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+
+[[!toc levels=2]]
+
+The Reader Monad
+================
+
+The goal for this part is to introduce the Reader Monad, and present
+two linguistics applications: binding and intensionality. Along the
+way, we'll continue to think through issues related to order, and a
+related notion of flow of information.
+
+At this point, we've seen monads in general, and three examples of
+monads: the identity monad (invisible boxes), the Maybe monad (option
+types), and the List monad.
+
+We've also seen an application of the Maybe monad to safe division.
+The starting point was to allow the division function to return an int
+option instead of an int. If we divide 6 by 2, we get the answer Just
+3. But if we divide 6 by 0, we get the answer Nothing.
+
+The next step was to adjust the other arithmetic functions to teach
+them what to do if they received Nothing instead of a boxed integer.
+This meant changing the type of their input from ints to int options.
+But we didn't need to do this piecemeal; rather, we "lift"ed the
+ordinary arithmetic operations into the monad using the various tools
+provided by the monad.
+
+We'll go over this lifting operation in detail in the next section.
+
+## Tracing the effect of safe-div on a larger computation
+
+So let's see how this works in terms of a specific computation.
+
+
+
+(+ 1 (* (/ 6 2) 4)) in tree format:
+
+ ___________
+ | |
+_|__ ___|___
+| | | |
++ 1 __|___ 4
+ | |
+ * __|___
+ | |
+ _|__ 2
+ | |
+ / 6
+
+
+No matter which arithmetic operation we begin with, this computation
+should eventually reduce to 13. Given a specific reduction strategy,
+we can watch the order in which the computation proceeds. Following
+on the lambda evaluator developed during the previous homework, let's
+adopt the following reduction strategy:
+
+ In order to reduce an expression of the form (head arg), do the following in order:
+ 1. Reduce head to h'
+ 2. Reduce arg to a'.
+ 3. If (h' a') is a redex, reduce it.
+
+There are many details left unspecified here, but this will be enough
+for today. The order in which the computation unfolds will be
+
+ 1. Reduce head (+ 1) to itself
+ 2. Reduce arg ((* ((/ 6) 2)) 3)
+ 1. Reduce head (* ((/ 6) 2))
+ 1. Reduce head *
+ 2. Reduce arg ((/ 6) 2)
+ 1. Reduce head (/ 6) to itself
+ 2. Reduce arg 2 to itself
+ 3. Reduce ((/ 6) 2) to 3
+ 3. Reduce (* 3) to itself
+ 2. Reduce arg 4 to itself
+ 3. Reduce ((* 3) 4) to 12
+ 3. Reduce ((+ 1) 12) to 13
+
+This reduction pattern follows the structure of the original
+expression exactly, at each node moving first to the left branch,
+processing the left branch, then moving to the right branch, and
+finally processing the results of the two subcomputation. (This is
+called depth-first postorder traversal of the tree.)
+
+[diagram with arrows traversing the tree]
+
+It will be helpful to see how the types change as we make adjustments.
+
+ type num = int
+ type contents = Num of num | Op2 of (num -> num -> num)
+ type tree = Leaf of contents | Branch of tree * tree
+
+Never mind that these types will allow us to construct silly
+arithmetric trees such as `+ *` or `2 3`. Note that during the
+reduction sequence, the result of reduction was in every case a
+well-formed subtree. So the process of reduction could be animated by
+replacing subtrees with the result of reduction on that subtree, till
+the entire tree is replaced by a single integer (namely, 13).
+
+Now we replace the number 2 with 0:
+
+
+
+ ___________
+ | |
+_|__ ___|___
+| | | |
++ 1 __|___ 4
+ | |
+ * __|___
+ | |
+ _|__ 0
+ | |
+ / 6
+
+
+When we reduce, we get quite a ways into the computation before things
+break down:
+
+ 1. Reduce head (+ 1) to itself
+ 2. Reduce arg ((* ((/ 6) 0)) 3)
+ 1. Reduce head (* ((/ 6) 0))
+ 1. Reduce head *
+ 2. Reduce arg ((/ 6) 0)
+ 1. Reduce head (/ 6) to itself
+ 2. Reduce arg 0 to itself
+ 3. Reduce ((/ 6) 0) to ACKKKK
+
+This is where we replace `/` with `safe-div`.
