-Here's what we did in seminar on Monday 9/13,
+These notes will recapitulate, make more precise, and to some degree expand what we did in the last hour of our first meeting, leading up to the definitions of the `factorial` and `length` functions.
-Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class. These notes expand on *a lot*, and some of this material will be reviewed next week.
+### Getting started ###
-Applications
-============
+We begin with a decidable fragment of arithmetic. Our language has some primitive literal values:
-We mentioned a number of linguistic and philosophical applications of the tools that we'd be helping you learn in the seminar. (We really do mean "helping you learn," not "teaching you." You'll need to aggressively browse and experiment with the material yourself, or nothing we do in a few two-hour sessions will succeed in inducing mastery of it.)
+ 0, 1, 2, 3, ...
-From linguistics
-----------------
+In fact we could get by with just the primitive literal `0` and the `succ` function, but we will make things a bit more convenient by allowing literal expressions of any natural number. We won't worry about numbers being too big for our finite computers to handle.
-* generalized quantifiers are a special case of operating on continuations
+We also have some predefined functions:
-* (Chris: fill in other applications...)
+ succ, +, *, pred, -
-* expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the [summary](/damn) for more explanation and elaboration
+Again, we might be able to get by with just `succ`, and define the others in terms of it, but we'll be a bit more relaxed. Since we want to stick with natural numbers, not the whole range of integers, we'll make `pred 0` just be `0`, and `2-4` also be `0`.
-From philosophy
----------------
+Here's another set of functions:
-* the natural semantics for positive free logic is thought by some to have objectionable ontological commitments; Jim says that thought turns on not understanding the notion of a "union type", and conflating the folk notion of "naming" with the technical notion of semantic value. We'll discuss this in due course.
+ ==, <, >, <=, >=, !=
-* those issues may bear on Russell's Gray's Elegy argument in "On Denoting"
+`==` is just what we non-programmers normally express by `=`. It's a relation that holds or not between two values. Here we'll treat it as a function that takes two values as arguments and returns a *boolean* value, that is a truth-value, as a result. The reason for using the doubled `=` symbol is that the single `=` symbol tends to get used in lots of different roles in programming, so we reserve `==` to express this meaning. I will deliberately try to minimize the uses of single `=` in this made-up language (but not eliminate it entirely), to reduce ambiguity and confusion. The `==` relation---or as we're treating it here, the `==` *function* that returns a boolean value---can at least take two numbers as arguments. Probably it makes sense for it to take other kinds of values as arguments, too. For example, it should operate on two truth-values as well. Maybe we'd want it to operate on a number and a truth-value, too? and always return false in that case? What about operating on two functions? Here we encounter the difficulty that the computer can't in general *decide* when two functions are equivalent. Let's not try to sort this all out just yet. We'll suppose that `==` can at least take two numbers as arguments, or two truth-values.
-* and on discussion of the difference between the meaning of "is beautiful" and "beauty," and the difference between the meaning of "that snow is white" and "the proposition that snow is white."
+As mentioned in class, we represent the truth-values like this:
-* the apparatus of monads, and techniques for statically representing the semantics of an imperatival language quite generally, are explicitly or implicitly invoked in dynamic semantics
+ 'true, 'false
-* the semantics for mutation will enable us to make sense of a difference between numerical and qualitative identity---for purely mathematical objects!
+These are instances of a broader class of literal values that I called *symbolic atoms*. We'll return to them shortly. The reason we write them with an initial `'` will also be explained shortly. For now, it's enough to note that the expression:
-* issues in that same neighborhood will help us better understand proposals like Kit Fine's that semantics is essentially coordinated, and that `R a a` and `R a b` can differ in interpretation even when `a` and `b` don't
+ 1 + 2 == 3
+evaluates to `'true`, and the expression:
+ 1 + 0 == 3
+evaluates to `'false`. Something else that evaluates to `'false` is the simple expression:
+ 'false
-Declarative/functional vs Imperatival/dynamic models of computation
-===================================================================
+That is, literal values are a limiting case of expression, that evaluate to just themselves. More complex expressions like `1 + 0` don't evaluate to themselves, but rather down to literal values.
-Many of you, like us, will have grown up thinking the paradigm of computation is a sequence of changes. Let go of that. It will take some care to separate the operative notion of "sequencing" here from other notions close to it, but once that's done, you'll see that languages that have no significant notions of sequencing or changes are Turing complete: they can perform any computation we know how to describe. In itself, that only puts them on equal footing with more mainstream, imperatival programming languages like C and Java and Python, which are also Turing complete. But further, the languages we want you to become familiar with can reasonably be understood to be more fundamental. They embody the elemental building blocks that computer scientists use when reasoning about and designing other languages.
+The functions `succ` and `pred` come before their arguments, like this:
-Jim offered the metaphor: think of imperatival languages, which include "mutation" and "side-effects" (we'll flesh out these keywords as we proceeed), as the pâté of computation. We want to teach you about the meat and potatoes, where as it turns out there is no sequencing and no changes. There's just the evaluation or simplification of complex expressions.
+ succ 1
-Now, when you ask the Scheme interpreter to simplify an expression for you, that's a kind of dynamic interaction between you and the interpreter. You may wonder then why these languages should not also be understood imperatively. The difference is that in a purely declarative or functional language, there are no dynamic effects in the language itself. It's just a static semantic fact about the language that one expression reduces to another. You may have verified that fact through your dynamic interactions with the Scheme interpreter, but that's different from saying that there are dynamic effects in the language itself.
+On the other hand, the functions `+`, `*`, `-`, `==`, and so on come in between their arguments, like this:
-What the latter would amount to will become clearer as we build our way up to languages which are genuinely imperatival or dynamic.
+ x < y
-Many of the slogans and keywords we'll encounter in discussions of these issues call for careful interpretation. They mean various different things.
+Functions of this latter sort are said to have an "infix" syntax. This is just a convenience for how we write them. Our language will have to keep rigorous track of which functions have infix syntax and which don't, but we'll just rely on context and our brains to make sense of this for now. Functions with the ordinary, non-infix syntax can take two arguments, as well. If we had defined the less-than relation (boolean function) in that style, we'd write it like this instead:
-For example, you'll encounter the claim that declarative languages are distinguished by their **referential transparency.** What's meant by this is not always exactly the same, and as a cluster, it's related to but not the same as this means for philosophers and linguists.
+ lessthan? (x, y)
-The notion of **function** that we'll be working with will be one that, by default, sometimes counts as non-identical functions that map all their inputs to the very same outputs. For example, two functions from jumbled decks of cards to sorted decks of cards may use different algorithms and hence be different functions.
+or perhaps like this:
-It's possible to enhance the lambda calculus so that functions do get identified when they map all the same inputs to the same outputs. This is called making the calculus **extensional**. Church called languages which didn't do this **intensional**. If you try to understand that kind of "intensionality" in terms of functions from worlds to extensions (an idea also associated with Church), you may hurt yourself. So too if you try to understand it in terms of mental stereotypes, another notion sometimes designated by "intension."
+ lessthan? x y
-It's often said that dynamic systems are distinguished because they are the ones in which **order matters**. However, there are many ways in which order can matter. If we have a trivalent boolean system, for example---easily had in a purely functional calculus---we might choose to give a truth-table like this for "and":
+We'll get more acquainted with the difference between these next week. For now, I'll just stick to the first form.
- true and true = true
+Another set of operations we have are:
-And then we'd notice that `* and false` has a different intepretation than `false and *`. (The same phenomenon is already present with the material conditional in bivalent logics; but seeing that a non-symmetric semantics for `and` is available even for functional languages is instructive.)
