-docs} provides the |Reader e a| type, which is just a convenient |newtype| wrapper around |(e -> a)|, along with an appropriate |Monad| instance and some |Reader|-specific utility functions such as |ask| (retrieve the environment), |asks| (retrieve a function of the environment), and |local| (run a subcomputation under a different environment). \item The |Control.Monad.Writer| module~\cite{writer-docs} provides the |Writer| monad, which allows information to be collected as a computation progresses. |Writer w a| is isomorphic to |(a,w)|, where the output value |a| is carried along with an annotation or ``log'' of type |w|, which must be an instance of |Monoid| (\pref{sec:monoid}); the special function |tell| performs logging. \item The |Control.Monad.State| module~\cite{state-docs} provides the |State s a| type, a |newtype| wrapper around |s -> (a,s)|. Something of type |State s a| represents a stateful computation which produces an |a| but can access and modify the state of type |s| along the way. The module also provides |State|-specific utility functions such as |get| (read the current state), |gets| (read a function of the current state), |put| (overwrite the state), and |modify| (apply a function to the state). \item The |Control.Monad.Cont| module~\cite{cont-docs} provides the |Cont| monad, which represents computations in continuation-passing style. It can be used to suspend and resume computations, and to implement non-local transfers of control, co-routines, other complex control structures---all in a functionally pure way. |Cont| has been called the ``mother of all monads''~\cite{piponi-mother-of-monads} because of its universal properties. \end{itemize} \begin{exercise} \label{ex:list-monad} Implement a |Monad| instance for the list constructor |[]|. Follow the types! \begin{solution} % XXX write me \end{solution} \end{exercise} \begin{exercise} \label{ex:reader-monad} Implement a |Monad| instance for |((->) e)|. \begin{solution} % XXX write me \end{solution} \end{exercise} \subsection{Intuition} \label{sec:monad-intuition} Let's look more closely at the type of |(>>=)|. The basic intuition is that it combines % \footnote{We should be careful to use the word % `combine' to describe what |(>>=)| does, not `sequence'; a common % misconception is that monads have something to do with sequencing % effects. In fact, monads have nothing inherently to do with % sequencing---no more than, say, function composition. Sequencing in % Haskell is driven by data dependencies: if value |a| is needed in % order to compute value |b|, then |a| must be computed before |b|. % Certain monads---notably |State| and |IO|---are structured in order % to enforce sequencing of effects via data dependencies, but this is % a particular consequence of the way they are defined, not because of % anything inherent in the |Monad| type class itself.} two computations into one larger computation. The first argument, |m a|, is the first computation. However, it would be boring if the second argument were just an |m b|; then there would be no way for the computations to interact with one another. So, the second argument to |(>>=)| has type |a -> m b|: a function of this type, given a \emph{result} of the first computation, can produce a second computation to be run. In other words, |x >>= k| is a computation which runs |x|, and then uses the result(s) of |x| to \emph{decide} what computation to run second, using the output of the second computation as the result of the entire computation. Intuitively, it is this ability to use the output from previous computations to decide what computations to run next that makes |Monad| more powerful than |Applicative|. The structure of an |Applicative| computation is fixed, whereas the structure of a |Monad| computation can change based on intermediate results. To see the increased power of |Monad| from a different point of view, let's see what happens if we try to implement |(>>=)| in terms of |fmap|, |pure|, and |(<*>)|. We are given a value |x| of type |m a|, and a function |k| of type |a -> m b|, so the only thing we can do is apply |k| to |x|. We can't apply it directly, of course; we have to use |fmap| to lift it over the |m|. But what is the type of |fmap k|? Well, it's |m a -> m (m b)|. So after we apply it to |x|, we are left with something of type |m (m b)|---but now we are stuck; what we really want is an |m b|, but there's no way to get there from here. We can \emph{add} |m|'s using |pure|, but we have no way to \emph{collapse} multiple |m|'s into one. This ability to collapse multiple |m|'s is exactly the ability provided by the function |join :: m (m a) -> m a|, and it should come as no surprise that an alternative definition of |Monad| can be given in terms of |join|, as shown in \pref{lst:monad-alt2-join}. \begin{listing}[htp] \begin{code} class Applicative m => Monad'' m where join :: m (m a) -> m a \end{code} \caption{An alternative definition of |Monad| in terms of |join|} \label{lst:monad-alt2-join} \end{listing} % XXX come back to this. Note that both definitions come from CT? In fact, in a category theory context, monads are often defined in terms of |return|, |fmap|, and |join| (usually called $\eta$, $T$, and $\mu$ in the mathematical literature). Haskell uses the equivalent formulation in terms of |(>>=)| instead of |join| since it is more convenient to use; however, sometimes it can be easier to think about |Monad| instances in terms of |join|, since it is a more ``atomic'' operation. (For example, |join| for the list monad is just |concat|.) \begin{exercise} Implement |(>>=)| in terms of |fmap| and |join|. \begin{solution} % XXX write me \end{solution} \end{exercise} \begin{exercise} Implement |join| in terms of |(>>=)|. \begin{solution} % XXX write me \end{solution} \end{exercise} \subsection{Utility functions} \label{sec:monad-utilities} The |Control.Monad| module~\cite{control-monad-docs} provides a large number of convenient utility functions, all of which can be implemented in terms of the basic |Monad| operations (|return| and |(>>=)| in particular). We have already seen one of them, namely, |join|. We also mention some other noteworthy ones here; implementing these utility functions oneself is a good exercise. For a more detailed guide to these functions, with commentary and example code, see Henk-Jan van Tuyl's tour~\cite{vanTuyl-monad-tour}. \begin{itemize} \item |liftM :: Monad m => (a -> b) -> m a -> m b|. This should be familiar; of course, it is just |fmap|. The fact that we have both |fmap| and |liftM| is an unfortunate consequence of the fact that the |Monad| type class does not require a |Functor| instance, even though mathematically speaking, every monad is a functor. However, |fmap| and |liftM| are essentially interchangeable, since it is a bug (in a social rather than technical sense) for any type to be an instance of |Monad| without also being an instance of |Functor|. \item |ap :: Monad m => m (a -> b) -> m a -> m b| should also be familiar: it is equivalent to |(<*>)|, justifying the claim that the |Monad| interface is strictly more powerful than |Applicative|. We can make any |Monad| into an instance of |Applicative| by setting |pure = return| and |(<*>) = ap|. \item |sequence :: Monad m => [m a] -> m [a]| takes a list of computations and combines them into one computation which collects a list of their results. It is again something of a historical accident that |sequence| has a |Monad| constraint, since it can actually be implemented only in terms of |Applicative|. There is an additional generalization of |sequence| to structures other than lists, which will be discussed in the section on |Traversable| (\pref{sec:traversable}). \item |replicateM :: Monad m => Int -> m a -> m [a]| is simply a combination of |replicate| and |sequence|. \item |when :: Monad m => Bool -> m () -> m ()| conditionally executes a computation, evaluating to its second argument if the test is |True|, and to |return ()| if the test is |False|. A collection of other sorts of monadic conditionals can be found in the |IfElse| package \cite{IfElse}. \item |mapM :: Monad m => (a -> m b) -> [a] -> m [b]| maps its first argument over the second, and |sequence|s the results. The |forM| function is just |mapM| with its arguments reversed; it is called |forM| since it models generalized |for| loops: the list |[a]| provides the loop indices, and the function |a -> m