last tweaks to monad page for now

[lambda.git] / topics / week7_introducing_monads.mdwn

diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn
index e4dae12..bc719e0 100644 (file)
--- a/topics/week7_introducing_monads.mdwn
+++ b/topics/week7_introducing_monads.mdwn
@@ -1,4 +1,4 @@
-<!-- λ Λ ∀ ≡ α β γ ρ ω Ω -->
+<!-- λ Λ ∀ ≡ α β γ ρ ω Ω ○ μ η δ ζ ξ ⋆ ★ • ∙ ● 𝟎 𝟏 𝟐 𝟘 𝟙 𝟚 𝟬 𝟭 𝟮 -->

 <!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
 
 Introducing Monads
 <!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
 
 Introducing Monads


@@ -22,7 +22,7 @@ any case, our emphasis will be on starting with the abstract structure
 of monads, followed by instances of monads from the philosophical and
 linguistics literature.
 
 of monads, followed by instances of monads from the philosophical and
 linguistics literature.
 

-> <small>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:
+> <small>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)
 [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 
 
     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
 
 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,
 
 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.
 
 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
 
 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:

 
 <code><u>int</u></code>
 
 
 <code><u>int</u></code>
 


@@ -75,7 +75,7 @@ A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Do
 <code>P -> <u>Q</u></code>
 
 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`.
 <code>P -> <u>Q</u></code>
 
 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:
+For instance, the following are Kleisli arrow types:

 
 <code>int -> <u>bool</u></code>
 
 
 <code>int -> <u>bool</u></code>
 


@@ -83,10 +83,17 @@ For instance, the following are Kleisli arrows:
 
 In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
 
 
 In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
 

-Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where if `α list` is our box type, we can write the second type 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:

 
 <code><u>int</u> -> <u>int list</u></code>
 
 
 <code><u>int</u> -> <u>int list</u></code>
 

+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:
+
+<code><u>int</u> -> <span class="box2">int</span></code>
+
+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 <code>int -> <u>int</u></code> (that is, `int -> int list`) or <code>int -> <u>bool</u></code>:
 
 <pre>\x. [x]
 Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are <code>int -> <u>int</u></code> (that is, `int -> int list`) or <code>int -> <u>bool</u></code>:
 
 <pre>\x. [x]


@@ -113,11 +120,11 @@ Here are the types of our crucial functions, together with our pronunciation, an
 
 <code>mid (/εmaidεnt@tI/): P -> <u>P</u></code>
 
 
 <code>mid (/εmaidεnt@tI/): P -> <u>P</u></code>
 

-> 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`. This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`.
+> 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`.

 
 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
 
 
 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
 

-> In Haskell, this is called `Control.Monad.ap` or `Control.Applicative.<*>`. In the class presentation Jim called it `●`.
+> 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.<*>`.

 
 <code>&lt;=&lt; or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
 
 
 <code>&lt;=&lt; or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
 


@@ -125,7 +132,7 @@ Here are the types of our crucial functions, together with our pronunciation, an
 
 <code>&gt;=&gt; (flip mcomp, should we call it mpmoc?): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
 
 
 <code>&gt;=&gt; (flip mcomp, should we call it mpmoc?): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
 

-> In Haskell, this is `Control.Monad.>=>`.
+> 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.

 
 <code>&gt;&gt;= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
 
 
 <code>&gt;&gt;= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
 


@@ -135,14 +142,11 @@ Here are the types of our crucial functions, together with our pronunciation, an
 
 > In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
 
 
 > 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".

 
 

-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.
+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.)

