X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek7_introducing_monads.mdwn;h=fc508a835f7802e8bb00e7f311080210ab0bfbe8;hp=718677e2bb61247dc69f97ffc017bd899609e1c6;hb=caa10a9060a7295040ab497c68621522c4628595;hpb=a31c51ce293108196a9f6ded726ac637ad52acb3 diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn index 718677e2..fc508a83 100644 --- a/topics/week7_introducing_monads.mdwn +++ b/topics/week7_introducing_monads.mdwn @@ -1,4 +1,4 @@ - + Introducing Monads @@ -19,10 +19,10 @@ The closest we will come to metaphorical talk is to suggest that monadic types place values inside of *boxes*, and that monads wrap and unwrap boxes to expose or enclose the values inside of them. In any case, our emphasis will be on starting with the abstract structure -of monads, followed by instances of monads from the philosophical and +of monads, followed in coming weeks by instances of monads from the philosophical and linguistics literature. -> After you've read this once and are coming back to re-read it to try to digest the details further, the "endofunctors" that slogan is talking about are a combination of our boxes and their associated maps. Their "monoidal" character is captured in the Monad Laws, where a "monoid"---don't confuse with a mon*ad*---is a simpler algebraic notion, meaning a universe with some associative operation that has an identity. For advanced study, here are some further links on the relation between monads as we're working with them and monads as they appear in category theory: +> After you've read this once and are coming back to re-read it to try to digest the details further, the "endofunctors" that slogan is talking about are a combination of our boxes and their associated maps. Their "monoidal" character is captured in the Monad Laws, where a "monoid"---don't confuse with a mon*ad*---is a simpler algebraic notion, meaning a universe with some associative operation that has an identity. For advanced study, here are some further links on the relation between monads as we're working with them and monads as they appear in Category Theory: [1](http://en.wikipedia.org/wiki/Outline_of_category_theory) [2](http://lambda1.jimpryor.net/advanced_topics/monads_in_category_theory/) [3](http://en.wikibooks.org/wiki/Haskell/Category_theory) @@ -41,7 +41,7 @@ type variables. For instance, we might have P_3 ≡ ∀α. α -> α P_4 ≡ ∀α. α -> β -etc. +and so on. A *box type* will be a type expression that contains exactly one free type variable. (You could extend this to expressions with more free variables; then you'd have @@ -54,13 +54,13 @@ to specify which one of them the box is capturing. But let's keep it simple.) So The idea is that whatever type the free type variable `α` might be instantiated to, we will have a "type box" of a certain sort that "contains" values of type `α`. For instance, -if `α list` is our box type, and `α` is the type `int`, then in this context, `int list` +if `α list` is our box type, and `α` instantiates to the type `int`, then in this context, `int list` is the type of a boxed integer. -Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with `α`s as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where `R` is some fixed type, `R -> α` is a box type. +Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with `α`s as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where `R` is some fixed type, `R -> α` will be one box type we work extensively with. Also, for clarity: the *box type* is the type `α list` (or as we might just say, the `list` type operator); the *boxed type* is some specific instantiation of the free type variable `α`. We'll often write boxed types as a box containing what the free -type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we would write: +type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we write: int @@ -74,8 +74,8 @@ A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Do P -> Q -That is, they are functions from values of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`. -For instance, the following are Kleisli arrows: +That is, they are functions from values of one type `P` to a boxed type `Q`, for some choice of box and of type expressions `P` and `Q`. +For instance, the following are Kleisli arrow types: int -> bool @@ -120,39 +120,41 @@ Here are the types of our crucial functions, together with our pronunciation, an mid (/εmaidεnt@tI/): P -> P -> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. In the class presentation Jim called it `𝟭`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".) This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`. +> This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`. It will be a way of boxing values with your box type that plays a distinguished role in the various Laws and interdefinitions we present below. + +> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. All of these names are somewhat unfortunate. First, it has little to do with `η`-reduction in the Lambda Calculus. Second, it has little to do with the `() : unit` value we discussed in earlier classes. Third, it has little to do with the `return` keyword in C and other languages; that's more closely related to continuations, which we'll discuss in later weeks. Finally, this