+Safe-div returns not an int, but an int option. If the division goes
+through, the result is Just n, where n is the integer result.
+But if the division attempts to divide by zero, the result is Nothing.
+
+We could try changing the type of the arithmetic operators from `int
+-> int -> int` to `int -> int -> int option`; but since we now have to
+anticipate the possibility that any argument might involve division by
+zero inside of it, it would be better to prepare for the possibility
+that any subcomputation might return here is the net result for our
+types. The easy way to do that is to change (only) the type of num
+from int to int option, leaving everying else the same:
+
+ type num = int option
+ type contents = Num of num | Op2 of (num -> num -> num)
+ type tree = Leaf of contents | Branch of tree * tree
+
+The only difference is that instead of defining our numbers to be
+simple integers, they are now int options; and so Op is an operator
+over int options.
+
+At this point, we bring in the monadic machinery. In particular, here
+is the ⧠and the map2 function from the notes on safe division:
+
+ ⧠(a: 'a) = Just a;;
+
+ map2 (g : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) =
+ match u with
+ | None -> None
+ | Some x ->
+ (match v with
+ | None -> None
+ | Some y -> Some (g x y));;
+
+Then we lift the entire computation into the monad by applying ⧠to
+the integers, and by applying `map2` to the operators:
+
+
+
+ ___________________
+ | |
+ ___|____ ____|_____
+ | | | |
+map2 + â§1 _____|_____ â§4
+ | |
+ map2 * ____|____
+ | |
+ ___|____ â§0
+ | |
+ map2 / â§6
+
+
+With these adjustments, the faulty computation now completes smoothly:
+
+ 1. Reduce head ((map2 +) â§1)
+ 2. Reduce arg (((map2 *) (((map2 /) â§6) â§2)) â§3)
+ 1. Reduce head ((map2 *) (((map2 /) â§6) â§2))
+ 1. Reduce head *
+ 2. Reduce arg (((map2 /) â§6) â§0)
+ 1. Reduce head ((map2 /) â§6)
+ 2. Reduce arg â§0
+ 3. Reduce (((map2 /) â§6) â§0) to Nothing
+ 3. Reduce ((map2 *) Nothing) to Nothing
+ 2. Reduce arg â§4
+ 3. Reduce (((map2 *) Nothing) â§4) to Nothing
+ 3. Reduce (((map2 +) â§1) Nothing) to Nothing
+
+As soon as we try to divide by 0, safe-div returns Nothing.
+Thanks to the details of map2, the fact that Nothing has been returned
+by one of the arguments of a map2-ed operator guarantees that the
+map2-ed operator will pass on the Nothing as its result. So the
+result of each enclosing computation will be Nothing, up to the root
+of the tree.
+
+It is unfortunate that we need to continue the computation after
+encountering our first Nothing. We know immediately at the result of
+the entire computation will be Nothing, yet we continue to compute
+subresults and combinations. It would be more efficient to simply
+jump to the top as soon as Nothing is encoutered. Let's call that
+strategy Abort. We'll arrive at an Abort operator later in the semester.
+
+So at this point, we can see how the Maybe/option monad provides
+plumbing that allows subcomputations to send information from one part
+of the computation to another. In this case, the safe-div function
+can send the information that division by zero has been attempted
+throughout the rest of the computation. If you think of the plumbing
+as threaded through the tree in depth-first, postorder traversal, then
+safe-div drops Nothing into the plumbing half way through the
+computation, and that Nothing travels through the rest of the plumbing
+till it comes out of the result faucet at the top of the tree.
+
+## Information flowing in the other direction: top to bottom
+
+We can think of this application as facilitating information flow.
+In the save-div example, a subcomputation created a message that
+propagated upwards to the larger computation:
+
+
+ message: Division by zero occurred!