+ and, or, not
-Another way in which order can matter that's present even in functional languages is that the interpretation of some complex expressions can depend on the order in which sub-expressions are evaluated. Evaluated in one order, the computations might never terminate (and so semantically we interpret them as having "the bottom value"---we'll discuss this). Evaluated in another order, they might have a perfectly mundane value. Here's an example, though we'll reserve discussion of it until later:
+The first two of these are infix functions that expect two boolean arguments, and gives a boolean result. The third is a function that expects only one boolean argument. Our earlier function `!=` means "doesn't equal", and:
- (\x. y) ((\x. x x) (\x. x x))
+ x != y
-Again, these facts are all part of the metatheory of purely functional languages. But *there is* a different sense of "order matters" such that it's only in imperatival languages that order so matters.
+will in general be just another way to write:
- x := 2
- x := x + 1
- x == 3
+ not (x == y)
-Here the comparison in the last line will evaluate to true.
+You see that you can use parentheses in the standard way.
- x := x + 1
- x := 2
- x == 3
+I've started throwing in some variables. We'll say variables are any expression that starts with a lower-case letter, then is followed by a sequence of 0 or more upper- or lower-case letters, or underscores (`_`). Then at the end you can optionally have a `?` or `!` or a sequence of `'`s, understood as "prime" symbols. Hence, all of these are legal variables:
-Here the comparison in the last line will evaluate to false.
+ x
+ x1
+ x_not_y
+ xUBERANT
+ x'
+ x''
+ x?
+ xs
-One of our goals for this course is to get you to understand *what is* that new
-sense such that only so matters in imperatival languages.
+We'll follow a *convention* of using variables with short names and a final `s` to represent collections like sequences (to be discussed below). But this is just a convention to help us remember what we're up to, not a strict rule of the language. We'll also follow a convention of only using variables ending in `?` to represent functions that return a boolean value. Thus, for example, `zero?` will be a function that expects a single number argument and returns a boolean corresponding to whether that number is `0`. `odd?` will be a function that expects a single number argument and returns a boolean corresponding to whether than number is odd. Above, I suggested we might use `lessthan?` to represent a function that expects *two* number arguments, and again returns a boolean result.
-Finally, you'll see the term **dynamic** used in a variety of ways in the literature for this course:
+We also conventionally reserve variables ending in `!` for a different special class of functions, that we will explain later in the course.
-* dynamic versus static typing
+In fact you can think of `succ` and `pred` and `not` and all the rest as also being variables; it's just that these variables have been pre-defined in our language to be bound to special functions we designated in advance. You can even think of `==` and `<` as being variables, too, bound to other functions. But I haven't given you rules yet which would make them legal variables, because they don't start with a lower-case letter. We can make the rules more liberal later.
-* dynamic versus lexical scoping
+Only a few things in our language aren't variables. These include the **keywords** like `let` and `case` and so on that we'll discuss below. You can't use `let` as a variable, else the syntax of our language would become too hard to mechanically parse. (And probably too hard for our meager brains to parse, too.)
-* dynamic versus static control operators
+The rule for symbolic atoms is that a single quote `'` followed by any single word that could be a legal variable is a symbolic atom. Thus `'false` is a symbolic atom, but so too are `'x` and `'succ`. For the time being, I'll restrict myself to only talking about the symbolic atoms `'true` and `'false`. These are a special subgroup of symbolic atoms that we call the *booleans* or *truth-values*. Nothing deep hangs on these being a subclass of a larger category in this way; it just seems elegant. Other languages sometimes make booleans their own special type, not a subclass of any other limited type. Others make them a subclass of the numbers (yuck). We will think of them this way.
-* finally, we're used ourselves to talking about dynamic versus static semantics
+Note that in symbolic atoms there is no closing `'`, just a `'` at the beginning. That's enough to make the whole word, up to the next space (or whatever) count as naming a symbolic atom.
-For the most part, these uses are only loosely connected to each other. We'll tend to use "imperatival" to describe the kinds of semantic properties made available in dynamic semantics, languages which have robust notions of sequencing changes, and so on.
+We call these things symbolic *atoms* because they aren't collections. Thus numbers are also atoms, just not symbolic ones. And functions are also atoms, but again, not symbolic ones.
-Map
-===
+Functions are another class of values we'll have in our language. They aren't "literal" values, though. Numbers and symbolic atoms are simple expressions in the language that evaluate to themselves. That's what we mean by calling them "literals." Functions aren't expressions in the language at all; they have to be generated from the evaluation of more complex expressions.
-<table>
-<tr>
-<td width=30%>Scheme (functional part)</td>
-<td width=30%>OCaml (functional part)</td>
-<td width=30%>C, Java, Pasval<br>
-Scheme (imperative part)<br>
-OCaml (imperative part)</td>
-<tr>
-<td width=30%>lambda calculus<br>
-combinatorial logic</td>
-<tr>
-<td colspan=3 align=center>--------------------------------------------------- Turing complete ---------------------------------------------------</td>
-<tr>
-<td width=30%>
-<td width=30%>more advanced type systems, such as polymorphic types
-<td width=30%>
-<tr>
-<td width=30%>
-<td width=30%>simply-typed lambda calculus (what linguists mostly use)
-<td width=30%>
-</table>
+(By the way, I really am serious about thinking of *the numbers themselves* as being expressions in this language; rather than some "numerals" that aren't themselves numbers. We can talk about this down the road. For now, don't worry about it too much.)
+I said we wanted to be starting with a fragment of arithmetic, so we'll keep the function values off-stage for the moment, and also all the symbolic atoms except for `'true` and `'false`. So we've got numbers, truth-values, and some functions and relations (that is, boolean functions) defined on them. We also help ourselves to a notion of bounded quantification, as in ∀`x < M.` φ, where `M` and φ are (simple or complex) expressions that evaluate to a number and a boolean, respectively. We limit ourselves to *bounded* quantification so that the fragment we're dealing with can be "effectively" or mechanically decided. (As we extend the language, we will lose that property, but it will be a topic for later discussion exactly when that happens.)
-Rosetta Stone
-=============
+As I mentioned in class, I will sometimes write ∀ x : ψ . φ in my informal metalanguage, where the ψ clause represents the quantifier's *restrictor*. Other people write this like `[`∀ x : ψ `]` φ, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim & Kratzer) use in writing λ x : ψ . φ, where ψ clause restricts the range of arguments over which the function designated by the λ-expression is defined. Later we will see the colon used in a somewhat similar (but also somewhat different) way in our programming languages. But that's just foreshadowing.
-Here's how it looks to say the same thing in various of these languages.
-1. Binding suitable values to the variables `three` and `two`, and adding them.
+### Let and lambda ###
- In Scheme:
+So we have bounded quantification as in ∀ `x < 10.` φ. Obviously we could also make sense of ∀ `x == 5.` φ in just the same way. This would evaluate φ but with the variable `x` now bound to the value `5`, ignoring whatever it may be bound to in broader contexts. I will express this idea in a more perspicuous vocabulary, like this: `let x be 5 in` φ. (I say `be` rather than `=` because, as I mentioned before, it's too easy for the `=` sign to get used for too many subtly different jobs.)
- (let* ((three 3))
- (let ((two 2))
- (+ three two)))
+As one of you was quick to notice in class, though, when I shift to the `let`-vocabulary, I no longer restricted myself to just the case where φ evaluates to a boolean. I also permitted myself expressions like this:
- In OCaml:
+ let x be 5 in x + 1
- let three = 3 in
- let two = 2 in
- three + two
+which evaluates to `6`. Okay, fair enough, so I am moving beyond the ∀ `x==5.` φ idea when I do this. But the rules for how to interpret this are just a straightforward generalization of our existing understanding for how to interpret bound variables. So there's nothing fundamentally novel here.
