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'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.
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
- ([xs, ys], [z:zs, ws]) match ([[], [1]], [[10, 20, 30], [0]])
+ ([xs, ys], [z & zs, ws]) match ([[], [1]], [[10, 20, 30], [0]])
in z & ys
Also, we will permit complex patterns in λ-expressions, too. So you can write:
Finally, we're in a position to revisit the two definitions of `length` that Jim presented in class. Here is the first:
`letrec`
- `length match` λ `xs. case xs of [] then 0; _:ys then 1 + length ys end`
+ `length match` λ `xs. case xs of [] then 0; _ & ys then 1 + length ys end`
`in length`
This function accept a sequence `xs`, and if it's 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").
Here's another way to define the `length` function:
`letrec`
- `aux match` λ `(n, xs). case xs of [] then n; _:ys then aux (n + 1, ys) end`
+ `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 our 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`.
This was a lot of material, and you may need to read it carefully and think about it, but none of it should seem profoundly different from things you're already accustomed to doing. What we worked our way up to was just the kind of recursive definitions of `factorial` and `length` that you volunteered in class, before learning any programming.
-You have all the materials you need now to do this week's [[assignment|assignment1]]. Some of you may find it easy. Many of you will not. But if you understand what we've done here, and give it your time and attention, we believe you can do it.
+You have all the materials you need now to do this week's [[assignment|/exercises/assignment1]]. Some of you may find it easy. Many of you will not. But if you understand what we've done here, and give it your time and attention, we believe you can do it.
There are also some [[advanced notes|week1 advanced notes]] extending this week's material.
### Summary ###
-Here is the hierarcy of **values** that we've talked about so far.
+Here is the hierarchy of **values** that we've talked about so far.
* Multivalues
* Singular values, including:
* Booleans (or truth-values)
* Functions: these are not literals, but instead have to be generated by evaluating complex expressions
* Containers, including:
- * the literals `[]` and `{}`
- * Non-empty sequences
- * Non-empty sets
+ * the **literal containers** `[]` and `{}`
+ * Non-empty sequences, built using `&`
+ * Non-empty sets, built using `&`
* Tuples proper and other containers, to be introduced later
We've also talked about a variety of **expressions** in our language, that evaluate down to various values (if their evaluation doesn't "crash" or otherwise go awry). These include:
* All of the literal atoms and literal containers
* Variables
-* Various complex expressions, built using `&` or λ or `let` or `letrec` or `case`
+* Complex expressions that apply `&` or some variable understood to be bound to a function to some arguments
+* Various other complex expressions involving the keywords λ or `let` or `letrec` or `case`
The special syntaxes `[10, 20, 30]` are just shorthand for the more offical syntax using `&` and `[]`, and likewise for `{10, 20, 30}`. The `if ... then ... else ...` syntax is just shorthand for a `case`-construction using the literal patterns `'true` and `'false`.
+We also talked about **patterns**. These aren't themselves expressions, but form part of some larger expressions.
+