+ ^
+ ___________ |
+ | | |
+_|__ ___|___ |
+| | | | |
++ 1 __|___ 4 |
+ | | |
+ * __|___ -----|
+ | |
+ _|__ 0
+ | |
+ / 6
+
+
+We might want to reverse the direction of information flow, making
+information available at the top of the computation available to the
+subcomputations:
+
+
+ [λx]
+ ___________ |
+ | | |
+_|__ ___|___ |
+| | | | |
++ 1 __|___ 4 |
+ | | |
+ * __|___ |
+ | | |
+ _|__ x <----|
+ | |
+ / 6
+
+
+We've seen exactly this sort of configuration before: it's exactly
+what we have when a lambda binds a variable that occurs in a deeply
+embedded position. Whatever the value of the argument that the lambda
+form combines with, that is what will be substituted in for free
+occurrences of that variable within the body of the lambda.
+
+## Binding
+
+So our next step is to add a (primitive) version of binding to our
+computation. We'll allow for just one binding dependency for now, and
+then generalize later.
+
+Binding is independent of the safe division, so we'll return to a
+situation in which the Maybe monad hasn't been added. One of the nice
+properties of programming with monads is that it is possible to add or
+subtract layers according to the needs of the moment. Since we need
+simplicity, we'll set the Maybe monad aside for now.
+
+So we'll go back to the point where the num type is simple int, not
+int options.
+
+ type num = int
+
+And we'll start with the computation the map2 or the ⧠from the option
+monad.
+
+As you might guess, the technique we'll use to arrive at binding will
+be to use the Reader monad, defined here in terms of m-identity and bind:
+
+ α ==> int -> α (* The ==> is a Kleisli arrow *)
+ â§a = \x.a
+ u >>= f = \x.f(ux)x
+ map f u = \x.f(ux)
+ map2 f u v = \x.f(ux)(vx)
+
+
+A boxed type in this monad will be a function from an integer to an
+object in the original type. The unit function ⧠lifts a value `a` to
+a function that expects to receive an integer, but throws away the
+integer and returns `a` instead (most values in the computation don't
+depend on the input integer).
+
+The bind function in this monad takes a monadic object `u`, a function
+`f` lifting non-monadic objects into the monad, and returns a function
+that expects an integer `x`. It feeds `x` to `u`, which delivers a
+result in the orginal type, which is fed in turn to `f`. `f` returns
+a monadic object, which upon being fed an integer, returns an object
+of the orginal type.
+
+The map2 function corresponding to this bind operation is given
+above. It should look familiar---we'll be commenting on this
+familiarity in a moment.
+
+Lifing the computation into the monad, we have the following adjusted
+types:
+
+type num = int -> int
+
+That is, `num` is once again replaced with the type of a boxed int.
+When we were dealing with the Maybe monad, a boxed int had type `int
+option`. In this monad, a boxed int has type `int -> int`.
+
+
+
+ __________________
+ | |
+ ___|____ ____|_____
+ | | | |
+map2 + â§1 ____|_____ â§4
+ | |
+ map2 * ___|____
+ | |
+ ___|____ x
+ | |
+ map2 / â§6
+
+
+It remains only to decide how the variable `x` will access the value input
+at the top of the tree. Since the input value is supposed to be the
+value put in place of the variable `x`. Like every leaf in the tree
+in argument position, the code we want in order to represent the
+variable will have the type of a boxed int, namely, `int -> int`. So
+we have the following:
+
+ x = (\fun (i:int) = i)
+
+That is, variables in this system denote the indentity function!
+
+The result of evaluating this tree will be a boxed integer: a function
+from any integer `x` to `(+ 1 (* (/ 6 x)) 4)`.
+
+Take a look at the definition of the reader monad again. The
+midentity takes some object `a` and returns `\x.a`. In other words,
+`â§a = Ka`, so `⧠= K`. Likewise, the reason the `map2` function
+looked familiar is that it is essentially the `S` combinator.