- Notice OCaml lets you write the `+` in between the `three` and `two`, as you're accustomed to. However most functions need to come leftmost, even if they're binary. And you can do this with `+` too, if you enclose it in parentheses so that the OCaml parser doesn't get confused by your syntax:
+We can have multiple `let`-expressions embedded, as in:
- let three = 3 in
- let two = 2 in
- ( + ) three two
+ let y be (let x be 5 in x + 1) in 2 * y
- In the lambda calculus: here we're on our own, we don't have predefined constants like `+` and `3` and `2` to work with. We've got to build up everything from scratch. We'll be seeing how to do that over the next weeks.
+ let x be 5 in let y be x + 1 in 2 * y
- But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in:
+both of which evaluate to `12`. When we have a stack of `let`-expressions as in the second example, I will write it like this:
- (((\three (\two ((... three) two))) ...) ...)
-
- In an ordinary imperatival language like C:
+ let
+ x be 5;
+ y be x + 1
+ in 2 * y
- int three = 3;
- int two = 2;
- three + two;
+It's okay to also write it all inline, like so: `let x be 5; y be x + 1 in 2 * y`. The `;` represents that we have a couple of `let`-bindings coming in sequence. The earlier bindings in the sequence are considered to be in effect for the later right-hand expressions in the sequence. Thus in:
-2. Mutation
+ let x be 0 in (let x be 5; y be x + 1 in 2 * y)
- In C this looks almost the same as what we had before:
+The `x + 1` that is evaluated to give the value that `y` gets bound to uses the (more local) binding of `x` to `5`, not the (previous, less local) binding of `x` to `0`. By the way, the parentheses in that displayed expression were just to focus your attention. It would have parsed and meant the same without them.
- int x = 3;
- x = 2;
+Now we can allow ourselves to introduce λ-expressions in the following way. If a λ-expression is applied to an argument, as in: `(`λ `x.` φ`) M`, for any (simple or complex) expressions φ and `M`, this means the same as: `let x be M in` φ. That is, the argument to the λ-expression provides (when evaluated) a value for the variable `x` to be bound to, and then the result of the whole thing is whatever φ evaluates to, under that binding to `x`.
- Here we first initialize `x` to hold the value 3; then we mutate `x` to hold a new value.
+If we restricted ourselves to only that usage of λ-expressions, that is when they were applied to all the arguments they're expecting, then we wouldn't have moved very far from the decidable fragment of arithmetic we began with.
- In (the imperatival part of) Scheme, this could be done as:
+However, it's tempting to help ourselves to the notion (at least partly) *unapplied* λ-expressions, too. If I can make sense of what:
- (let ((x (box 3)))
- (set-box! x 2))
+`(`λ `x. x + 1) 5`
- In general, mutating operations in Scheme are named with a trailing `!`. There are other imperatival constructions, though, like `(print ...)`, that don't follow that convention.
+means, then I can make sense of what:
- In (the imperatival part of) OCaml, this could be done as:
+`(`λ `x. x + 1)`
- let x = ref 3 in
- x := 2
+means, too. It's just *the function* that waits for an argument and then returns the result of `x + 1` with `x` bound to that argument.
- Of course you don't need to remember any of this syntax. We're just illustrating it so that you see that in Scheme and OCaml it looks somewhat different than we had above. The difference is much more obvious than it is in C.
+This does take us beyond our (first-order) fragment of arithmetic, at least if we allow the bodies and arguments of λ-expressions to be any expressible value, including other λ-expressions. But we're having too much fun, so why should we hold back?
- In the lambda calculus: sorry, you can't do mutation. At least, not natively. Later in the term we'll be learning how in fact, really, you can embed mutation inside the lambda calculus even though the lambda calculus has no primitive facilities for mutation.
+So now we have a new kind of value our language can work with, alongside numbers and booleans. We now have function values, too. We can bind these function values to variables just like other values:
+`let id be` λ `x. x; y be id 5 in y`
+evaluates to `5`. In reaching that result, the variable `id` was temporarily bound to the identity function, that expects an argument, binds it to the variable `x`, and then returns the result of evaluating `x` under that binding.
+This is what is going on, behind the scenes, with all the expressions like `succ` and `+` that I said could really be understood as variables. They have just been pre-bound to certain agreed-upon functions rather than others.
-3. Anonymous functions
+### Containers ###
- Functions are "first-class values" in the lambda calculus, in Scheme, and in OCaml. What that means is that they can be arguments to other functions. They can be the results of the application of other functions to some arguments. They can be stored in data structures. And so on.
+So far, we've only been talking about *atomic* values. Our language will also have some *container* values, that have other values as members. One example are **ordered sequences**, like:
- First, we'll show what "anonymous" functions look like. These are functions that have not been bound as values to any variables. That is, there are no variables whose value they are.
+ [10, 20, 30]
- In the lambda calculus:
+This is a sequence of length 3. It's the result of *cons*ing the value `10` onto the front of the shorter, length-2 sequence `[20, 30]`. In this made-up language, we'll represent the sequence-consing operation like this:
- (\x M)
+ 10 & [20, 30]
- is always anonymous! Here `M` stands for any expression of the language, simple or complex. It's only when you do
+If you want to know why we call it "cons", that's because this is what the operation is called in Scheme, and they call it that as shorthand for "constructing" the longer list (they call it a "list" rather than a "sequence") out of the components `10` and `[20, 30]`. The name is a bit unfortunate, though, because other structured values besides lists also get "constructed", but we don't say "cons" about them. Still, this is the tradition. Let's just take "cons" to be a nonsense label with an interesting back-history.
- ((\y N) (\x M))
+The sequence `[20, 30]` in turn is the result of:
- that `(\x M)` has a "name" (it's named `y` during the evaluation of `N`).
+ 20 & [30]
- In Scheme, the same thing is written:
+and the sequence `[30]` is the result of consing `30` onto the empty sequence `[]`. Note that the sequence `[30]` is not the same as the number `30`. The former is a container value, with one element. The latter is an atomic value, and as such won't have any elements. If you try to do this:
- (lambda (x) M)
+ [30] + 1
- Not very different, right? For example, if `M` stands for `(+ 3 x)`, then this is an anonymous function that adds 3 to whatever argument it's given:
+it won't work. We haven't discussed what happens with illegal expressions like that, or like `'true + 1`. For the time being, I'll just say these "don't work", or that they "crash". We'll discuss the variety of ways these illegalities might be handled later.
- (lambda (x) (+ 3 x))
+Also, if you try to do this:
- Scheme uses a lot of parentheses, and they are always significant, never optional. In `(+ 3 x)` the parentheses mean "apply the function `+` to the arguments `3` and `x`. In `(lambda (x) ...)` the parentheses have a different meaning: they mark where the anonymous function you're defining begins and ends, and so on. As you'll see, parentheses have yet further roles in Scheme. I know it's confusing.
+ 20 & 30
- In OCaml, we write our anonymous function like this:
+it won't work. The consing operator `&` always requires a container (here, a sequence) on its right-hand side. And `30` is not a container.