b| specifies the ``body'' of the loop for each index. \item |(=<<) :: Monad m => (a -> m b) -> m a -> m b| is just |(>>=)| with its arguments reversed; sometimes this direction is more convenient since it corresponds more closely to function application. \item |(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c| is sort of like function composition, but with an extra |m| on the result type of each function, and the arguments swapped. We'll have more to say about this operation later. There is also a flipped variant, |(<=<)|. \item The |guard| function is for use with instances of |MonadPlus|, which is discussed at the end of the |Monoid| section. \end{itemize} Many of these functions also have ``underscored'' variants, such as |sequence_| and |mapM_|; these variants throw away the results of the computations passed to them as arguments, using them only for their side effects. Other monadic functions which are occasionally useful include |filterM|, |zipWithM|, |foldM|, and |forever|. \subsection{Laws} \label{sec:monad-laws} There are several laws that instances of |Monad| should satisfy \cite{monad-laws}. The standard presentation is shown in \pref{lst:monad-laws}. \begin{listing}[htp] \begin{code} return a >>= k = k a m >>= return = m m >>= (\x -> k x >>= h) = (m >>= k) >>= h fmap g xs = xs >>= return . g = liftM g xs \end{code} \caption{The |Monad| laws} \label{lst:monad-laws} \end{listing} The first and second laws express the fact that |return| behaves nicely: if we inject a value |a| into a monadic context with |return|, and then bind to |k|, it is the same as just applying |k| to |a| in the first place; if we bind a computation |m| to |return|, nothing changes. The third law essentially says that |(>>=)| is associative, sort of. The last law ensures that |fmap| and |liftM| are the same for types which are instances of both |Functor| and |Monad|---which, as already noted, should be every instance of |Monad|. However, the presentation of the above laws, especially the third, is marred by the asymmetry of |(>>=)|. It's hard to look at the laws and see what they're really saying. I prefer a much more elegant version of the laws, which is formulated in terms of |(>=>)|.\footnote{I like to pronounce this operator ``fish'', but that's probably not the canonical pronunciation\dots} Recall that |(>=>)| ``composes'' two functions of type |a -> m b| and |b -> m c|. You can think of something of type |a -> m b| (roughly) as a function from |a| to |b| which may also have some sort of effect in the context corresponding to |m|. (Note that |return| is such a function.) |(>=>)| lets us compose these ``effectful functions'', and we would like to know what properties |(>=>)| has. The monad laws reformulated in terms of |(>=>)| are shown in \pref{lst:monad-laws-fish}. \begin{listing}[htp] \begin{code} return >=> g = g g >=> return = g (g >=> h) >=> k = g >=> (h >=> k) \end{code} \caption{The |Monad| laws, reformulated in terms of |(>=>)|} \label{lst:monad-laws-fish} \end{listing} Ah, much better! The laws simply state that |return| is the identity of |(>=>)|, and that |(>=>)| is associative.\footnote{As fans of category theory will note, these laws say precisely that functions of type |a -> m b| are the arrows of a category with |(>=>)| as composition! Indeed, this is known as the \emph{Kleisli category} of the monad |m|. It will come up again when we discuss |Arrow|s.} Working out the equivalence between these two formulations, given the definition |g >=> h = \x -> g x >>= h|, is left as an exercise. There is also a formulation of the monad laws in terms of |fmap|, |return|, and |join|; for a discussion of this formulation, see the Haskell wikibook page on category theory~\cite{wikibook-ct}. \subsection{|do| notation} \label{sec:do-notation} Haskell's special |do| notation supports an ``imperative style'' of programming by providing syntactic sugar for chains of monadic expressions. The genesis of the notation lies in realizing that something like |a >>= \x -> b >> c >>= \y -> d | can be more readably written by putting successive computations on separate lines: \begin{code} a >>= \x -> b >> c >>= \y -> d \end{code} This emphasizes that the overall computation consists of four computations |a|, |b|, |c|, and |d|, and that |x| is bound to the result of |a|, and |y| is bound to the result of |c| (|b|, |c|, and |d| are allowed to refer to |x|, and |d| is allowed to refer to |y| as well). From here it is not hard to imagine a nicer notation: \begin{code} do { x <- a ; b ; y <- c ; d } \end{code} (The curly braces and semicolons may optionally be omitted; the Haskell parser uses layout to determine where they should be inserted.) This discussion should make clear that |do| notation is just syntactic sugar. In fact, |do| blocks are recursively translated into monad operations (almost) as shown in \pref{lst:do-translation}. \begin{listing}[htp] \begin{align*} |do | e &\longrightarrow e \\ |do | \{e; \mathit{stmts}\} &\longrightarrow e | >> do | \{ \mathit{stmts} \} \\ |do | \{v | <- | e; \mathit{stmts}\} &\longrightarrow e | >>= \|v | -> do | \{stmts\} \\ |do | \{ |let | \mathit{decls}; \mathit{stmts}\} &\longrightarrow |let | \mathit{decls} | in do | \{\mathit{stmts}\} \end{align*} \caption{Desugaring of |do| blocks (almost)} \label{lst:do-translation} \end{listing} This is not quite the whole story, since $v$ might be a pattern instead of a variable. For example, one can write \begin{code} do (x:xs) <- foo bar x \end{code} but what happens if |foo| produces an empty list? Well, remember that ugly |fail| function in the |Monad| type class declaration? That's what happens. See section 3.14 of the Haskell Report for the full details~\cite{haskell-report}. See also the discussion of |MonadPlus| and |MonadZero| (\pref{sec:other-monoidal}). A final note on intuition: |do| notation plays very strongly to the ``computational context'' point of view rather than the ``container'' point of view, since the binding notation |x <- m| is suggestive of ``extracting'' a single |x| from |m| and doing something with it. But |m| may represent some sort of a container, such as a list or a tree; the meaning of |x <- m| is entirely dependent on the implementation of |(>>=)|. For example, if |m| is a list, |x <- m| actually means that |x| will take on each value from the list in turn. \subsection{Monad transformers} \label{sec:monad-transformers} One would often like to be able to combine two monads into one: for example, to have stateful, nondeterministic computations (|State| + |[]|), or computations which may fail and can consult a read-only environment (|Maybe| + |Reader|), and so on. Unfortunately, monads do not compose as nicely as applicative functors (yet another reason to use |Applicative| if you don't need the full power that |Monad| provides), but some monads can be combined in certain ways. The monad transformer library~\cite{mtl} provides a number of \emph{monad transformers}, such as |StateT|, |ReaderT|, |ErrorT| \cite{error-docs}, and (soon) |MaybeT|, which can be applied to other monads to produce a new monad with the effects of both. For example, |StateT s Maybe| is an instance of |Monad|; computations of type |StateT s Maybe a| may fail, and have access to a mutable state of type |s|. These transformers can be multiply stacked. One thing to keep in mind while using monad transformers is that the order of composition matters. For example, when a |StateT s Maybe a| computation fails, the state ceases being updated; on the other hand, the state of a |MaybeT (State s) a| computation may continue to be modified even after the computation has failed. (This may seem backwards, but it is correct. Monad transformers build composite monads ``inside out''; for example, |MaybeT (State s) a| is isomorphic to |s -> (Maybe a, s)|. Lambdabot has an indispensable |@unmtl| command which you can use to ``unpack'' a monad transformer stack in this way.) All monad transformers should implement the |MonadTrans| type class (\pref{lst:monadtrans}), defined in |Control.Monad.Trans|. It allows arbitrary computations in the base monad |m| to be ``lifted'' into computations in the transformed monad |t m|. (Note that type application associates to the left, just like function application, so |t m a = (t m) a|. As an exercise, you may wish to work out |t|'s kind, which is rather more interesting than most of the kinds we've seen up to this point.) However, you should only have to think about |MonadTrans| when defining your own monad transformers, not when using predefined ones. \begin{listing}[htp] \begin{code} class MonadTrans t where lift :: Monad m => m a -> t m a \end{code} \caption{The |MonadTrans| type class} \label{lst:monadtrans} \end{listing} There are also type classes such as |MonadState|, which provides state-specific methods like |get| and |put|, allowing you to conveniently use these methods not only with |State|, but with any monad which is an instance of |MonadState|---including % |MaybeT (State s)|, % |StateT s (ReaderT r IO)|, and so on. Similar type classes exist for |Reader|, |Writer|, |Cont|, |IO|, and others.