+
+We've seen this before as a strategy for translating a binding
+construct into a set of combinators. To remind you, here is a part of
+the general scheme for translating a lambda abstract into Combinatory
+Logic. The translation function `[.]` translates a lambda term into a
+term in Combinatory Logic:
+
+ [(MN)] = ([M] [N])
+ [\a.a] = I
+ [\a.M] = K[M] (assuming a not free in M)
+ [\a.(MN)] = S[\a.M][\a.N]
+
+The reason we can make do with this subset of the full translation
+scheme is that we're making the simplifying assumption that there is
+at most a single lambda involved. So once again we have the identity
+function I as the translation of the bound variable, K as the function
+governing expressions that don't contain an instance of the bound
+variable, S as the operation that manages the combination of complex
+expressions.
+
+## Jacobson's Variable Free Semantics as a Reader Monad
+
+We've designed the presentation above to make it as easy as possible
+to show that Jacobson's Variable Free Semantics (e.g., Jacobson 1999,
+[Towards a Variable-Free
+Semantics](http://www.springerlink.com/content/j706674r4w217jj5/))
+implements a reader monad.
+
+More specifically, it will turn out that Jacobson's geach combinator
+*g* is exactly our `lift` operator, and her binding combinator *z* is
+exactly our `bind` (though with the arguments reversed).
+
+Jacobson's system contains two main combinators, *g* and *z*. She
+calls *g* the Geach rule, and *z* performs binding. Here is a typical
+computation. This implementation is based closely on email from Simon
+Charlow, with beta reduction as performed by the on-line evaluator:
+
+
+; Jacobson's analysis of "Everyone_i thinks he_i left"
+let g = \f u. \x. f (u x) in
+let z = \f u. \x. f (u x) x in
+let he = \x. x in
+let everyone = \P. FORALL x (P x) in
+
+everyone (z thinks (g left he))
+
+~~> FORALL x (thinks (left x) x)
+
+
+Two things to notice: First, pronouns once again denote identity
+functions, just as we saw in the reader monad in the previous section.
+
+Second, *g* plays the role of transmitting a binding dependency for an
+embedded constituent to a containing constituent.
+
+The basic recipe in Jacobson's system is that you transmit the
+dependence of a pronoun upwards through the tree using *g* until just
+before you are about to combine with the binder, when you finish off
+with *z*. Here is an example with a longer chain of *g*'s:
+
+
+everyone (z thinks (g (t bill) (g said (g left he))))
+
+~~> FORALL x (thinks (said (left x) bill) x)
+
+
+If you compare Jacobson's values for *g* and *z* to the functions in
+the reader monad given above, you'll see that Jacobson's *g*
+combinator is exactly our `map` operator. Furthermore, Jacobson's *z*
+combinator is identital to `>>=`, except with the order of the
+arguments reversed (i.e., `(z f u) == (u >>= f)`).
+
+(The `t` combinator in the derivations above is given by `t x =
+\xy.yx`; it handles situations in which English word order reverses
+the usual function/argument order.)
+
+In other words,
+Jacobson's variable-free semantics is essentially a Reader monad.
+
+One of the peculiar aspects of Jacobson's system is that binding is
+accomplished not by applying *z* to the element that will (in some
+pre-theoretic sense) bind the pronoun, here, *everyone*, but rather by
+applying *z* instead to the predicate that will take *everyone* as an
+argument, here, *thinks*.
+
+
+## The Reader monad for intensionality
+
+[This section has not been revised since 2010, so there may be a few
+places where it doesn't follow the convetions we've adopted this time;
+nevertheless, it should be followable.]
+
+Now we'll look at using the Reader monad to do intensional function application.
+
+In Shan (2001) [Monads for natural
+language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that
+making expressions sensitive to the world of evaluation is conceptually
+the same thing as making use of the Reader monad.
+This technique was beautifully re-invented
+by Ben-Avi and Winter (2007) in their paper [A modular
+approach to
+intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf),
+though without explicitly using monads.
+
+All of the code in the discussion below can be found here: [[code/intensionality-monad.ml]].
+To run it, download the file, start OCaml, and say
+
+ # #use "intensionality-monad.ml";;
+
+Note the extra `#` attached to the directive `use`.
+
+First, the familiar linguistic problem:
+
+ Bill left.
+ Cam left.
+ Ann believes [Bill left].
+ Ann believes [Cam left].