- fun x -> (3 + x)
+We've said that:
- or:
+ [10, 20, 30]
- fun x -> (( + ) 3 x)
+is the same as;
- In OCaml, parentheses only serve a grouping function and they often can be omitted. Or more could be added. For instance, we could equally well say:
+ 10 & (20 & (30 & []))
- fun x -> ( + ) 3 x
+and the latter can also be written without the parentheses. Our language knows that `&` should always be understood as "implicitly associating to the right", that is, that:
- or:
+ 10 & 20 & 30 & []
- (fun x -> (( + ) (3) (x)))
+should be interpreted like the expression displayed before. Other operators like `-` should be understood as "implicitly associating to the left." If we write:
- As we saw above, parentheses can often be omitted in the lambda calculus too. But not in Scheme. Every parentheses has a specific role.
+ 30 - 2 - 1
-4. Supplying an argument to an anonymous function
+we presumably want it to be understood as:
- Just because the functions we built aren't named doesn't mean we can't do anything with them. We can give them arguments. For example, in Scheme we can say:
+ (30 - 2) - 1
- ((lambda (x) (+ 3 x)) 2)
+not as:
- The outermost parentheses here mean "apply the function `(lambda (x) (+ 3 x))` to the argument `2`.
+ 30 - (2 - 1)
- In OCaml:
+Other operators don't implicitly associate at all. For example, you may understand the expression:
- (fun x -> ( + ) 3 x) 2
+ 10 < x < 20
+because we have familiar conventions about what it means. But what it means is not:
-5. Binding variables to values with "let"
+ (10 < x) < 20
- Let's go back and re-consider this Scheme expression:
+The result of the parenthesized expression is either `'true` or `'false`, assuming `x` evaluates to a number. But `'true < 20` doesn't mean anything, much less what we expect `10 < x < 20` to mean. So `<` doesn't implicitly associate to the left. Neither does it implicitly associate to the right. If you want expressions like `10 < x < 20` to be meaningful, they will need their own special rules.
- (let* ((three 3))
- (let ((two 2))
- (+ three two)))
+Sequences are containers that keep track of the order of their arguments, and also those arguments' multiplicity (how many times each one appears). Other containers might also keep track of these things, and more structural properties too, or they might keep track of less. Let's say we also have **set containers** too, like this:
- Scheme also has a simple `let` (without the ` *`), and it permits you to group several variable bindings together in a single `let`- or `let*`-statement, like this:
+ {10, 20, 30}
- (let* ((three 3) (two 2))
- (+ three two))
+Whereas the sequences `[10, 20, 10]`, `[10, 20]`, and `[20, 10]` are three different sequences, `{10, 20, 10}`, `{10, 20}`, and `{20, 10}` would just be different ways of expressing a single set.
- Often you'll get the same results whether you use `let*` or `let`. However, there are cases where it makes a difference, and in those cases, `let*` behaves more like you'd expect. So you should just get into the habit of consistently using that. It's also good discipline for this seminar, especially while you're learning, to write things out the longer way, like this:
+We can let the `&` operator do extra-duty, and express the "consing" relation for sets, too:
- (let* ((three 3))
- (let ((two 2))
- (+ three two)))
+ 10 & {20}
- However, here you've got the double parentheses in `(let* ((three 3)) ...)`. They're doubled because the syntax permits more assignments than just the assignment of the value `3` to the variable `three`. Myself I tend to use `[` and `]` for the outer of these parentheses: `(let* [(three 3)] ...)`. Scheme can be configured to parse `[...]` as if they're just more `(...)`.
+evaluates to `{10, 20}`, and so too does:
- Someone asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:
+ 10 & {10, 20}
- (let* [(three (+ 1 2))]
- (let [(two 2)]
- (+ three two)))
-
- The question also came up whether the `(+ 1 2)` computation would be performed before or after it was bound to the variable `three`. That's a terrific question. Let's say this: both strategies could be reasonable designs for a language. We are going to discuss this carefully in coming weeks. In fact Scheme and OCaml make the same design choice. But you should think of the underlying form of the `let`-statement as not settling this by itself.
+As I mentioned in class, we'll let `&&` express the operation by which two sequences are appended or concatenated to each other:
- Repeating our starting point for reference:
+ [10, 20] && [30, 40, 50]
- (let* [(three 3)]
- (let [(two 2)]
- (+ three two)))
+evaluates to `[10, 20, 30, 40, 50]`. For sets, we'll let `and` and `or` and `-` do extra duty, and express set intersection, set union, and set subtraction, when their arguments are sets. If the arguments of `and` and `or` are booleans, on the other hand, or the arguments of `-` are numbers, then they express the functions we were understanding them to express before.
- Recall in OCaml this same computation was written:
+In addition to sequences, there's another kind of expression that might initially be confused with them. We might call these **tuples** or **multivalues**. They are written surrounded by parentheses rather than square brackets. Here's an example:
- let three = 3 in
- let two = 2 in
- ( + ) three two
+`(0, 'true,` λ`x. x)`
-6. Binding with "let" is the same as supplying an argument to a lambda
+That's a tuple with 3 elements (also called a "triple").
- The preceding expression in Scheme is exactly equivalent to:
+In the programming languages and other formal systems we'll be looking at, tuples and sequences are often understood and handled differently. This is because we apply different assumptions to them. In the case of a sequence, it's assumed that they will have homogeneously-typed elements, and that their length will be irrelevant to their own type. So you can have the sequence:
- (((lambda (three) (lambda (two) (+ three two))) 3) 2)
+ [20, 30]
- The preceding expression in OCaml is exactly equivalent to:
+and the sequence:
- (fun three -> (fun two -> ( + ) three two)) 3 2
+ [30]
- Read this several times until you understand it.
+and even the sequence:
-7. Functions can also be bound to variables (and hence, cease being "anonymous").
+ []
- In Scheme:
+and these will all be of the same type, namely a sequence of numbers. You can have sequences with other types of elements, too, for example a sequence of booleans:
- (let* [(bar (lambda (x) B))] M)
+ ['true, 'false, 'true]
- then wherever `bar` occurs in `M` (and isn't rebound by a more local "let" or "lambda"), it will be interpreted as the function `(lambda (x) B)`.
+or a sequence of sequences of numbers:
- Similarly, in OCaml:
+ [[10, 20], [], [30]]
- let bar = fun x -> B in
- M
+An excellent question that came up in class is "How do we tell whether `[]` expresses the empty sequence of numbers or the empty sequence of something else?" We will discuss that question in later weeks. It's central to some of the developments we'll be exploring. For now, just put that question on a mental shelf and assume that somehow this just works out right.
- This in Scheme:
+Now whereas sequences expect homogenously-typed elements, and their length is irrelevant to their own type, mulivalues or tuples are the opposite in both respects. They may have elements of heterogenous type, as our example:
- (let* [(bar (lambda (x) B))] (bar A))
+`(0, 'true,` λ`x. x)`
- as we've said, means the same as:
+did. They need not, but they may. Also, the type of a multivalue or tuple does depend on its length, and moreover on the specific types of each of its elements. A tuple of length 2 (also called a "pair") whose first element is a number and second element is a boolean is a different type of thing that a tuple whose first element is a boolean and whose second element is a number. Most functions expecting the first as an argument will crash if you give them the second instead.
- ((lambda (bar) (bar A)) (lambda (x) B))
+Earlier I said that we can call these things "multivalues or tuples". Here I'll make a technical comment, that in fact I'll understand these slightly differently. Really I'll understand the bare expression `(10, x)` to express a multivalue, and to express a tuple proper, you'll have to write `Pair (10, x)` or something like that. The difference between these is that only the tuple is itself a single value that can be bound to a single variable. The multivalue isn't a single value at all, but rather a plurality of values. This is a bit subtle, and other languages we're looking at this term don't always make this distinction. But the result is that they have to say complicated things elsewhere. If we permit ourselves this fine distinction here, many other things downstream will go more smoothly than they do in the languages that don't make it. Ours is just a made-up language, but I've thought this through carefully, so humor me. We haven't yet introduced the apparatus to make sense of expressions like `Pair (10, x)`, so for the time being I'll just restrict myself to multivalues, not to tuples proper. The result will be that while we can say:
- which, as we'll see, is equivalent to:
+ let x be [10, 20] in ...