\footnote{The only problem with this scheme is the quadratic number of instances required as the number of standard monad transformers grows---but as the current set of standard monad transformers seems adequate for most common use cases, this may not be that big of a deal.} There are two excellent references on monad transformers. Martin Grabm\"uller's \emph{Monad Transformers Step by Step} \cite{grabmueller-06-step-by-step} is a thorough description, with running examples, of how to use monad transformers to elegantly build up computations with various effects. Cale Gibbard's article on how to use monad transformers~\cite{gibbard-monad-trans} is more practical, describing how to structure code using monad transformers to make writing it as painless as possible. Another good starting place for learning about monad transformers is a blog post by Dan Piponi \cite{piponi-grok-transformers}. \subsection{MonadFix} \label{sec:monadfix} The |MonadFix| class describes monads which support the special fixpoint operation |mfix :: (a -> m a) -> m a|, which allows the output of monadic computations to be defined via recursion. This is supported in GHC and Hugs by a special ``recursive do'' notation, |mdo|. For more information, see Levent Erk\"ok's thesis, \emph{Value Recursion in Monadic Computations}~\cite{erkok-monadfix}. \subsection{Further reading} \label{sec:monad-further-reading} Philip Wadler was the first to propose using monads to structure functional programs~\cite{wadler-92-essence}. His paper is still a readable introduction to the subject. Much of the monad transformer library (mtl)~\cite{mtl}, including the |Reader|, |Writer|, |State|, and other monads, as well as the monad transformer framework itself, was inspired by Mark Jones's classic paper \emph{Functional Programming with Overloading and Higher-Order Polymorphism}~\cite{jones-95-fp}. It's still very much worth a read---and highly readable---after almost fifteen years. There are, of course, numerous monad tutorials of varying quality \cite{newbern-all-about-monads, gibbard-monads-as-containers, kow-understanding-monads, rossato-monadic-way, piponi-you-could-have-invented, pimlott-monster, wagner-understanding-monads, kidd-monads-15-minutes, gibbard-monads-as-computation, practical-monads}. A few of the best include Cale Gibbard's \emph{Monads as containers}~\cite{gibbard-monads-as-containers} and \emph{Monads as computation}~\cite{gibbard-monads-as-computation}; Jeff Newbern's \emph{All About Monads}~\cite{newbern-all-about-monads}, a comprehensive guide with lots of examples; and Dan Piponi's \emph{You could have invented monads!}, which features great exercises~\cite{piponi-you-could-have-invented}. If you just want to know how to use |IO|, you could consult the \emph{Introduction to IO}~\cite{intro-to-IO}. Even this is just a sampling; a more complete list can be found on the Haskell wiki~\cite{monad-tutorial-timeline}. (All these monad tutorials have prompted some parodies~\cite{dons-kow-think-of-a-monad} as well as other kinds of backlash~\cite{gordon-tutorials-awful, monad-tutorial-fallacy}.) Other good monad references which are not necessarily tutorials include Henk-Jan van Tuyl's tour of the functions in |Control.Monad|~\cite{vanTuyl-monad-tour}, Dan Piponi's ``field guide''~\cite{piponi-field-guide}, and Tim Newsham's \emph{What's a Monad?}~\cite{newsham-whats-a-monad}. There are also many blog posts which have been written on various aspects of monads; a collection of links can be found on the Haskell wiki~\cite{wiki-monad-blog-articles}. One of the quirks of the |Monad| class and the Haskell type system is that it is not possible to straightforwardly declare |Monad| instances for types which require a class constraint on their data, even if they are monads from a mathematical point of view. For example, |Data.Set| requires an |Ord| constraint on its data, so it cannot be easily made an instance of |Monad|. A solution to this problem was first described by Eric Kidd~\cite{ekidd-07-data-set-monad}, and later made into a library by Ganesh Sittampalam and Peter Gavin~\cite{sittampalam-rmonad}. There are many good reasons for eschewing |do| notation; some have gone so far as to consider it harmful~\cite{do-considered-harmful}. Monads can be generalized in various ways; for an exposition of one possibility, \term{parameterized monads}, see Robert Atkey's paper on the subject \cite{atkey-parameterised}, or Dan Piponi's exposition \cite{piponi-beyond-monads}. For the categorically inclined, monads can be viewed as monoids~\cite{piponi-monoid-monad} and also as closure operators~\cite{dominus-triples-closure}. Derek Elkins's article in Issue 13 of the Monad.Reader~\cite{elkins-calculating-monads} contains an exposition of the category-theoretic underpinnings of some of the standard |Monad| instances, such as |State| and |Cont|. There is also an alternative way to compose monads, using coproducts, as described by L\"uth and Ghani~\cite{luth-ghani-monad-coproducts}, although this method has not (yet?) seen widespread use. Links to many more research papers related to monads can be found on the Haskell wiki~\cite{monads-arrows-research-papers}. \section{Monoid} \label{sec:monoid} A monoid is a set $S$ together with a binary operation $\oplus$ which combines elements from $S$. The $\oplus$ operator is required to be associative (that is, $(a \oplus b) \oplus c = a \oplus (b \oplus c)$, for any $a,b,c$ which are elements of $S$), and there must be some element of $S$ which is the identity with respect to $\oplus$. (If you are familiar with group theory, a monoid is like a group without the requirement that inverses exist.) For example, the natural numbers under addition form a monoid: the sum of any two natural numbers is a natural number; $(a+b)+c = a+(b+c)$ for any natural numbers $a$, $b$, and~$c\,$; and zero is the additive identity. The integers under multiplication also form a monoid, as do natural numbers under $\max$, Boolean values under conjunction and disjunction, lists under concatenation, functions from a set to itself under composition\dots. Monoids show up all over the place, once you know to look for them. \subsection{Definition} \label{sec:monoid-def} The definition of the |Monoid| type class (defined in |Data.Monoid|)~\cite{monoid-docs} is shown in \pref{lst:monoid}. \begin{listing}[htp] \begin{code} class Monoid a where mempty :: a mappend :: a -> a -> a mconcat :: [a] -> a mconcat = foldr mappend mempty \end{code} \caption{The |Monoid| type class} \label{lst:monoid} \end{listing} The |mempty| value specifies the identity element of the monoid, and |mappend| is the binary operation. The default definition for |mconcat| ``reduces'' a list of elements by combining them all with |mappend|, using a right fold. It is only in the |Monoid| class so that specific instances have the option of providing an alternative, more efficient implementation; usually, you can safely ignore |mconcat| when creating a |Monoid| instance, since its default definition will work just fine. The |Monoid| methods are rather unfortunately named; they are inspired by the list instance of |Monoid|, where indeed |mempty = []| and |mappend = (++)|, but this is misleading since many monoids have little to do with appending~\cite{brian-hurt-discussion}. \subsection{Laws} \label{sec:monoid-laws} Of course, every |Monoid| instance should actually be a monoid in the mathematical sense, which implies the laws shown in \pref{lst:monoid-laws}. \begin{listing} \begin{code} mempty `mappend` x = x x `mappend` mempty = x (x `mappend` y) `mappend` z = x `mappend` (y `mappend` z) \end{code} \caption{The |Monoid| laws} \label{lst:monoid-laws} \end{listing} \subsection{Instances} \label{sec:monoid-instances} There