+
+We want an analysis on which the first three sentences can be true at
+the same time that the last sentence is false. If sentences denoted
+simple truth values or booleans, we have a problem: if the sentences
+*Bill left* and *Cam left* are both true, they denote the same object,
+and Ann's beliefs can't distinguish between them.
+
+The traditional solution to the problem sketched above is to allow
+sentences to denote a function from worlds to truth values, what
+Montague called an intension. So if `s` is the type of possible
+worlds, we have the following situation:
+
+
+
+Extensional types Intensional types Examples
+-------------------------------------------------------------------
+
+S t s->t John left
+DP e s->e John
+VP e->t (s->e)->s->t left
+Vt e->e->t (s->e)->(s->e)->s->t saw
+Vs t->e->t (s->t)->(s->e)->s->t thought
+
+
+This system is modeled on the way Montague arranged his grammar.
+There are significant simplifications compared to Montague: for
+instance, determiner phrases are thought of here as corresponding to
+individuals rather than to generalized quantifiers.
+
+The main difference between the intensional types and the extensional
+types is that in the intensional types, the arguments are functions
+from worlds to extensions: intransitive verb phrases like "left" now
+take so-called "individual concepts" as arguments (type s->e) rather than plain
+individuals (type e), and attitude verbs like "think" now take
+propositions (type s->t) rather than truth values (type t).
+In addition, the result of each predicate is an intension.
+This expresses the fact that the set of people who left in one world
+may be different than the set of people who left in a different world.
+
+Normally, the dependence of the extension of a predicate to the world
+of evaluation is hidden inside of an evaluation coordinate, or built
+into the the lexical meaning function, but we've made it explicit here
+in the way that the intensionality monad makes most natural.
+
+The intensional types are more complicated than the extensional
+types. Wouldn't it be nice to make the complicated types available
+for those expressions like attitude verbs that need to worry about
+intensions, and keep the rest of the grammar as extensional as
+possible? This desire is parallel to our earlier desire to limit the
+concern about division by zero to the division function, and let the
+other functions, like addition or multiplication, ignore
+division-by-zero problems as much as possible.
+
+So here's what we do:
+
+In OCaml, we'll use integers to model possible worlds. Characters (characters in the computational sense, i.e., letters like `'a'` and `'b'`, not Kaplanian characters) will model individuals, and OCaml booleans will serve for truth values:
+
+ type s = int;;
+ type e = char;;
+ type t = bool;;
+
+ let ann = 'a';;
+ let bill = 'b';;
+ let cam = 'c';;
+
+ let left1 (x:e) = true;;
+ let saw1 (x:e) (y:e) = y < x;;
+
+ left1 ann;; (* true *)
+ saw1 bill ann;; (* true *)
+ saw1 ann bill;; (* false *)
+
+So here's our extensional system: everyone left, including Ann;
+and Ann saw Bill (`saw1 bill ann`), but Bill didn't see Ann. (Note that the word
+order we're using is VOS, verb-object-subject.)
+
+Now we add intensions. Because different people leave in different
+worlds, the meaning of *leave* must depend on the world in which it is
+being evaluated:
+
+ let left (x:e) (w:s) = match (x, w) with ('c', 2) -> false | _ -> true;;
+ left ann 1;; (* true: Ann left in world 1 *)
+ left cam 2;; (* false: Cam didn't leave in world 2 *)
+
+This new definition says that everyone always left, except that
+in world 2, Cam didn't leave.
+
+Note that although this general *left* is sensitive to world of
+evaluation, it does not have the fully intensionalized type given in
+the chart above, which was `(s->e)->s->t`. This is because
+*left* does not exploit the additional resolving power provided by
+making the subject an individual concept. In semantics jargon, we say
+that *leave* is extensional with respect to its first argument.
+
+Therefore we will adopt the general strategy of defining predicates
+in a way that they take arguments of the lowest type that will allow
+us to make all the distinctions the predicate requires. When it comes
+time to combine this predicate with monadic arguments, we'll have to
+make use of various lifting predicates.
