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.
### Getting started ###
We begin with a decidable fragment of arithmetic. Our language has some primitive literal values:
0, 1, 2, 3, ...
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.
We also have some predefined functions:
succ, +, *, pred, -
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`.
Here's another set of functions:
==, <, >, <=, >=, !=
`==` 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.
As mentioned in class, we represent the truth-values like this:
'true, 'false
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:
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
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.
The functions `succ` and `pred` come before their arguments, like this:
succ 1
On the other hand, the functions `+`, `*`, `-`, `==`, and so on come in between their arguments, like this:
x < y
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:
lessthan? (x, y)
or perhaps like this:
lessthan? x y
We'll get more acquainted with the difference between these next week. For now, I'll just stick to the first form.
Another set of operations we have are:
and, or, not
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
will in general be just another way to write:
not (x == y)
You see that you can use parentheses in the standard way.
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:
x
x1
x_not_y
xUBERANT
x'
x''
x?
xs
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.
We also conventionally reserve variables ending in `!` for a different special class of functions, that we will explain later in the course.
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.
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.)
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.
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.
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.
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.
(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.)
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.
### Let and lambda ###
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.)
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:
let x be 5 in x + 1
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.
We can have multiple `let`-expressions embedded, as in:
let y be (let x be 5 in x + 1) in 2 * y
let x be 5 in let y be x + 1 in 2 * y
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:
let
x be 5;
y be x + 1
in 2 * y
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:
let x be 0 in (let x be 5; y be x + 1 in 2 * y)
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.
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`.
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.
However, it's tempting to help ourselves to the notion (at least partly) *unapplied* λ-expressions, too. If I can make sense of what:
`(`λ `x. x + 1) 5`
means, then I can make sense of what:
`(`λ `x. x + 1)`
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.
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?
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`
will evaluate 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.
### Containers ###
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:
[10, 20, 30]
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:
10 & [20, 30]
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.
The sequence `[20, 30]` in turn is the result of:
20 & [30]
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:
[30] + 1
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.
Also, if you try to do this:
20 & 30
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.
We've said that:
[10, 20, 30]
is the same as;
10 & (20 & (30 & []))
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:
10 & 20 & 30 & []
should be interpreted like the expression displayed before. Other operators like `-` should be understood as "implicitly associating to the left." If we write:
30 - 2 - 1
we presumably want it to be understood as:
(30 - 2) - 1
not as:
30 - (2 - 1)
Other operators don't implicitly associate at all. For example, you may understand the expression:
10 < x < 20
because we have familiar conventions about what it means. But what it means is not:
(10 < x) < 20
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.
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:
{10, 20, 30}
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.
We can let the `&` operator do extra-duty, and express the "consing" relation for sets, too:
10 & {20}
would evaluate to `{10, 20}`, and so too would:
10 & {10, 20}
As I mentioned in class, we'll let `&&` express the operation by which two sequences are appended or concatenated to each other:
[10, 20] && [30, 40, 50]
will evaluate 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.
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:
`(0, 'true,` λ`x. x)`
That's a tuple with 3 elements (also called a "triple").
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:
[20, 30]
and the sequence:
[30]
and even the sequence:
[]
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:
['true, 'false, 'true]
or a sequence of sequences of numbers:
[[10, 20], [], [30]]
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.
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:
`(0, 'true,` λ`x. x)`
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.
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:
let x be [10, 20] in ...
that is, sequences are first-class values in our language, we can't say:
let x be (10, 'true) in ...
or even:
let x be (10, 20) in ...
However, intuitively it ought to make sense to say:
let (x, y) be (10, 'true) in ...
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:
let x be 10; y be 'true in ...
but in other examples it will be substantially more convenient to be able to bind `x` and `y` simultaneously. Here's an example:
`let`
`f be` λ `x. (x, 2*x)`
`(x, y) be f 10`
`in [x, y]`
which will evaluate 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.
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 match 10
in ...
rather than:
let
x be 10
in ...
### Patterns ###
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:
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. 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:
let
(x, 10) match (2, 10)
in x
which will evaluate to `2`. What if you did, instead:
let
(x, 10) match (2, 100)
in x
or, more perversely:
let
(x, 10) match 2
in x
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.
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:
let
x & xs match [10, 20, 30]
in (x, xs)
will evaluate to the multivalue `(10, [20, 30])`.
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:
let
x & xs match {10, 20, 30}
in (x, xs)
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.
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.
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:
[x1, x2]
is the same as:
x1 & x2 & []
we might as well allow this for patterns, too, so that:
[x1, x2]
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.
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:
let
[(x, y), (z:zs, w)] match [([], 'true), ([10, 20, 30], 'false)]
in (z, y)
Also, we will permit complex patterns in λ-expressions, too. So you can write:
λ`(x, y).` φ
as well as:
λ`x.` φ
You can even write:
λ `[x, 10].` φ
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".
Thus, you can now do things like this:
`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)`
which will evaluate `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)`.
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 `_`:
`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.
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.
### Recursive let ###
*More coming*
### Comparing recursive-style and iterative-style definitions ###
*More coming*