are quite a few interesting |Monoid| instances defined in |Data.Monoid|. \begin{itemize} \item |[a]| is a |Monoid|, with |mempty = []| and |mappend = (++)|. It is not hard to check that % |(x ++ y) ++ z = x ++ (y ++ z)| for any lists |x|, |y|, and |z|, and that the empty list is the identity: % |[] ++ x = x ++ [] = x|. \item As noted previously, we can make a monoid out of any numeric type under either addition or multiplication. However, since we can't have two instances for the same type, |Data.Monoid| provides two |newtype| wrappers, |Sum| and |Product|, with appropriate |Monoid| instances. % XXX mention 'newtype' package here? \begin{code} > getSum (mconcat . map Sum $ [1..5]) 15 > getProduct (mconcat . map Product $ [1..5]) 120 \end{code} This example code is silly, of course; we could just write % |sum [1..5]| and |product [1..5]|. Nevertheless, these instances are useful in more generalized settings, as we will see in the discussion of |Foldable| (\pref{sec:foldable}). \item |Any| and |All| are |newtype| wrappers providing |Monoid| instances for |Bool| (under disjunction and conjunction, respectively). \item There are three instances for |Maybe|: a basic instance which lifts a |Monoid| instance for |a| to an instance for |Maybe a|, and two |newtype| wrappers |First| and |Last| for which |mappend| selects the first (respectively last) non-|Nothing| item. \item |Endo a| is a newtype wrapper for functions |a -> a|, which form a monoid under composition. \item There are several ways to ``lift'' |Monoid| instances to instances with additional structure. We have already seen that an instance for |a| can be lifted to an instance for |Maybe a|. There are also tuple instances: if |a| and |b| are instances of |Monoid|, then so is |(a,b)|, using the monoid operations for |a| and |b| in the obvious pairwise manner. Finally, if |a| is a |Monoid|, then so is the function type |e -> a| for any |e|; in particular, % |g `mappend` h| is the function which applies both |g| and |h| to its argument and then combines the results using the underlying |Monoid| instance for |a|. This can be quite useful and elegant~\cite{swierstra-monoid-pointfree}. \item The type |Ordering = LT || EQ || GT| is a |Monoid|, defined in such a way that % |mconcat (zipWith compare xs ys)| computes the lexicographic ordering of |xs| and |ys|. In particular, % |mempty = EQ|, and |mappend| evaluates to its leftmost non-|EQ| argument (or |EQ| if both arguments are |EQ|). This can be used together with the function instance of |Monoid| to do some clever things~\cite{gibbard08monoids-reddit}. \item There are also |Monoid| instances for several standard data structures in the containers library~\cite{containers-library}, including |Map|, |Set|, and |Sequence|. \end{itemize} |Monoid| is also used to enable several other type class instances. As noted previously, we can use |Monoid| to make |((,) e)| an instance of |Applicative|, as shown in \pref{lst:pair-applicative}. \begin{listing}[htp] \begin{code} instance Monoid e => Applicative ((,) e) where pure x = (mempty, x) (u, f) <*> (v, x) = (u `mappend` v, f x) \end{code} \caption{An |Applicative| instance for |((,) e)| using |Monoid|} \label{lst:pair-applicative} \end{listing} |Monoid| can be similarly used to make |((,) e)| an instance of |Monad| as well; this is known as the \term{writer monad}. As we've already seen, |Writer| and |WriterT| are a newtype wrapper and transformer for this monad, respectively. |Monoid| also plays a key role in the |Foldable| type class (\pref{sec:foldable}). \subsection{Other monoidal classes: Alternative, MonadPlus, ArrowPlus} \label{sec:other-monoidal} The |Alternative| type class~\cite{alternative-docs}, shown in \pref{lst:alternative}, is for |Applicative| functors which also have a monoid structure. \begin{listing}[htp] \begin{code} class Applicative f => Alternative f where empty :: f a (<|>) :: f a -> f a -> f a \end{code} \caption{The |Alternative| type class} \label{lst:alternative} \end{listing} Of course, instances of |Alternative| should satisfy the monoid laws. %% |optional|, |some|, |many| ??? Likewise, |MonadPlus|~\cite{monadplus-docs}, shown in \pref{lst:monadplus}, is for |Monad|s with a monoid structure. \begin{listing} \begin{code} class Monad m => MonadPlus m where mzero :: m a mplus :: m a -> m a -> m a \end{code} \caption{The |MonadPlus| type class} \label{lst:monadplus} \end{listing} The |MonadPlus| documentation states that it is intended to model monads which also support ``choice and failure''; in addition to the monoid laws, instances of |MonadPlus| are expected to satisfy \begin{code} mzero >>= f = mzero v >> mzero = mzero \end{code} which explains the sense in which |mzero| denotes failure. Since |mzero| should be the identity for |mplus|, the computation |m1 `mplus` m2| succeeds (evaluates to something other than |mzero|) if either |m1| or |m2| does; so |mplus| represents choice. The |guard| function can also be used with instances of |MonadPlus|; it requires a condition to be satisfied and fails (using |mzero|) if it is not. A simple example of a |MonadPlus| instance is |[]|, which is exactly the same as the |Monoid| instance for |[]|: the empty list represents failure, and list concatenation represents choice. In general, however, a |MonadPlus| instance for a type need not be the same as its |Monoid| instance; |Maybe| is an example of such a type. A great introduction to the |MonadPlus| type class, with interesting examples of its use, is Doug Auclair's Monad.Reader article~\cite{auclair08monadplus}. There used to be a type class called |MonadZero| containing only |mzero|, representing monads with failure. The |do|-notation requires some notion of failure to deal with failing pattern matches. Unfortunately, |MonadZero| was scrapped in favor of adding the |fail| method to the |Monad| class. If we are lucky, someday |MonadZero| will be restored, and |fail| will be banished to the bit bucket where it belongs~\cite{monad-plus-reform}. The idea is that any |do|-block which uses pattern matching (and hence may fail) would require a |MonadZero| constraint; otherwise, only a |Monad| constraint would be required. Finally, |ArrowZero| and |ArrowPlus|~\cite{arrowplus-docs}, shown in \pref{lst:arrowplus}, represent |Arrow|s (\pref{sec:arrow}) with a monoid structure. \begin{listing} \begin{code} class Arrow (~>) => ArrowZero (~>) where zeroArrow :: b ~> c class ArrowZero (~>) => ArrowPlus (~>) where (<+>) :: (b ~> c) -> (b ~> c) -> (b ~> c) \end{code} \caption{The |ArrowZero| and |ArrowPlus| type classes} \label{lst:arrowplus} \end{listing} \subsection{Further reading} \label{sec:monoid-further-reading} Monoids have gotten a fair bit of attention recently, ultimately due to a blog post by Brian Hurt~\cite{hurt-random-thoughts}, in which he complained about the fact that the names of many Haskell type classes (|Monoid| in particular) are taken from abstract mathematics. This resulted in a long haskell-cafe thread~\cite{brian-hurt-discussion} arguing the point and discussing monoids in general. However, this was quickly followed by several blog posts about |Monoid|.\footnote{May its name live forever.