+
+Likewise, although *see* depends on the world of evaluation, it is
+extensional in both of its syntactic arguments:
+
+ let saw x y w = (w < 2) && (y < x);;
+ saw bill ann 1;; (* true: Ann saw Bill in world 1 *)
+ saw bill ann 2;; (* false: no one saw anyone in world 2 *)
+
+This (again, partially) intensionalized version of *see* coincides
+with the `saw1` function we defined above for world 1; in world 2, no
+one saw anyone.
+
+Just to keep things straight, let's review the facts:
+
+
+ World 1: Everyone left.
+ Ann saw Bill, Ann saw Cam, Bill saw Cam, no one else saw anyone.
+ World 2: Ann left, Bill left, Cam didn't leave.
+ No one saw anyone.
+
+
+Now we are ready for the intensionality monad:
+
+
+type 'a intension = s -> 'a;;
+let unit x = fun (w:s) -> x;;
+(* as before, bind can be written more compactly, but having
+ it spelled out like this will be useful down the road *)
+let bind u f = fun (w:s) -> let a = u w in let u' = f a in u' w;;
+
+
+Then the individual concept `unit ann` is a rigid designator: a
+constant function from worlds to individuals that returns `'a'` no
+matter which world is used as an argument. This is a typical kind of
+thing for a monad unit to do.
+
+Then combining a predicate like *left* which is extensional in its
+subject argument with an intensional subject like `unit ann` is simply bind
+in action:
+
+ bind (unit ann) left 1;; (* true: Ann left in world 1 *)
+ bind (unit cam) left 2;; (* false: Cam didn't leave in world 2 *)
+
+As usual, bind takes a monad box containing Ann, extracts Ann, and
+feeds her to the extensional *left*. In linguistic terms, we take the
+individual concept `unit ann`, apply it to the world of evaluation in
+order to get hold of an individual (`'a'`), then feed that individual
+to the extensional predicate *left*.
+
+We can arrange for a transitive verb that is extensional in both of
+its arguments to take intensional arguments:
+
+ let lift2' f u v = bind u (fun x -> bind v (fun y -> f x y));;
+
+This is almost the same `lift2` predicate we defined in order to allow
+addition in our division monad example. The difference is that this
+variant operates on verb meanings that take extensional arguments but
+returns an intensional result. Thus the original `lift2` predicate
+has `unit (f x y)` where we have just `f x y` here.
+
+The use of `bind` here to combine *left* with an individual concept,
+and the use of `lift2'` to combine *see* with two intensional
+arguments closely parallels the two of Montague's meaning postulates
+(in PTQ) that express the relationship between extensional verbs and
+their uses in intensional contexts.
+
+
+lift2' saw (unit bill) (unit ann) 1;; (* true *)
+lift2' saw (unit bill) (unit ann) 2;; (* false *)
+
+
+Ann did see bill in world 1, but Ann didn't see Bill in world 2.
+
+Finally, we can define our intensional verb *thinks*. *Think* is
+intensional with respect to its sentential complement, though still extensional
+with respect to its subject. (As Montague noticed, almost all verbs
+in English are extensional with respect to their subject; a possible
+exception is "appear".)
+
+ let thinks (p:s->t) (x:e) (w:s) =
+ match (x, p 2) with ('a', false) -> false | _ -> p w;;
+
+Ann disbelieves any proposition that is false in world 2. Apparently,
+she firmly believes we're in world 2. Everyone else believes a
+proposition iff that proposition is true in the world of evaluation.
+
+ bind (unit ann) (thinks (bind (unit bill) left)) 1;;
+
+So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).
+
+ bind (unit ann) (thinks (bind (unit cam) left)) 1;;
+
+But in world 1, Ann doesn't believe that Cam left (even though he
+did leave in world 1: `bind (unit cam) left 1 == true`). Ann's thoughts are hung up on
+what is happening in world 2, where Cam doesn't leave.
+
+*Small project*: add intersective ("red") and non-intersective
+ adjectives ("good") to the fragment. The intersective adjectives
+ will be extensional with respect to the nominal they combine with
+ (using bind), and the non-intersective adjectives will take
+ intensional arguments.
+
+notes: cascade, general env