 
 

-Haskell's name "bind" for `>>=` is not well chosen from our perspective, but this is too deeply entrenched by now. We've at least preprended an `m` to the front of it.
-
-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.) In the handout we called `mid` `𝟭`. But now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".)
-
-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: <code>k <=< j ≡ \a. (j a >>= k)</code>.
+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: <code>k <=< j ≡ \a. (j a >>= k)</code>. 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
 
 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


@@ -169,8 +173,28 @@ has to obey the following Map Laws:
        Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
        have to obey the following MapN Laws:
 
        Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
        have to obey the following MapN Laws:
 

-    TODO LAWS
-
+    1. <code>mid (id : P->P) : <u>P</u> -> <u>P</u></code> 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 (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`.
+
+<!-- Probably there's a shorter proof, but:
+   mid T m$ xs m$ mid f
+== mid T m$ ((mid id) m$ xs) m$ mid f, by 1
+== mid (○) m$ mid T m$ mid id m$ xs m$ mid f, by 3
+== mid ($id) m$ (mid (○) m$ mid T) m$ xs m$ mid f, by 4
+== mid (○) m$ mid ($id) m$ mid (○) m$ mid T m$ xs m$ mid f, by 3
+== mid ((○) ($id)) m$ mid (○) m$ mid T m$ xs m$ mid f, by 2
+== mid ((○) ($id) (○)) m$ mid T m$ xs m$ mid f, by 2
+== mid id m$ mid T m$ xs m$ mid f, by definitions of ○ and $
+== mid T m$ xs m$ mid f, by 1
+== mid ($f) m$ (mid T m$ xs), by 4
+== mid (○) m$ mid ($f) m$ mid T m$ xs, by 3
+== mid ((○) ($f)) m$ mid T m$ xs, by 2
+== mid ((○) ($f) T) m$ xs, by 2
+== mid f m$ xs, by definitions of ○ and $ and T == flip ($)
+-->

 
 *   ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
        is in addition an associative `mcomp` having `mid` as its left and
 
 *   ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
        is in addition an associative `mcomp` having `mid` as its left and


@@ -182,7 +206,7 @@ has to obey the following Map Laws:
 
     You could just as well express the Monad laws using `>=>`:
 
 
     You could just as well express the Monad laws using `>=>`:
 

-        l >=> (k >=> j) == (l >=> k) >-> j
+        l >=> (k >=> j) == (l >=> k) >=> j

         k >=> mid == k
         mid >=> k == k
 
         k >=> mid == k
         mid >=> k == k
 


@@ -200,20 +224,35 @@ has to obey the following Map Laws:
     > <pre>map f ○ mid == mid ○ f<br>map f ○ join == join ○ map (map f)</pre>
     > The Monad Laws then take the form:
     > <pre>join ○ (map join) == join ○ join<br>join ○ mid == id == join ○ map mid</pre>
     > <pre>map f ○ mid == mid ○ f<br>map f ○ join == join ○ map (map f)</pre>
     > The Monad Laws then take the form:
     > <pre>join ○ (map join) == join ○ join<br>join ○ mid == id == join ○ map mid</pre>

-    > 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 instead mapped a second box around the elements of the original (with `map mid`, leaving the original box on the outside), and then merged them.<p>
+    > 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.<p>

     > The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
     > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre></small>
 
 
     > The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
     > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre></small>
 
 

+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 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), 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`).
+
+<!--
+Some global transformations that we work with in semantics, like Veltman's test functions, can't directly be expressed in terms of the  primitive Monad operations? For example, there's no `j` such that `xs >>= j == mzero` if `xs` anywhere contains the value `1`.
+-->
+

 
 ## Interdefinitions and Subsidiary notions##
 
 We said above that various of these box type operations can be defined in terms of others. Here is a list of various ways in which they're related. We try to stick to the consistent typing conventions that:
 
 <pre>
 
 ## Interdefinitions and Subsidiary notions##
 
 We said above that various of these box type operations can be defined in terms of others. Here is a list of various ways in which they're related. We try to stick to the consistent typing conventions that:
 
 <pre>

-f : α -> β; g and h have types of the same format (note that α and β are permitted to be, but needn't be, boxed types)
-j : α -> <u>β</u>; k and l have types of the same format
-u : <u>α</u>; v and xs and ys have types of the same format
+f : α -> β;  g and h have types of the same form
+             also sometimes these will have types of the form α -> β -> γ
+             note that α and β are permitted to be, but needn't be, boxed types
+j : α -> <u>β</u>; k and l have types of the same form
+u : <u>α</u>;      v and xs and ys have types of the same form
+

 w : <span class="box2">α</span>
 </pre>
 
 w : <span class="box2">α</span>
 </pre>
 


@@ -239,9 +278,9 @@ Here are some other monadic notion that you may sometimes encounter:
 
 * <code>mzero</code> is a value of type <code><u>α</u></code> that is exemplified by `Nothing` for the box type `Maybe α` and by `[]` for the box type `List α`. It has the behavior that `anything m$ mzero == mzero == mzero m$ anything == mzero >>= anything`. In Haskell, this notion is called `Control.Applicative.empty` or `Control.Monad.mzero`.
 