- ((lambda (x) B) A)
+that is, sequences are first-class values in our language, we can't say:
- and that means the same as:
+ let x be (10, 'true) in ...
- (let* [(x A)] B)
+or even:
- in other words: evaluate `B` with `x` assigned to the value `A`.
+ let x be (10, 20) in ...
- Similarly, this in OCaml:
+However, intuitively it ought to make sense to say:
- let bar = fun x -> B in
- bar A
+ let (x, y) be (10, 'true) in ...
- is equivalent to:
+That should just bind the variable `x` to the value `10` and the variable `y` to the value `'true`, and go on to evaluate the rest of the expression with those bindings in place. In this particular example, we could equally have said:
- (fun x -> B) A
+ let x be 10; y be 'true in ...
- and that means the same as:
+but in other examples it will be substantially more convenient to be able to bind `x` and `y` simultaneously. Here's an example:
- let x = A in
- B
+`let`
+ `f be` λ `x. (x, 2*x)`
+ `(x, y) be f 10`
+`in [x, y]`
-8. Pushing a "let"-binding from now until the end
+which evaluates to `[10, 20]`. Note that we have the function `f` returning two values, rather than just one, just by having its body evaluate to a multivalue rather than to a single value.
- What if you want to do something like this, in Scheme?
+It's a little bit awkward to say `let (x, y) be ...`, so I propose we instead always say `let (x, y) match ...`. (This will be even more natural as we continue generalizing what we've done here, as we will in the next section.) For consistency, we'll say `match` instead of `be` in all cases, so that we write even this:
- (let* [(x A)] ... for the rest of the file or interactive session ...)
+ let
+ x match 10
+ in ...
- or this, in OCaml:
+rather than:
- let x = A in
- ... for the rest of the file or interactive session ...
+ let
+ x be 10
+ in ...
- Scheme and OCaml have syntactic shorthands for doing this. In Scheme it's written like this:
- (define x A)
- ... rest of the file or interactive session ...
- In OCaml it's written like this:
+### Patterns ###
- let x = A;;
- ... rest of the file or interactive session ...
+What we just introduced is what's known in programming circles as a "pattern". Patterns can look superficially like expressions, but the context in which they appear determines that they are interpreted as patterns not as expressions. The left-hand sides of the binding lists of a `let`-expression are always patterns. Simple variables are patterns. Interestingly, literal values are also patterns. So you can say things like this:
- It's easy to be lulled into thinking this is a kind of imperative construction. *But it's not!* It's really just a shorthand for the compound "let"-expressions we've already been looking at, taking the maximum syntactically permissible scope. (Compare the "dot" convention in the lambda calculus, discussed above.)
+ let
+ 0 match 0;
+ [] match [];
+ 'true match 'true
+ in ...
+(`[]` is also a literal value, like `0` and `'true`.) This isn't very useful in this example, but it will enable us to do interesting things later. So variables are patterns and literal values are patterns. Also, a multivalue of any pattern is a pattern. (Strictly speaking, it's only a multipattern, but I won't fuss about this here.) That's why we can have `(x, y)` on the left-hand side of a `let`-binding: it's a pattern, just like `x` is. Notice that `(x, 10)` is also a pattern. So we can say this:
-9. Some shorthand
+ let
+ (x, 10) match (2, 10)
+ in x
- OCaml permits you to abbreviate:
+which evaluates to `2`. What if you did, instead:
- let bar = fun x -> B in
- M
+ let
+ (x, 10) match (2, 100)
+ in x
- as:
+or, more perversely:
- let bar x = B in
- M
+ let
+ (x, 10) match 2
+ in x
- It also permits you to abbreviate:
+Those will be pattern-matching failures. The pattern has to "fit" the value its being matched against, and that requires having the same structure, and also having the same literal values in whatever positions the pattern specifies literal values. A pattern-matching failure in a `let`-expression makes the whole expression crash. Shortly though we'll consider `case`-expressions, which can recover from pattern-match failures in a useful way.
- let bar = fun x -> B;;
+We can also allow ourselves some other kinds of complex patterns. For example, if `p` and `ps` are two patterns, then `p & ps` will also be a pattern, that can match non-empty sequences and sets. When this pattern is matched against a non-empty sequence, we take the first value in the sequence and match it against the pattern `p`; we take the rest of the sequence and match it against the pattern `ps`. (If either of those results in a pattern-matching failure, then `p & ps` fails to match too.) For example:
- as:
+ let
+ x & xs match [10, 20, 30]
+ in (x, xs)
- let bar x = B;;
+evaluates to the multivalue `(10, [20, 30])`.
- Similarly, Scheme permits you to abbreviate:
+When the pattern `p & ps` is matched against a non-empty set, we just arbitrarily choose one value in the set match it against the pattern `p`; and match the rest of the set, with that value removed, against the pattern `ps`. You cannot control what order the values are chosen in. Thus:
- (define bar (lambda (x) B))
+ let
+ x & xs match {10, 20, 30}
+ in (x, xs)
- as:
+might evaluate to `(20, {10, 30})` or to `(30, {10, 20})` or to `(10, {30, 20})`, or to one of these on Mondays and another on Tuesdays, and never to the third. You cannot control it or predict it. It's good style to only pattern match against sets when the final result will be the same no matter in what order the values are selected from the set.
- (define (bar x) B)
+A question that came up in class was whether `x + y` could also be a pattern. In this language (and most languages), no. The difference between `x & xs` and `x + y` is that `&` is a *constructor* whereas `+` is a *function*. We will be talking about this more in later weeks. For now, just take it that `&` is special. Not every way of forming a complex expression corresponds to a way of forming a complex pattern.
- and this is the form you'll most often see Scheme definitions written in.
+Since as we said, `x & xs` is a pattern, we can let `x1 & x2 & xs` be a pattern as well, the same as `x1 & (x2 & xs)`. And since when we're dealing with expressions, we said that:
- However, conceptually you should think backwards through the abbreviations and equivalences we've just presented.
+ [x1, x2]
- (define (bar x) B)
+is the same as:
- just means:
+ x1 & x2 & []
- (define bar (lambda (x) B))
+we might as well allow this for patterns, too, so that:
- which just means:
+ [x1, x2]
- (let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...)
+is a pattern, meaning the same as `x1 & x2 & []`. Note that while `x & xs` matches *any* non-empty sequence, of length one or more, `[x1, x2]` only matches sequences of length exactly two.
- which just means:
+For the time being, these are the only patterns we'll allow. But since the definition of patterns is recursive, this permits very complex patterns. What would this evaluate to:
- (lambda (bar) ... rest of the file or interactive session ...) (lambda (x) B)
+ let
+ [(x, y), (z:zs, w)] match [([], 'true), ([10, 20, 30], 'false)]
+ in (z, y)
- or in other words, interpret the rest of the file or interactive session with `bar` assigned the function `(lambda (x) B)`.
+Also, we will permit complex patterns in λ-expressions, too. So you can write:
+λ`(x, y).` φ
-10. Shadowing
+as well as:
- You can override a binding with a more inner binding to the same variable. For instance the following expression in OCaml:
+λ`x.` φ
- let x = 3 in
- let x = 2 in
- x
+You can even write:
- will evaluate to 2, not to 3. It's easy to be lulled into thinking this is the same as what happens when we say in C:
+λ `[x, 10].` φ
- int x = 3;
- x = 2;
-
- <em>but it's not the same!</em> In the latter case we have mutation, in the former case we don't. You will learn to recognize the difference as we proceed.