} First, Dan Piponi wrote a great introductory post, ``Haskell Monoids and their Uses''~\cite{piponi-monoids-uses}. This was quickly followed by Heinrich Apfelmus's ``Monoids and Finger Trees''~\cite{apfelmus-finger-trees}, an accessible exposition of Hinze and Paterson's classic paper on 2-3 finger trees~\cite{hinze-paterson-06-finger-trees}, which makes very clever use of |Monoid| to implement an elegant and generic data structure. Dan Piponi then wrote two fascinating articles about using |Monoids| (and finger trees) to perform fast incremental regular expression matching~\cite{piponi-incremental-regex, piponi-beyond-regex}. In a similar vein, David Place's article on improving |Data.Map| in order to compute incremental folds~\cite{place08refold-map} is also a good example of using |Monoid| to generalize a data structure. Some other interesting examples of |Monoid| use include building elegant list sorting combinators~\cite{gibbard08monoids-reddit}, collecting unstructured information~\cite{yorgey08partial-knowledge}, and a brilliant series of posts by Chung-Chieh Shan and Dylan Thurston using |Monoid|s to elegantly solve a difficult combinatorial puzzle~\cite{shan-wordnumbers-1, shan-wordnumbers-2, shan-wordnumbers-3, shan-wordnumbers-4}. As unlikely as it sounds, monads can actually be viewed as a sort of monoid, with |join| playing the role of the binary operation and |return| the role of the identity; see Dan Piponi's blog post~\cite{piponi-monoid-monad}. \section{Foldable} \label{sec:foldable} The |Foldable| class, defined in the |Data.Foldable| module~\cite{foldable-docs}, abstracts over containers which can be ``folded'' into a summary value. This allows such folding operations to be written in a container-agnostic way. \subsection{Definition} \label{sec:foldable-definition} The definition of the |Foldable| type class is shown in \pref{lst:foldable}. \begin{listing}[htp] \begin{code} class Foldable t where fold :: Monoid m => t m -> m foldMap :: Monoid m => (a -> m) -> t a -> m foldr :: (a -> b -> b) -> b -> t a -> b foldl :: (a -> b -> a) -> a -> t b -> a foldr1 :: (a -> a -> a) -> t a -> a foldl1 :: (a -> a -> a) -> t a -> a \end{code} \caption{The |Foldable| type class} \label{lst:foldable} \end{listing} This may look complicated, but in fact, to make a |Foldable| instance you only need to implement one method: your choice of |foldMap| or |foldr|. All the other methods have default implementations in terms of these, and are presumably included in the class in case more efficient implementations can be provided. \subsection{Instances and examples} \label{sec:foldable-instances} The type of |foldMap| should make it clear what it is supposed to do: given a way to convert the data in a container into a |Monoid| (a function |a -> m|) and a container of |a|'s (|t a|), |foldMap| provides a way to iterate over the entire contents of the container, converting all the |a|'s to |m|'s and combining all the |m|'s with |mappend|. \pref{lst:foldMap-examples} shows two examples: a simple implementation of |foldMap| for lists, and a binary tree example provided by the |Foldable| documentation. \begin{listing}[htp] \begin{code} instance Foldable [] where foldMap g = mconcat . map g data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a) instance Foldable Tree where foldMap f Empty = mempty foldMap f (Leaf x) = f x foldMap f (Node l k r) = foldMap f l ++ f k ++ foldMap f r where (++) = mappend \end{code} \caption{Two |foldMap| examples} \label{lst:foldMap-examples} \end{listing} The |foldr| function has a type similar to the |foldr| found in the |Prelude|, but more general, since the |foldr| in the |Prelude| works only on lists. The |Foldable| module also provides instances for |Maybe| and |Array|; additionally, many of the data structures found in the standard containers library~\cite{containers-library} (for example, |Map|, |Set|, |Tree|, and |Sequence|) provide their own |Foldable| instances. \subsection{Derived folds} \label{sec:folds} Given an instance of |Foldable|, we can write generic, container-agnostic functions such as the examples shown in \pref{lst:foldable-examples}. \begin{listing}[htp] \begin{code} -- Compute the size of any container. containerSize :: Foldable f => f a -> Int containerSize = getSum . foldMap (const (Sum 1)) -- Compute a list of elements of a container satisfying a predicate. filterF :: Foldable f => (a -> Bool) -> f a -> [a] filterF p = foldMap (\a -> if p a then [a] else []) -- Get a list of all the Strings in a container which include the -- letter a. aStrings :: Foldable f => f String -> [String] aStrings = filterF (elem 'a') \end{code} \caption{|Foldable| examples} \label{lst:foldable-examples} \end{listing} The |Foldable| module also provides a large number of predefined folds, many of which are generalized versions of |Prelude| functions of the same name that only work on lists: |concat|, |concatMap|, |and|, |or|, |any|, |all|, |sum|, |product|, |maximum|(|By|), |minimum|(|By|), |elem|, |notElem|, and |find|. The reader may enjoy coming up with elegant implementations of these functions using |fold| or |foldMap| and appropriate |Monoid| instances. There are also generic functions that work with |Applicative| or |Monad| instances to generate some sort of computation from each element in a container, and then perform all the side effects from those computations, discarding the results: |traverse_|, |sequenceA_|, and others. The results must be discarded because the |Foldable| class is too weak to specify what to do with them: we cannot, in general, make an arbitrary |Applicative| or |Monad| instance into a |Monoid|. If we do have an |Applicative| or |Monad| with a monoid structure---that is, an |Alternative| or a |MonadPlus|---then we can use the |asum| or |msum| functions, which can combine the results as well. Consult the |Foldable| documentation~\cite{foldable-docs} for more details on any of these functions. Note that the |Foldable| operations always forget the structure of the container being folded. If we start with a container of type |t a| for some |Foldable t|, then |t| will never appear in the output type of any operations defined in the |Foldable| module. Many times this is exactly what we want, but sometimes we would like to be able to generically traverse a container while preserving its structure---and this is exactly what the |Traversable| class provides, which will be discussed in the next section. \subsection{Further reading} \label{sec:foldable-further-reading} The |Foldable| class had its genesis in McBride and Paterson's paper introducing |Applicative|~\cite{mcbride08applicative}, although it has been fleshed out quite a bit from the form in the paper. An interesting use of |Foldable| (as well as |Traversable|) can be found in Janis Voigtl\"ander's paper \emph{Bidirectionalization for free!}~\cite{voigtlaender-09-bidirectionalization}. \section{Traversable} \label{sec:traversable} \subsection{Definition} \label{sec:traversable-definition} The |Traversable| type class, defined in the |Data.Traversable| module~\cite{traversable-docs}, is shown in \pref{lst:traversable}. \begin{listing}[htp] \begin{code} class (Functor t, Foldable t) => Traversable t where traverse :: Applicative f => (a -> f b) -> t a -> f (t b) sequenceA :: Applicative f => t (f a) -> f (t a) mapM :: Monad m => (a -> m b) -> t a -> m (t b) sequence :: Monad m => t (m a) -> m (t a) \end{code} \caption{The |Traversable| type class} \label{lst:traversable} \end{listing} As you can see, every |Traversable| is also a foldable functor. Like |Foldable|, there is a lot in this type class, but making instances is actually rather easy: one need only implement |traverse| or |sequenceA|; the other methods all have default implementations in terms of these functions. A good exercise is to figure out what the default implementations should be: given either |traverse| or |sequenceA|, how would you define the other three methods? (Hint for |mapM|: |Control.Applicative| exports the |WrapMonad| newtype, which makes any |Monad| into an |Applicative|. The |sequence| function can be implemented in terms of |mapM|.) \subsection{Intuition} \label{sec:traversable-intuition} The key method of the |Traversable| class, and the source of its unique power, is |sequenceA|. Consider its type: \begin{code} sequenceA :: Applicative f => t (f a) -> f (t a) \end{code} This answers the fundamental question: when can we commute two functors? For example, can we turn a tree of lists into a list of trees? (Answer: yes, in two ways. Figuring out what they are, and why, is left as an exercise. A much more challenging question is whether a list of trees can be turned into a tree of lists.) The ability to compose two monads depends crucially on this ability to commute functors. Intuitively, if we want to build a composed monad % |M a = m (n a)| out of monads |m| and |n|, then to be able to implement |join :: M (M a) -> M a|, that is, % |join :: m (n (m (n a))) -> m (n a)|, we have to be able to commute the |n| past the |m| to get |m (m (n (n a)))|, and then we can use the |join|s for |m| and |n| to produce something of type |m (n a)|. See Mark Jones's paper for more details~\cite{jones-95-fp}. \subsection{Instances and examples} \label{sec:traversable-instances} What's an example of a |Traversable| instance? \pref{lst:traversable-example} shows an example instance for the same |Tree| type used as an example in the previous |Foldable| section. It is instructive to compare this instance with a |Functor| instance for |Tree|, which is also shown. \begin{listing}[htp] \begin{code} data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a) instance