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=60a36aaded782146433a0564da73dc66467942d1;hb=caa10a9060a7295040ab497c68621522c4628595;hpb=c8fd5635d402bd3ae8073725257bef20e161256e diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn index 60a36aad..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 @@ -83,17 +83,16 @@ For instance, the following are Kleisli arrows: In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type Q is bool). -Note that the left-hand schema `P` is permitted to itself be a boxed -type. That is, if `α list` is our box type, and `P` is to boxed type -`int list`, we can write the boxed type that has `P` as its left-hand -side as +Note that either of the schemas `P` or `Q` are permitted to themselves be boxed +types. That is, if `α list` is our box type, we can write the second type as: int -> int list -If it's clear that we're uniformly talking about the same box type (in -this example, `α list`), we can equivalently write +And also what the rhs there is a boxing of is itself a boxed type (with the same kind of box):, so we can write it as: -int -> int +int -> int + +We have to be careful though not to to unthinkingly equivocate between different kinds of boxes. Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are int -> int (that is, `int -> int list`) or int -> bool: @@ -121,39 +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$` as an infix operator, reminiscent of `$` which is just ordinary function application (also expressed by mere 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* @@ -176,10 +177,26 @@ 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`. In examples we'll be working with at first, order _never_ matters; but down the road, sometimes it will. This Law states a class of cases where it's guaranteed not to. - 5. A consequence of the laws already stated is that when the functions are `mid`'d, the order of `m$`ing doesn't matter either: TODO - + 3. The `map2`ing of composition onto boxes `fs` and `gs` of functions, when `m$`'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 (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`. + + * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there is in addition an associative `mcomp` having `mid` as its left and @@ -203,7 +220,9 @@ has to obey the following Map Laws: u >>= mid == u mid a >>= k == k a - Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context. + (Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context.) Some authors try to make the first of those Laws look more symmetrical by writing it as: + + (A >>= \a -> B) >>= \b -> C == A >>= (\a -> B >>= \b -> C) > In Category Theory discussion, the Monad Laws are instead expressed in terms of `join` (which they call `μ`) and `mid` (which they call `η`). These are assumed to be "natural transformations" for their box type, which means that they satisfy these equations with that box type's `map`: > 
map f ○ mid == mid ○ f
map f ○ join == join ○ map (map f)
@@ -212,15 +231,16 @@ has to obey the following Map Laws: > The first of these says that if you have a triply-boxed type, and you first merge the inner two boxes (with `map join`), and then merge the resulting box with the outermost box, that's the same as if you had first merged the outer two boxes, and then merged the resulting box with the innermost box. The second law says that if you take a box type and wrap a second box around it (with `mid`) and then merge them, that's the same as if you had done nothing, or if you had instead wrapped a second box around each element of the original (with `map mid`, leaving the original box on the outside), and then merged them.

> The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type α: > 

μ ○ M(μ) == μ ○ μ
μ ○ η == id == μ ○ M(η)
 + > A word of advice: if you're doing any work in this conceptual neighborhood and need a Greek letter, don't use μ. In addition to the preceding usage, there's also a use in recursion theory (for the minimization operator), in type theory (as a fixed point operator for types), and in the λμ-calculus, which is a formal system that deals with _continuations_, which we will focus on later in the course. So μ already exhibits more ambiguity than it can handle. As hinted in last week's homework and explained in class, the operations available in a Mappable system exactly preserve the "structure" of the boxed type they're operating on, and moreover are only sensitive to what content is in the corresponding original position. If you say `map f [1,2,3]`, then what ends up in the first position of the result depends only on how `f` and `1` combine. -For MapNable operations, on the other hand, the structure of the result may instead by a complex function of the structure of the original arguments. But only of their structure, not of their contents. And if you say `map2 f [10,20] [1,2,3]`, what ends up in the first position of the result depends only on how `f` and `10` and `1` combine. +For MapNable operations, on the other hand, the structure of the result may instead be a complex function of the structure of the original arguments. But only of their structure, not of their contents. And if you say `map2 f [10,20] [1,2,3]`, what ends up in the first position of the result depends only on how `f` and `10` and `1` combine. With `map`, you can supply an `f` such that `map f [3,2,0,1] == [[3,3,3],[2,2],[],[1]]`. But you can't transform `[3,2,0,1]` to `[3,3,3,2,2,1]`, and you can't do that with MapNable operations, either. That would involve the structure of the result (here, the length of the list) being sensitive to the content, and not merely the structure, of the original. -For Monads (Composables), you can perform more radical transformations of that sort. For example, `join (map (\x. dup x x) [3,2,0,1])` would give us `[3,3,3,2,2,1]` (for a suitable definition of `dup`). +For Monads (Composables), on the other hand, you can perform more radical transformations of that sort. For example, `join (map (\x. dup x x) [3,2,0,1])` would give us `[3,3,3,2,2,1]` (for a suitable definition of `dup`).