+just be sure to always supply that function with arguments that are two-element sequences whose second element is `10`. If you don't, you will have a pattern-matching failure and the interpretation of your expression will "crash".
- The OCaml expression just means:
+Thus, you can now do things like this:
- (fun x -> ((fun x -> x) 2) 3)
+`let`
+ `f match` λ`(x, y). (x, x + y, x + 2*y, x + 3*y);`
+ `(a, b, c, d) match f (10, 1)`
+`in (b, d)`
- and there's no more mutation going on there than there is in:
+which evaluates `f (10, 1)` to `(10, 11, 12, 13)`, which it will match against the complex pattern `(a, b, c, d)`, binding all four of the contained variables, and then evaluate `(b, d)` under those bindings, giving us the result `(11, 13)`.
- <pre>
- <code>∀x. (F x or ∀x (not (F x)))</code>
- </pre>
+Notice that in the preceding expression, the variables `a` and `c` were never used. We're allowed to do that, but there's also a special syntax to indicate that we want to throw away a value like this. We use the special pattern `_`:
- When a previously-bound variable is rebound in the way we see here, that's called **shadowing**: the outer binding is shadowed during the scope of the inner binding.
+`let`
+ `f match` λ`(x, y). (x, x + y, x + 2*y, x + 3*y);`
+ `(_, b, _, d) match f (10, 1)`
+`in (b, d)`
+The role of `_` here is just to occupy a slot in the complex pattern `(_, b, _, d)`, to make it a multivalue of four values, rather than one of only two.
-Some more comparisons between Scheme and OCaml
-----------------------------------------------
+One last wrinkle. What if you tried to make a pattern like this: `[x, x]`, where some variable occurs multiple times. This is known as a "non-linear pattern". Some languages permit these (and require that the values being bound against `x` in the two positions be equal). Many languages don't permit that. Let's agree not to do this.
-11. Simple predefined values
+### Case and if/then/else ###
- Numbers in Scheme: `2`, `3`
- In OCaml: `2`, `3`
+In class we introduced this form of complex expression:
- Booleans in Scheme: `#t`, `#f`
- In OCaml: `true`, `false`
+`if` φ `then` ψ `else` χ
- The eighth letter in the Latin alphabet, in Scheme: `#\h`
- In OCaml: `'h'`
+Here φ should evaluate to a boolean, and ψ and χ should evaluate to the same type. The result of the whole expression will be the same as ψ, if φ evaluates to `'true`, else to the result of χ.
-12. Compound values
+We said that that could be taken as shorthand for the following `case`-expression:
- These are values which are built up out of (zero or more) simple values.
+`case` φ `of`
+ `'true then` ψ`;`
+ `'false then` χ
+`end`
- Ordered pairs in Scheme: `'(2 . 3)`
- In OCaml: `(2, 3)`
+The `case`-expression has a list of patterns and expressions. Its initial expression φ is evaluated and then attempted to be matched against each of the patterns in turn. When we reach a pattern that can be matched---that doesn't result in a match-failure---then we evaluate the expression after the `then`, using the variable bindings in effect from the immediately preceding match. (Any match that fails has no effect on future variable bindings.) That is the result of the whole `case`-expression; we don't attempt to do any further pattern-matching after finding a pattern that succeeds.
- Lists in Scheme: `'(2 3)`
- In OCaml: `[2; 3]`
- We'll be explaining the difference between pairs and lists next week.
+If a `case`-expression gets to the end of its list of patterns, and *none* of them have matched its initial expression, the result is a pattern-matching failure. So it's good style to always include a final pattern that's guaranteed to match anything. You could use a simple variable for this, or the special pattern `_`:
- The empty list, in Scheme: `'()`
- In OCaml: `[]`
+ case 4 of
+ 1 then 'true;
+ 2 then 'true;
+ x then 'false
+ end
- The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`
- In OCaml: `"h"`
+ case 4 of
+ 1 then 'true;
+ 2 then 'true;
+ _ then 'false
+ end
- A longer string, in Scheme: `"horse"`
- In OCaml: `"horse"`
+will both evaluate to `'false`, without any pattern-matching failure.
- A shorter string, in Scheme: `""`
- In OCaml: `""`
+There's a superficial similarity between the `let`-constructions and the `case`-constructions. Each has a list whose left-hand sides are patterns and right-hand sides are expressions. Each also has an additional expression that stands out in a special position: in `let`-expressions at the end, in `case`-expressions at the beginning. But the relations of these different elements to each other is different. In `let`-expressions, the right-hand sides of the list supply the values that get bound to the variables in the patterns on the left-hand sides. Also, each pattern in the list will get matched, unless there's a pattern-match failure before we get to it. In `case`-expressions, on the other hand, it's the initial expression that supplies the value (or multivalues) that we attempt to match against the pattern, and we stop as soon as we reach a pattern that we can successfully match against. Then the variables in that pattern are bound when evaluating the corresponding right-hand side expression.
-13. Function application
- Binary functions in OCaml: `foo 2 3`
-
- Or: `( + ) 2 3`
+### Recursive let ###
- These are the same as: `((foo 2) 3)`. In other words, functions in OCaml are "curried". `foo 2` returns a `2`-fooer, which waits for an argument like `3` and then foos `2` to it. `( + ) 2` returns a `2`-adder, which waits for an argument like `3` and then adds `2` to it.
+Given all these tools, we're (almost) in a position to define functions like the `factorial` and `length` functions we defined in class.
- In Scheme, on the other hand, there's a difference between `((foo 2) 3)` and `(foo 2 3)`. Scheme distinguishes between unary functions that return unary functions and binary functions. For our seminar purposes, it will be easiest if you confine yourself to unary functions in Scheme as much as possible.
+Here's an attempt to define the `factorial` function:
- Additionally, as said above, Scheme is very sensitive to parentheses and whenever you want a function applied to any number of arguments, you need to wrap the function and its arguments in a parentheses.
+`let`
+ `factorial match` λ `n. if n == 0 then 1 else n * factorial (n-1)`
+`in factorial`
+or, using `case`:
-What "sequencing" is and isn't
-------------------------------
+`let`
+ `factorial match` λ `n. case n of 0 then 1; _ then n * factorial (n - 1) end`
+`in factorial`
-We mentioned before the idea that computation is a sequencing of some changes. I said we'd be discussing (fragments of, and in some cases, entire) languages that have no native notion of change.
+But there's a problem here. What value does `factorial` have when evaluating the expression `factorial (n - 1)`?
-Neither do they have any useful notion of sequencing. But what this would be takes some care to identify.
+As we said in class, the natural precedent for this with non-function variables would go something like this:
-First off, the mere concatenation of expressions isn't what we mean by sequencing. Concatenation of expressions is how you build syntactically complex expressions out of simpler ones. The complex expressions often express a computation where a function is applied to one (or more) arguments,
+ let
+ x match 0;
+ y match x + 1;
+ x match x + 1;
+ z match 2 * x
+ in (y, z)
-Second, the kind of rebinding we called "shadowing" doesn't involve any changes or sequencing. All the precedence facts about that kind of rebinding are just consequences of the compound syntactic structures in which it occurs.
+We'd expect this to evaluate to `(1, 2)`, and indeed it does. That's because the `x` in the `x + 1` on the right-hand side of the third binding (`x match x + 1`) is evaluated under the scope of the first binding, of `x` to `0`.