Traversable Tree where traverse g Empty = pure Empty traverse g (Leaf x) = Leaf <$> g x traverse g (Node l x r) = Node <$> traverse g l <*> g x <*> traverse g r instance Functor Tree where fmap g Empty = Empty fmap g (Leaf x) = Leaf $ g x fmap g (Node l x r) = Node (fmap g l) (g x) (fmap g r) \end{code} %$ \caption{An example |Tree| instance of |Traversable|} \label{lst:traversable-example} \end{listing} It should be clear that the |Traversable| and |Functor| instances for |Tree| are almost identical; the only difference is that the |Functor| instance involves normal function application, whereas the applications in the |Traversable| instance take place within an |Applicative| context, using |(<$>)| and |(<*>)|. In fact, this will be %$ true for any type. Any |Traversable| functor is also |Foldable|, and a |Functor|. We can see this not only from the class declaration, but by the fact that we can implement the methods of both classes given only the |Traversable| methods. A good exercise is to implement |fmap| and |foldMap| using only the |Traversable| methods; the implementations are surprisingly elegant. The |Traversable| module provides these implementations as |fmapDefault| and |foldMapDefault|. The standard libraries provide a number of |Traversable| instances, including instances for |[]|, |Maybe|, |Map|, |Tree|, and |Sequence|. Notably, |Set| is not |Traversable|, although it is |Foldable|. \subsection{Further reading} \label{sec:traversable-further-reading} The |Traversable| class also had its genesis in McBride and Paterson's |Applicative| paper~\cite{mcbride08applicative}, and is described in more detail in Gibbons and Oliveira, \emph{The Essence of the Iterator Pattern}~\cite{gibbons-07-iterator}, which also contains a wealth of references to related work. \section{Category} \label{sec:category} |Category| is another fairly new addition to the Haskell standard libraries; you may or may not have it installed depending on the version of your |base| package. It generalizes the notion of function composition to general ``morphisms''. The definition of the |Category| type class (from |Control.Category|~\cite{category-docs}) is shown in \pref{lst:category}. For ease of reading, note that I have used an infix type constructor |(~>)|, much like the infix function type constructor |(->)|. This syntax is not part of Haskell 98. The second definition shown is the one used in the standard libraries. For the remainder of this document, I will use the infix type constructor |(~>)| for |Category| as well as |Arrow|. %Editor: Sounded like category and arrow are equiv % \begin{listing}[htp] \begin{code} class Category (~>) where id :: a ~> a (.) :: (b ~> c) -> (a ~> b) -> (a ~> c) -- The same thing, with a normal (prefix) type constructor class Category cat where id :: cat a a (.) :: cat b c -> cat a b -> cat a c \end{code} \caption{The |Category| type class} \label{lst:category} \end{listing} Note that an instance of |Category| should be a type constructor which takes two type arguments, that is, something of kind |* -> * -> *|. It is instructive to imagine the type constructor variable |cat| replaced by the function constructor |(->)|: indeed, in this case we recover precisely the familiar identity function |id| and function composition operator |(.)| defined in the standard |Prelude|. Of course, the |Category| module provides exactly such an instance of |Category| for |(->)|. But it also provides one other instance, shown in \pref{lst:kleisli-category}, which should be familiar from the previous discussion of the |Monad| laws. |Kleisli m a b|, as defined in the |Control.Arrow| module, is just a |newtype| wrapper around % |a -> m b|. \begin{listing} \begin{code} newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b } instance Monad m => Category (Kleisli m) where id = Kleisli return Kleisli g . Kleisli h = Kleisli (h >=> g) \end{code} \caption{The |Kleisli| |Category| instance} \label{lst:kleisli-category} \end{listing} The only law that |Category| instances should satisfy is that |id| and |(.)| should form a monoid---that is, |id| should be the identity of |(.)|, and |(.)| should be associative. Finally, the |Category| module exports two additional operators: |(<<<)|, which is just a synonym for |(.)|, and |(>>>)|, which is |(.)| with its arguments reversed. (In previous versions of the libraries, these operators were defined as part of the |Arrow| class.) \subsection{Further reading} \label{sec:category-further-reading} The name |Category| is a bit misleading, since the |Category| class cannot represent arbitrary categories, but only categories whose objects are objects of |Hask|, the category of Haskell types. For a more general treatment of categories within Haskell, see the category-extras package~\cite{category-extras}. For more about category theory in general, see the excellent Haskell wikibook page~\cite{wikibook-ct}, Steve Awodey's new book~\cite{awodey-06-ct}, Benjamin Pierce's \emph{Basic category theory for computer scientists}~\cite{pierce-91-ct}, or Barr and Wells's category theory lecture notes~\cite{barr-wells-ct}. Benjamin Russell's blog post \cite{russell-motivating-ct} is another good source of motivation and category theory links. You certainly don't need to know any category theory to be a successful and productive Haskell programmer, but it does lend itself to much deeper appreciation of Haskell's underlying theory. \section{Arrow} \label{sec:arrow} The |Arrow| class represents another abstraction of computation, in a similar vein to |Monad| and |Applicative|. However, unlike |Monad| and |Applicative|, whose types only reflect their output, the type of an |Arrow| computation reflects both its input and output. Arrows generalize functions: if |(~>)| is an instance of |Arrow|, a value of type |b ~> c| can be thought of as a computation which takes values of type |b| as input, and produces values of type |c| as output. In the |(->)| instance of |Arrow| this is just a pure function; in general, however, an arrow may represent some sort of ``effectful'' computation. \subsection{Definition} \label{sec:arrow-definition} The definition of the |Arrow| type class, from |Control.Arrow|~\cite{arrow-docs}, is shown in \pref{lst:arrow}. \begin{listing}[htp] \begin{code} class Category (~>) => Arrow (~>) where arr :: (b -> c) -> (b ~> c) first :: (b ~> c) -> ((b, d) ~> (c, d)) second :: (b ~> c) -> ((d, b) ~> (d, c)) (***) :: (b ~> c) -> (b' ~> c') -> ((b, b') ~> (c, c')) (&&&) :: (b ~> c) -> (b ~> c') -> (b ~> (c, c')) \end{code} \caption{The |Arrow| type class} \label{lst:arrow} \end{listing} The first thing to note is the |Category| class constraint, which means that we get identity arrows and arrow composition for free: given two arrows |g :: b ~> c| and |h :: c ~> d|, we can form their composition |g >>> h :: b ~> d|.\footnote{In versions of the |base| package prior to version 4, there is no |Category| class, and the |Arrow| class includes the arrow composition operator |(>>>)|. It also includes |pure| as a synonym for |arr|, but this was removed since it conflicts with the |pure| from |Applicative|.