 
 * <code>mzero</code> is a value of type <code><u>α</u></code> that is exemplified by `Nothing` for the box type `Maybe α` and by `[]` for the box type `List α`. It has the behavior that `anything m$ mzero == mzero == mzero m$ anything == mzero >>= anything`. In Haskell, this notion is called `Control.Applicative.empty` or `Control.Monad.mzero`.
 

-* Haskell has a notion `>>` definable as `\u v. mid (const id) m$ u m$ v`. It works like this: `u >> v == u >>= const v`. This is often useful, and won't in general be identical to just `v`. For example, using the box type `List α`, `[1,2,3] >> [4,5] == [4,5,4,5,4,5]`. But in the special case of `mzero`, it is a consequence of what we said above that `anything >> mzero == mzero`. Haskell also calls `>>` `Control.Applicative.*>`.
+* Haskell has a notion `>>` definable as `\u v. map (const id) u m$ v`, or as `\u v. u >>= const v`. This is often useful, and `u >> v` won't in general be identical to just `v`. For example, using the box type `List α`, `[1,2,3] >> [4,5] == [4,5,4,5,4,5]`. But in the special case of `mzero`, it is a consequence of what we said above that `anything >> mzero == mzero`. Haskell also calls `>>` `Control.Applicative.*>`.

 
 

-* Haskell has a correlative notion `Control.Applicative.<*`, definable as `\u v. mid const m$ u m$ v`. For example, `[1,2,3] <* [4,5] == [1,1,2,2,3,3]`. You might expect Haskell to call `<*` `<<`, but they don't. They used to use `<<` for `flip (>>)` instead, but now they seem not to use it anymore. Maybe in the future they'll call `<*` `<<`.
+* Haskell has a correlative notion `Control.Applicative.<*`, definable as `\u v. map const u m$ v`. For example, `[1,2,3] <* [4,5] == [1,1,2,2,3,3]`. You might expect Haskell to call `<*` `<<`, but they don't. They used to use `<<` for `flip (>>)` instead, but now they seem not to use `<<` anymore.

 
 * <code>mapconst</code> is definable as `map ○ const`. For example `mapconst 4 [1,2,3] == [4,4,4]`. Haskell calls `mapconst` `<$` in `Data.Functor` and `Control.Applicative`. They also use `$>` for `flip mapconst`, and `Control.Monad.void` for `mapconst ()`.
 
 
 * <code>mapconst</code> is definable as `map ○ const`. For example `mapconst 4 [1,2,3] == [4,4,4]`. Haskell calls `mapconst` `<$` in `Data.Functor` and `Control.Applicative`. They also use `$>` for `flip mapconst`, and `Control.Monad.void` for `mapconst ()`.
 


@@ -255,8 +294,8 @@ then a boxed `α` is ... a `bool`. That is, <code><u>α</u> == α</code>.
 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
 definitions:
 
 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
 definitions:
 

-    mid ≡ \p. p
-    mcomp ≡ \f g x.f (g x)
+    mid ≡ \p. p, that is, our familiar combinator I
+    mcomp ≡ \f g x. f (g x), that is, ordinary function composition (○) (aka the B combinator)

 
 Identity is a monad.  Here is a demonstration that the laws hold:
 
 
 Identity is a monad.  Here is a demonstration that the laws hold:
 


@@ -306,7 +345,7 @@ For example:
 
 `j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
 
 
 `j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
 

-Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
+Contrast that to `m$` (`mapply`), which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:

 
     let js = [(\a->a*a),(\a->a+a)] in
     let xs = [7, 5] in
 
     let js = [(\a->a*a),(\a->a+a)] in
     let xs = [7, 5] in