-Third, the kinds of bindings we see in:
+We should expect the `factorial` variable in the right-hand side of our attempted definition to behave the same way. It will evaluate to whatever value it has before reaching this `let`-expression. We actually haven't said what is the result of trying to evaluate unbound variables, as in:
- (define foo A)
- (foo 2)
+ let
+ x match y + 0
+ in x
-Or even:
+Let's agree not to do that. We can consider such expressions only under the implied understanding that they are parts of larger expressions that assign a value to `y`, as for example in:
- (define foo A)
- (define foo B)
- (foo 2)
+ let
+ y match 1
+ in let
+ x match y + 0
+ in x
-don't involve any changes or sequencing in the sense we're trying to identify. As we said, these programs are just syntactic variants of (single) compound syntactic structures involving "let"s and "lambda"s.
+Hence, let's understand our attempted definition of `factorial` to be part of such a larger expression:
-Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:
+`let`
+ `factorial match` λ `n. n`
+`in let`
+ `factorial match` λ `n. case n of 0 then 1; _ then n * factorial (n - 1) end`
+`in factorial 4`
- (begin A B C)
+This would evaluate to what `4 * factorial 3` does, but with the `factorial` in the expression bound to the identity function λ `n. n`. In other words, we'd get the result `12`, not the correct answer `24`.
-In OCaml it looks like this:
+For the time being, we will fix this solution by just introducing a special new construction `letrec` that works the way we want. Now in:
- begin A; B; C end
+`let`
+ `factorial match` λ `n. n`
+`in letrec`
+ `factorial match` λ `n. case n of 0 then 1; _ then n * factorial (n - 1) end`
+`in factorial 4`
-Or this:
+the initial binding of `factorial` to the identity function gets ignored, and the `factorial` in the right-hand side of our definition is interpreted to mean the very same function that we are hereby binding to `factorial`. Exactly how this works is a deep and exciting topic, that we will be looking at very closely in a few weeks. For the time being, let's just accept that `letrec` does what we intuitively want when defining functions recursively.
- (A; B; C)
+**It's important to make sure you say letrec when that's what you want.** You may not *always* want `letrec`, though, if you're ever re-using variables (or doing other things) that rely on the bindings occurring in a specified order. With `letrec`, all the bindings in the construction happen simultaneously. This is why you can say, as Jim did in class:
-In the presence of imperatival elements, sequencing order is very relevant. For example, these will behave differently:
+`letrec`
+ `even? match` λ `n. case n of 0 then 'true; _ then odd? (n-1) end`
+ `odd? match` λ `n. case n of 0 then 'false; _ then even? (n-1) end`
+`in (even?, odd?)`
- (begin (print "under") (print "water"))
-
- (begin (print "water") (print "under"))
+Here neither the `even?` nor the `odd?` pattern is matched before the other. They, and also the `odd?` and the `even?` variables in their right-hand side expressions, are all bound at once.
-And so too these:
+As we said, this is deep and exciting, and it will make your head spin before we're done examining it. But let's trust `letrec` to do its job, for now.
- begin x := 3; x := 2; x end
- begin x := 2; x := 3; x end
+### Comparing recursive-style and iterative-style definitions ###
-However, if A and B are purely functional, non-imperatival expressions, then:
+Finally, we're in a position to revisit the two definitions of `length` that Jim presented in class. Here is the first:
- begin A; B; C end
+`letrec`
+ `length match` λ `xs. case xs of [] then 0; _:ys then 1 + length ys end`
+`in length`
-just evaluates to C (so long as A and B evaluate to something at all). So:
+This function accept a sequence `xs`, and if its empty returns `0`, else it says that its length is `1` plus whatever is the length of its remainder when you take away the first element. In programming circles, this remainder is commonly called the sequence's "tail" (and the first element is its "head").
- begin A; B; C end
+Thus if we evaluated `length [10, 20, 30]`, that would give the same result as `1 + length [20, 30]`, which would give the same result as `1 + (1 + length [30])`, which would give the same result as `1 + (1 + (1 + length []))`. But `length []` is `0`, so our original expression evaluates to `1 + (1 + (1 + 0))`, or `3`.
-contributes no more to a larger context in which it's embedded than C does. This is the sense in which functional languages have no serious notion of sequencing.
+Here's another way to define the `length` function:
-We'll discuss this more as the seminar proceeds.
+`letrec`
+ `aux match` λ `(n, xs). case xs of [] then n; _:ys then aux (n + 1, ys) end`
+`in` λ `xs. aux (0, xs)`
+This may be a bit confusing. What we have here is a helper function `aux` (for "auxiliary") that accepts *two* arguments, the first being a counter of how long we've counted in the sequence so far, and the second argument being how much more of the sequence we have to inspect. If the sequence we have to inspect is empty, then we're finished and we can just return out counter. (Note that we don't return `0`.) If not, then we add `1` to the counter, and proceed to inspect the tail of the sequence, ignoring the sequence's first element. After the `in`, we can't just return the `aux` function, because it expects two arguments, whereas `length` should just be a function of a single argument, the sequence whose length we're inquiring about. What we do instead is return a λ-generated function, that expects a single sequence argument `xs`, and then returns the result of calling `aux` with that sequence together with an initial counter of `0`.
+So for example, if we evaluated `length [10, 20, 30]`, that would give the same result as `aux (0, [10, 20, 30])`, which would give the same result as `aux (1, [20, 30])`, which would give the same result as `aux (2, [30])`, which would give the same result as `aux(3, [])`, which would give `3`. (This should make it clear why when `aux` is called with the empty sequence, it returns the result `n` rather than `0`.)
-Basics of Lambda Calculus
-=========================
+Programmers will sometimes define functions in the second style because it can be evaluated more efficiently than the first style. You don't need to worry about things like efficiency in this seminar. But you should become acquainted with, and comfortable with, both styles of recursive definition.
-The lambda calculus we'll be focusing on for the first part of the course has no types. (Some prefer to say it instead has a single type---but if you say that, you have to say that functions from this type to this type also belong to this type. Which is weird.)
+It may be helpful to contrast these recursive-style definitons to the way one would more naturally define the `length` function in an imperatival language. This uses some constructs we haven't explained yet, but I trust their meaning will be intuitively clear enough.
-Here is its syntax:
+`let`
+ `empty? match` λ `xs.` *this definition left as an exercise*;
+ `tail match` λ `xs.` *this definition left as an exercise*;
+ `length match` λ `xs. let`
+ `n := 0;`
+ `while not (empty? xs) do`
+ `n := n + 1;`
+ `xs := tail xs`
+ `end`
+ `in n`
+`in length`
-<blockquote>
-<strong>Variables</strong>: <code>x</code>, <code>y</code>, <code>z</code>...
-</blockquote>
+Here there is no recursion. Rather what happens is that we *initialize* the variable `n` with the value `0`, and then so long as our sequence variable `xs` is non-empty, we *increment* that variable `n`, and *overwrite* the variable `xs` with the tail of the sequence that it is then bound to, and repeat in a loop (the `while ... do ... end` construction). This is similar to what happens in our second definition of `length`, using `aux`, but here it happens using *mutation* or *overwriting* the values of variables, and a special looping construction, whereas in the preceding definitions we achieved the same effect instead with recursion.
-Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:
+We will be looking closely at mutation later in the term. For the time being, our focus will instead be on the recursive and *immutable* style of doing things. Meaning no variables get overwritten.
-<blockquote>
-<strong>Abstract</strong>: <code>(λa M)</code>
-</blockquote>
+It's helpful to observe that in expressions like:
-We'll tend to write <code>(λa M)</code> as just `(\a M)`, so we don't have to write out the markup code for the <code>λ</code>. You can yourself write <code>(λa M)</code> or `(\a M)` or `(lambda a M)`.