doesn't perfectly align with other uses of "pure" in the literature. `mid`'d values _will_ generally be "pure" in the other senses, but other boxed values can be too. + +> For all these reasons, we're thinking it will be clearer in our discussion to use a different name. In the class presentation Jim called it `𝟭`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".) m$ or mapply (/εm@plai/): P -> Q -> P -> Q -> We'll use `m$` as a left-associative infix operator, reminiscent of (the right-associative) `$` which is just ordinary function application (also expressed by mere left-associative juxtaposition). In the class presentation Jim called `m$` `●`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`. +> We'll use `m$` as a left-associative infix operator, reminiscent of (the right-associative) `$` which is just ordinary function application (also expressed by mere left-associative juxtaposition). In the class presentation Jim called `m$` `⚫`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`. <=< or mcomp : (Q -> R) -> (P -> Q) -> (P -> R) > In Haskell, this is `Control.Monad.<=<`. ->=> (flip mcomp, should we call it mpmoc?): (P -> Q) -> (Q -> R) -> (P -> R) +>=> or flip mcomp : (P -> Q) -> (Q -> R) -> (P -> R) + +> In Haskell, this is `Control.Monad.>=>`. We will move freely back and forth between using `<=<` (aka `mcomp`) and using `>=>`, which +is just `<=<` with its arguments flipped. `<=<` has the virtue that it corresponds more +closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue +that its types flow more naturally from left to right. -> In Haskell, this is `Control.Monad.>=>`. In the class handout, we gave the types for `>=>` twice, and once was correct but the other was a typo. The above is the correct typing. +> In the class handout, we gave the types for `>=>` twice, and once was correct but the other was a typo. The above is the correct typing. >>= or mbind : (Q) -> (Q -> R) -> (R) -=<< (flip mbind, should we call it mdnib?) (Q -> R) -> (Q) -> (R) +> Haskell uses the symbol `>>=` but calls it "bind". This is not well chosen from the perspective of formal semantics, since it's only loosely connected with what we mean by "binding." But the name is too deeply entrenched to change. We've at least preprended an "m" to the front of "bind". In some presentations this operation is called `★`. + +=<< or flip mbind : (Q -> R) -> (Q) -> (R) join: P -> P > In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`. -Haskell uses the symbol `>>=` but calls it "bind". This is not well chosen from the perspective of formal semantics, but it's too deeply entrenched to change. We've at least preprended an "m" to the front of "bind". - -Haskell's names "return" and "pure" for `mid` are even less well chosen, and we think it will be clearer in our discussion to use a different name. (Also, in other theoretical contexts this notion goes by other names, anyway, like `unit` or `η` --- having nothing to do with `η`-reduction in the Lambda Calculus.) - The menagerie isn't quite as bewildering as you might suppose. Many of these will be interdefinable. For example, here is how `mcomp` and `mbind` are related: k <=< j ≡ \a. (j a >>= k). We'll state some other interdefinitions below. -We will move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which -is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more -closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue -that its types flow more naturally from left to right. - These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system: * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable* @@ -176,8 +178,8 @@ has to obey the following Map Laws: 1. mid (id : P->P) : P -> P is a left identity for `m$`, that is: `(mid id) m$ xs = xs` 2. `mid (f a) = (mid f) m$ (mid a)` 3. The `map2`ing of composition onto boxes `fs` and `gs` of functions, when `m$`'d to a box `xs` of arguments == the `m$`ing of `fs` to the `m$`ing of `gs` to xs: `(mid (○) m$ fs m$ gs) m$ xs = fs m$ (gs m$ xs)`. - 4. When the arguments are `mid`'d, the order of `m$`ing doesn't matter: `fs m$ (mid x) = mid ($x) m$ fs`. (Note that it's `mid ($x)`, or `mid (\f. f x)` that gets `m$`d onto `fs`, not the original `mid x`.) Here's an example where the order *does* matter: `[succ,pred] m$ [1,2] == [2,3,0,1]`, but `[($1),($2)] m$ [succ,pred] == [2,0,3,1]`. This Law states a class of cases where the order is guaranteed not to matter. - 5. A consequence of the laws already stated is that when the functions are `mid`'d, the order of `m$`ing doesn't matter either: `mid f m$ xs == map (flip ($)) xs m$ mid f`. + 4. When the arguments (the right-hand operand of `m$`) are an `mid`'d value, the order of `m$`ing doesn't matter: `fs m$ (mid x) = mid ($x) m$ fs`. (Though note that it's `mid ($x)`, or `mid (\f. f x)` that gets `m$`d onto `fs`, not the original `mid x`.) Here's an example where the order *does* matter: `[succ,pred] m$ [1,2] == [2,3,0,1]`, but `[($1),($2)] m$ [succ,pred] == [2,0,3,1]`. This Law states a class of cases where the order is guaranteed not to matter. + 5. A consequence of the laws already stated is that when the _left_-hand operand of `m$` is a `mid`'d value, the order of `m$`ing doesn't matter either: `mid f m$ xs == map (flip ($)) xs m$ mid f`.