} As should be a familiar pattern by now, the only methods which must be defined when writing a new instance of |Arrow| are |arr| and |first|; the other methods have default definitions in terms of these, but are included in the |Arrow| class so that they can be overridden with more efficient implementations if desired. \subsection{Intuition} \label{sec:arrow-intuition} Let's look at each of the arrow methods in turn. Ross Paterson's web page on arrows~\cite{arrow-page} has nice diagrams which can help build intuition. \begin{itemize} \item The |arr| function takes any function |b -> c| and turns it into a generalized arrow |b ~> c|. The |arr| method justifies the claim that arrows generalize functions, since it says that we can treat any function as an arrow. It is intended that the arrow |arr g| is ``pure'' in the sense that it only computes |g| and has no ``effects'' (whatever that might mean for any particular arrow type). \item The |first| method turns any arrow from |b| to |c| into an arrow from |(b,d)| to |(c,d)|. The idea is that |first g| uses |g| to process the first element of a tuple, and lets the second element pass through unchanged. For the function instance of |Arrow|, of course, |first g (x,y) = (g x, y)|. \item The |second| function is similar to |first|, but with the elements of the tuples swapped. Indeed, it can be defined in terms of |first| using an auxiliary function |swap|, defined by |swap (x,y) = (y,x)|. \item The |(***)| operator is ``parallel composition'' of arrows: it takes two arrows and makes them into one arrow on tuples, which has the behavior of the first arrow on the first element of a tuple, and the behavior of the second arrow on the second element. The mnemonic is that |g *** h| is the \emph{product} (hence |*|) of |g| and |h|. For the function instance of |Arrow|, % we define |(g *** h) (x,y) = (g x, h y)|. The default implementation of |(***)| is in terms of |first|, |second|, and sequential arrow composition |(>>>)|. The reader may also wish to think about how to implement |first| and |second| in terms of |(***)|. \item The |(&&&)| operator is ``fanout composition'' of arrows: it takes two arrows |g| and |h| and makes them into a new arrow |g &&& h| which supplies its input as the input to both |g| and |h|, returning their results as a tuple. The mnemonic is that |g &&& h| performs both |g| \emph{and} |h| (hence |&|) on its input. For functions, we define % |(g &&& h) x = (g x, h x)|. \end{itemize} \subsection{Instances} \label{sec:arrow-instances} The |Arrow| library itself only provides two |Arrow| instances, both of which we have already seen: |(->)|, the normal function constructor, and |Kleisli m|, which makes functions of type |a -> m b| into |Arrow|s for any |Monad m|. These instances are shown in \pref{lst:arrow-instances}. \begin{listing} \begin{code} instance Arrow (->) where arr g = g first g (x,y) = (g x, y) newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b } instance Monad m => Arrow (Kleisli m) where arr f = Kleisli (return . f) first (Kleisli f) = Kleisli (\ ~(b,d) -> do c <- f b return (c,d) ) \end{code} \caption{The |(->)| and |Kleisli m| instances of |Arrow|} \label{lst:arrow-instances} \end{listing} \subsection{Laws} \label{sec:arrow-laws} There are quite a few laws that instances of |Arrow| should satisfy~\cite{hughes-00-generalising, lindley-arrow-calculus, paterson-03-programming-arrows}; they are shown in \pref{lst:arrow-laws}. Note that the version of the laws shown in \pref{lst:arrow-laws} is slightly different than the laws given in the first two above references, since several of the laws have now been subsumed by the |Category| laws (in particular, the requirements that |id| is the identity arrow and that |(>>>)| is associative). The laws shown here follow those in Paterson~\cite{paterson-03-programming-arrows}, which uses the |Category| class. \begin{listing}[htp] \begin{code} arr id = id arr (h . g) = arr g >>> arr h first (arr g) = arr (g *** id) first (g >>> h) = first g >>> first h first g >>> arr (id *** h) = arr (id *** h) >>> first g first g >>> arr fst = arr fst >>> g first (first g) >>> arr assoc = arr assoc >>> first g assoc ((x,y),z) = (x,(y,z)) \end{code} \caption{The |Arrow| laws} \label{lst:arrow-laws} \end{listing} The reader is advised not to lose too much sleep over the |Arrow| laws,\footnote{Unless category-theory-induced insomnolence is your cup of tea.} since it is not essential to understand them in order to program with arrows. There are also laws that |ArrowChoice|, |ArrowApply|, and |ArrowLoop| instances should satisfy; the interested reader should consult Paterson~\cite{paterson-03-programming-arrows}. \subsection{ArrowChoice} \label{sec:arrowchoice} Computations built using the |Arrow| class, like those built using the |Applicative| class, are rather inflexible: the structure of the computation is fixed at the outset, and there is no ability to choose between alternate execution paths based on intermediate results. The |ArrowChoice| class provides exactly such an ability; it is shown in \pref{lst:arrowchoice}. \begin{listing}[htp] \begin{code} class Arrow (~>) => ArrowChoice (~>) where left :: (b ~> c) -> (Either b d ~> Either c d) right :: (b ~> c) -> (Either d b ~> Either d c) (+++) :: (b ~> c) -> (b' ~> c') -> (Either b b' ~> Either c c') (|||) :: (b ~> d) -> (c ~> d) -> (Either b c ~> d) \end{code} \caption{The |ArrowChoice| type class} \label{lst:arrowchoice} \end{listing} A comparison of |ArrowChoice| to |Arrow| will reveal a striking parallel between |left|, |right|, |(+++)|, |(||||||)| and |first|, |second|, |(***)|, |(&&&)|, respectively. Indeed, they are dual: |first|, |second|, |(***)|, and |(&&&)| all operate on product types (tuples), and |left|, |right|, |(+++)|, and |(||||||)| are the corresponding operations on sum types. In general, these operations create arrows whose inputs are tagged with |Left| or |Right|, and can choose how to act based on these tags. \begin{itemize} \item If |g| is an arrow from |b| to |c|, then |left g| is an arrow from |Either b d| to |Either c d|. On inputs tagged with |Left|, the |left g| arrow has the behavior of |g|; on inputs tagged with |Right|, it behaves as the identity. \item The |right| function, of course, is the mirror image of |left|. The arrow |right g| has the behavior of |g| on inputs tagged with |Right|. \item The |(+++)| operator performs ``multiplexing'': |g +++ h| behaves as |g| on inputs tagged with |Left|, and as |h| on inputs tagged with |Right|. The tags are preserved. The |(+++)| operator is the \emph{sum} (hence |+|) of two arrows, just as |(***)| is the product. \item The |(||||||)| operator is ``merge'' or ``fanin'': the arrow |g |||||| h| behaves as |g| on inputs tagged with |Left|, and |h| on inputs tagged with |Right|, but the tags are discarded (hence, |g| and |h| must have the same output type). The mnemonic is that |g |||||| h| performs either |g| \emph{or} |h| on its input. \end{itemize} The |ArrowChoice| class allows computations to choose among a finite number of execution paths, based on intermediate results. The possible execution paths must be known in advance, and explicitly assembled with |(+++)| or |(||||||)|. However, sometimes more flexibility is needed: we would like to be able to \emph{compute} an arrow from intermediate results, and use this computed arrow to continue the computation. This is the power given to us by |ArrowApply|. \subsection{ArrowApply} \label{sec:arrowapply} The |ArrowApply| type class is shown in \pref{lst:arrowapply}. \begin{listing}[htp] \begin{code} class Arrow (~>) => ArrowApply (~>) where app :: (b ~> c, b) ~> c \end{code} \caption{The |ArrowApply| type class} \label{lst:arrowapply} \end{listing} If we have computed an arrow as the output of some previous computation, then |app| allows us to apply that arrow to an input, producing its output as the output of |app|. As an exercise, the reader may wish to use |app| to implement an alternative ``curried'' version, |app2 :: b ~> ((b ~> c) ~> c)|. This notion of being able to \emph{compute} a new computation may sound familiar: this is exactly what the monadic bind operator |(>>=)| does. It should not particularly come as a surprise that |ArrowApply| and |Monad| are exactly equivalent in expressive power. In particular, |Kleisli m| can be made an instance of |ArrowApply|, and any instance of |ArrowApply| can be made a |Monad| (via the |newtype| wrapper |ArrowMonad|). As an exercise, the reader may wish to try implementing these instances, shown in \pref{lst:arrowapp-monad}. \begin{listing}[htp] \begin{code} instance Monad m => ArrowApply (Kleisli m) where app = -- exercise newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b) instance ArrowApply a => Monad (ArrowMonad a) where return = -- exercise (ArrowMonad a) >>= k = -- exercise \end{code} \caption{Equivalence of |ArrowApply| and |Monad|} \label{lst:arrowapp-monad} \end{listing} \subsection{ArrowLoop} \label{sec:arrowloop} The |ArrowLoop| type class is shown in \pref{lst:arrowloop}; it describes arrows that can use recursion to compute results, and is used to desugar the |rec| construct in arrow notation (described below). Taken by itself, the type of the |loop| method does not seem to tell us much. Its intention, however, is a generalization of the |trace| function which is also shown. The |d| component of the first arrow's output is fed back in as its own input. In other words, the arrow |loop g| is obtained by recursively ``fixing'' the second component of the input to |g|. \begin{listing}[htp] \begin{code} class Arrow a => ArrowLoop a where loop :: a (b, d) (c, d) -> a b c trace :: ((b,d) -> (c,d)) -> b -> c trace f b = let (c,d) = f (b,d) in c \end{code} \caption{The |ArrowLoop| type class} \label{lst:arrowloop} \end{listing} It can be a bit difficult to grok what the |trace| function is doing. How can |d| appear on the left and right sides of the |let|? Well, this is Haskell's laziness at work. There is not space here for a full explanation; the interested reader is encouraged to study the standard |fix| function, and to read Paterson's arrow tutorial \cite{paterson-03-programming-arrows}. \subsection{Arrow notation} \label{sec:arrow-notation} Programming directly with the arrow combinators can be painful, especially when writing complex computations which need to retain simultaneous reference to a number of intermediate results. With nothing but the arrow combinators, such intermediate results must be kept in nested tuples, and it is up to the programmer to remember which intermediate results are in which components, and to swap, reassociate, and generally mangle tuples as necessary. This problem is solved by the special arrow notation supported by GHC, similar to |do| notation for monads, that allows names to be assigned to intermediate results while building up arrow computations. An example arrow implemented using arrow notation, taken from Paterson~\cite{paterson-03-programming-arrows}, is shown in \pref{lst:arrow-notation-example}. This arrow is intended to represent a recursively defined counter circuit with a reset line. \begin{listing}[htp] \begin{code} class ArrowLoop (~>) => ArrowCircuit (~>) where delay :: b -> (b ~> b) counter :: ArrowCircuit (~>) => Bool ~> Int counter = proc reset -> do rec output <- idA -< if reset then 0 else next next <- delay 0 -< output + 1 idA -< output \end{code} \caption{An example arrow using arrow notation} \label{lst:arrow-notation-example} \end{listing} There is not space here for a full explanation of arrow notation; the interested reader should consult Paterson's paper introducing the notation~\cite{paterson-01-arrow-notation}, or his later tutorial which presents a simplified version~\cite{paterson-03-programming-arrows}. \subsection{Further reading} \label{sec:arrow-further-reading} An excellent starting place for the student of arrows is the web page put together by Paterson~\cite{arrow-page}, which contains an introduction and many references. Some key papers on arrows include Hughes's original paper introducing arrows, \emph{Generalising Monads to Arrows}~\cite{hughes-00-generalising}, and Paterson's paper on arrow notation~\cite{paterson-01-arrow-notation}. Both Hughes and Paterson later wrote accessible tutorials intended for a broader audience~\cite{paterson-03-programming-arrows, hughes-05-programming-arrows}. Although Hughes's goal in defining the |Arrow| class was to generalize |Monad|s, and it has been said that |Arrow| lies ``between |Applicative| and |Monad|'' in power, they are not directly comparable. The precise relationship remained in some confusion until analyzed by Lindley, Wadler, and Yallop~\cite{lindley-idioms}, who also invented a new calculus of arrows, based on the lambda calculus, which considerably simplifies the presentation of the arrow laws \cite{lindley-arrow-calculus}. Some examples of |Arrow|s include Yampa~\cite{yampa}, the Haskell XML Toolkit~\cite{hxt}, and the functional GUI library Grapefruit \cite{grapefruit}. Some extensions to arrows have been explored; for example, the |BiArrow|s of Alimarine \etal, for two-way instead of one-way computation~\cite{alimarine-05-biarrows}. Links to many additional research papers relating |Arrow|s can be found on the Haskell wiki~\cite{monads-arrows-research-papers}. \section{Comonad} \label{sec:comonad} The final type class we will examine is |Comonad|. The |Comonad| class is the categorical dual of |Monad|; that is, |Comonad| is like |Monad| but with all the function arrows flipped. It is not actually in the standard Haskell libraries, but it has seen some interesting uses recently, so we include it here for completeness. \subsection{Definition} \label{sec:comonad-definition} The |Comonad| type class, defined in the |Control.Comonad| module of the category-extras library~\cite{category-extras}, is shown in \pref{lst:comonad}. \begin{listing}[htp] \begin{code} class Functor f => Copointed f where extract :: f a -> a class Copointed w => Comonad w where duplicate :: w a -> w (w a) extend :: (w a -> b) -> w a -> w b \end{code} \caption{The |Comonad| type class} \label{lst:comonad} \end{listing} As you can see, |extract| is the dual of |return|, |duplicate| is the dual of |join|, and |extend| is the dual of |(>>=)| (although its arguments are in a different order). The definition of |Comonad| is a bit redundant (after all, the |Monad| class does not need |join|), but this is so that a |Comonad| can be defined by |fmap|, |extract|, and \emph{either} |duplicate| or |extend|. Each has a default implementation in terms of the other. A prototypical example of a |Comonad| instance is shown in \pref{lst:stream-comonad}. \begin{listing} \begin{code} -- Infinite lazy streams data Stream a = Cons a (Stream a) instance Functor Stream where fmap g (Cons x xs) = Cons (g x) (fmap g xs) instance Copointed Stream where extract (Cons x _) = x -- 'duplicate' is like the list function 'tails' -- 'extend' computes a new Stream from an old, where the element -- at position n is computed as a function of everything from -- position n onwards in the old Stream instance Comonad Stream where duplicate s@(Cons x xs) = Cons s (duplicate xs) extend g s@(Cons x xs) = Cons (g s) (extend g xs) -- = fmap g (duplicate s) \end{code} \caption{A |Comonad| instance for |Stream|} \label{lst:stream-comonad} \end{listing} \subsection{Further reading} \label{sec:comonad-further-reading} Dan Piponi explains in a blog post what cellular automata have to do with comonads~\cite{piponi-automata-comonadic}. In another blog post, Conal Elliott has examined a comonadic formulation of functional reactive programming~\cite{conal-functional-behavior}. Sterling Clover's blog post \emph{Comonads in everyday life}~\cite{clover-comonads} explains the relationship between comonads and zippers, and how comonads can be used to design a menu system for a web site. Uustalu and Vene have a number of papers exploring ideas related to comonads and functional programming \cite{uustalu-comonadic-computation, uustalu-dual-substitution, capretta-recursive-coalgebras, uustalu-recursion-schemes, uutsalu-06-essence}. \section{Acknowledgements} A special thanks to all of those who taught me about standard Haskell type classes and helped me develop good intuition for them, particularly Jules Bean (quicksilver), Derek Elkins (ddarius), Conal Elliott (conal), Cale Gibbard (Cale), David House, Dan Piponi (sigfpe), and Kevin Reid (kpreid). I also thank the many people who provided a mountain of helpful feedback and suggestions on a first draft of the Typeclassopedia: David Amos, Kevin Ballard, Reid Barton, Doug Beardsley, Joachim Breitner, Andrew Cave, David Christiansen, Gregory Collins, Mark Jason Dominus, Conal Elliott, Yitz Gale, George Giorgidze, Steven Grady, Travis Hartwell, Steve Hicks, Philip H\"olzenspies, Edward Kmett, Eric Kow, Serge Le Huitouze, Felipe Lessa, Stefan Ljungstrand, Eric Macaulay, Rob MacAulay, Simon Meier, Eric Mertens, Tim Newsham, Russell O'Connor, Conrad Parker, Walt Rorie-Baety, Colin Ross, Tom Schrijvers, Aditya Siram, C. Smith, Martijn van Steenbergen, Joe Thornber, Jared Updike, Rob Vollmert, Andrew Wagner, Louis Wasserman, and Ashley Yakeley, as well as a few only known to me by their IRC nicks: b\_jonas, maltem, tehgeekmeister, and ziman. I have undoubtedly omitted a few inadvertently, which in no way diminishes my gratitude. Finally, I would like to thank Wouter Swierstra for his fantastic work editing the Monad.Reader, and my wife Joyia for her patience during the process of writing the Typeclassopedia. \section{About the author} Brent Yorgey~\cite{brent-blog, brent-homepage} is a third-year Ph.D. student in the programming languages group at the University of Pennsylvania~\cite{plclub}. He still enjoys teaching, creating EDSLs, playing Bach fugues, musing upon category theory, and cooking tasty lambda-treats for the denizens of \#haskell. \raggedright \bibliography{typeclassopedia} \Closesolutionfile{sols} \clearpage \section{Solutions to exercises} \label{sec:solutions} \input{sols} \end{document}