-
-<blockquote>
-<strong>Application</strong>: <code>(M N)</code>
-</blockquote>
-
-Some authors reserve the term "term" for just variables and abstracts. We won't participate in that convention; we'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
-
-Examples of expressions:
-
- x
- (y x)
- (x x)
- (\x y)
- (\x x)
- (\x (\y x))
- (x (\x x))
- ((\x (x x)) (\x (x x)))
-
-The lambda calculus has an associated proof theory. For now, we can regard the proof theory as having just one rule, called the rule of **beta-reduction** or "beta-contraction". Suppose you have some expression of the form:
-
- ((\a M) N)
-
-that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.
-
-The rule of beta-reduction permits a transition from that expression to the following:
-
- M [a:=N]
-
-What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
-
-What is a free occurrence?
-
-> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
-
-> If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`.
-
-> An occurrence of a variable is **free** if it's not bound.
-
-For instance:
-
-
-> T is defined to be `(x (\x (\y (x (y z)))))`
-
-The first occurrence of `x` in T is free. The `\x` we won't regard as being an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free.
-
-Here's an example of beta-reduction:
-
- ((\x (y x)) z)
-
-beta-reduces to:
-
- (y z)
-
-We'll write that like this:
-
- ((\x (y x)) z) ~~> (y z)
-
-Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write:
-
- M ~~> N
-
-We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol <code><big><big>→</big></big></code> for one-step contraction, and the symbol <code><big><big>↠</big></big></code> for zero-or-more step reduction. Hindley and Seldin use <code><big><big><big>⊳</big></big></big><sub>1</sub></code> and <code><big><big><big>⊳</big></big></big></code>.
-
-When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **beta-convertible**. We'll write that like this:
-
- M <~~> N
-
-This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.)
-
-In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol <code>≡</code> for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing metatheory for the lambda calculus. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in:
-
-> T is defined to be `(M N)`.
-
-We'll regard the following two expressions:
-
- (\x (x y))
-
- (\z (z y))
-
-as syntactically equivalent, since they only involve a typographic change of a bound variable. Read Hankin section 2.3 for discussion of different attitudes one can take about this.
-
-Note that neither of those expressions are identical to:
-
- (\x (x w))
-
-because here it's a free variable that's been changed. Nor are they identical to:
-
- (\y (y y))
-
-because here the second occurrence of `y` is no longer free.
-
-There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we've linked to about the lambda calculus. We expect you have a good intuitive understanding of what to do already, though, even if you're not able to articulate it rigorously.
-
-
-Shorthand
----------
-
-The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)
-
-
-**Dot notation** Dot means "put a left paren here, and put the right
-paren as far the right as possible without creating unbalanced
-parentheses". So:
-
- (\x (\y (x y)))
-
-can be abbreviated as:
-
- (\x (\y. x y))
-
-and:
-
- (\x (\y. (z y) z))
-
-would abbreviate:
-
- (\x (\y ((z y) z)))
-
-This on the other hand:
-
- (\x (\y. z y) z)
-
-would abbreviate:
-
- (\x (\y (z y)) z)
-
-**Parentheses** Outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:
-
- (\x. x y)
-
-as:
-
- \x. x y
-
-but you should include the parentheses in:
-
- (\x. x y) z
-
-and:
-
- z (\x. x y)
-
-**Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
-
- (\x y. M)
-
-Similarly, `(\x (\y (\z M)))` can be abbreviated as:
-
- (\x y z. M)
-
-
-Lambda terms represent functions
---------------------------------
-
-All (recursively computable) functions can be represented by lambda
-terms (the untyped lambda calculus is Turing complete). For some lambda terms, it is easy to see what function they represent:
-
-> `(\x x)` represents the identity function: given any argument `M`, this function
-simply returns `M`: `((\x x) M) ~~> M`.
-
-> `(\x (x x))` duplicates its argument:
-`((\x (x x)) M) ~~> (M M)`
-
-> `(\x (\y x))` throws away its second argument:
-`(((\x (\y x)) M) N) ~~> M`
-
-and so on.
-
-It is easy to see that distinct lambda expressions can represent the same
-function, considered as a mapping from input to outputs. Obviously:
-
- (\x x)
-
-and:
-
- (\z z)
-
-both represent the same function, the identity function. However, we said above that we would be regarding these expressions as synactically equivalent, so they aren't yet really examples of *distinct* lambda expressions representing a single function. However, all three of these are distinct lambda expressions:
-
- (\y x. y x) (\z z)
-
- (\x. (\z z) x)
-
- (\z z)
-
-yet when applied to any argument M, all of these will always return M. So they have the same extension. It's also true, though you may not yet be in a position to see, that no other function can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.
-
-The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as proof-theoretically equivalent even though they're not syntactically identical. However, the proof theory we've given so far doesn't permit you to reduce the second expression to the third. So these lambda expressions are non-equivalent.
-
-There's an extension of the proof-theory we've presented so far which does permit this further move. And in that extended proof theory, all computable functions with the same extension do turn out to be equivalent (convertible). However, at that point, we still won't be working with the traditional mathematical notion of a function as a set of ordered pairs. One reason is that the latter but not the former permits uncomputable functions. A second reason is that the latter but not the former prohibits functions from applying to themselves. We discussed this some at the end of Monday's meeting (and further discussion is best pursued in person).
-
-
-
-Booleans and pairs
-==================
-
-Our definition of these is reviewed in [[Assignment1]].
-
-
-It's possible to do the assignment without using a Scheme interpreter, however
-you should take this opportunity to [get Scheme installed on your
-computer](/how_to_get_the_programming_languages_running_on_your_computer), and
-[get started learning Scheme](/learning_scheme). It will help you test out
-proposed answers to the assignment.
-
-
-
-
-
-1. Declarative vs imperatival models of computation.
-2. Variety of ways in which "order can matter."
-3. Variety of meanings for "dynamic."
-4. Schoenfinkel, Curry, Church: a brief history
-5. Functions as "first-class values"
-6. "Curried" functions
-
-1. Beta reduction
-1. Encoding pairs (and triples and ...)
-1. Encoding booleans
+ let
+ x match 0;
+ y match x + 1;
+ x match x + 1;
+ z match 2 * x
+ in (y, z)
+the variable `x` has not been *overwritten* (mutated). Rather, we have *two* variables `x` and its just that the second one is *hiding* the first so long as its scope is in effect. Once its scope expires, the original variable `x` is still in place, with its orginal binding. A different example should help clarify this. What do you think this:
+ let
+ x match 0;
+ (y, z) match let
+ x match x + 1
+ in (x, 2*x)
+ in ([y, z], x)
+evaluates to? Well, the right-hand side of the second binding will evaluate to `(1, 2)`, because it uses the outer binding of `x` to `0` for the right-hand side of its own binding `x match x + 1`. That gives us a new binding of `x` to `1`, which is in place when we evaluate `(x, 2*x)`. That's why the whole right-hand side of the second binding evaluates to `(1, 2)`. So `y` gets bound to `1` and `z` to `2`. But now what is `x` bound to in the final line? The binding of `x` to `1` was in place only until we got to `(x, 2*x)`. After that its scope expired, and the original binding of `x` to `0` reappears. So the final line evaluates to `([1, 2], 0)`.
+This is very like what happens in ordinary predicate logic if you say: &exists; `x. F x and (` ∀ `x. G x ) and H x`. The `x` in `F x` and in `H x` are governed by the outermost quantifier, and only the `x` in `G x` is governed by the inner quantifier.