`(λa M)`

as just `(\a M)`, so we don't hav
`(M N)`

-Some authors reserve the term "term" for just variables and abstracts. We'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
Examples of expressions:
@@ -76,7 +76,11 @@ 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.
+The first occurrence of `x` in T is free. The `\x` we won't regard as containing 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.
+
+To read further:
+
+* [[!wikipedia Free variables and bound variables]]
Here's an example of beta-reduction:
@@ -102,7 +106,7 @@ When M and N are such that there's some P that M reduces to by zero or more step
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 `≡`

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:
+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 `≡`

for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing the metatheory. 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)`.
@@ -126,6 +130,8 @@ 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.
+* [More discussion in week 2 notes](/week2/#index1h1)
+
Shorthand
---------
@@ -193,8 +199,7 @@ Similarly, `(\x (\y (\z M)))` can be abbreviated as:
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:
+The untyped lambda calculus is Turing complete: all (recursively computable) functions can be represented by lambda terms. 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`.
@@ -228,7 +233,7 @@ yet when applied to any argument M, all of these will always return M. So they h
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).
+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 many 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).
@@ -307,7 +312,7 @@ Finally, you'll see the term **dynamic** used in a variety of ways in the litera
* dynamic versus static typing
-* dynamic versus lexical scoping
+* dynamic versus lexical [[!wikipedia Scope (programming) desc="scoping"]]
* dynamic versus static control operators
@@ -315,6 +320,16 @@ Finally, you'll see the term **dynamic** used in a variety of ways in the litera
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.
+To read further about the relation between declarative or functional programming, on the one hand, and imperatival programming on the other, you can begin here:
+
+* [[!wikipedia Declarative programming]]
+* [[!wikipedia Functional programming]]
+* [[!wikipedia Purely functional]]
+* [[!wikipedia Referential transparency (computer science)]]
+* [[!wikipedia Imperative programming]]
+* [[!wikipedia Side effect (computer science) desc="Side effects"]]
+
+
Map
===
@@ -326,7 +341,7 @@ Map
Scheme (imperative part)OCaml (imperative part) -lambda calculus

+untyped lambda calculus

combinatorial logic --------------------------------------------------- Turing complete --------------------------------------------------- @@ -341,6 +356,7 @@ combinatorial logic + Rosetta Stone ============= @@ -350,28 +366,65 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo * [Try Scheme in your web browser](http://tryscheme.sourceforge.net/) + + +1. Function application and parentheses + + In Scheme and the lambda calculus, the functions you're applying always go to the left. So you write `(foo 2)` and also `(+ 2 3)`. + + Mostly that's how OCaml is written too: + + foo 2 + + But a few familiar binary operators can be written infix, so: + + 2 + 3 + + You can also write them operator-leftmost, if you put them inside parentheses to help the parser understand you: + + ( + ) 2 3 + + I'll mostly do this, for uniformity with Scheme and the lambda calculus. + + In OCaml and the lambda calculus, this: + + foo 2 3 + + means the same as: + + ((foo 2) 3) + + These functions 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. For further reading: + +* [[!wikipedia Currying]] + + 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. + + 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. So you have to write `(foo 2)`; if you only say `foo 2`, Scheme won't understand you. -1. Binding suitable values to the variables `three` and `two`, and adding them. + Scheme uses a lot of parentheses, and they are always significant, never optional. Often the parentheses mean "apply this function to these arguments," as just described. But in a moment we'll see other constructions in Scheme where the parentheses have different roles. They do lots of different work in Scheme. + + +2. Binding suitable values to the variables `three` and `two`, and adding them. In Scheme: (let* ((three 3)) - (let ((two 2)) + (let* ((two 2)) (+ three two))) - In OCaml: + Most of the parentheses in this construction *aren't* playing the role of applying a function to some arguments---only the ones in `(+ three two)` are doing that. - let three = 3 in - let two = 2 in - three + two - 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: + In OCaml: let three = 3 in let two = 2 in ( + ) three two - 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. + 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. But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in: @@ -406,23 +459,27 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo 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. - 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. + 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. 3. Anonymous functions - 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. + Functions are "first-class values" in the lambda calculus, in Scheme, and in OCaml. What that means is that they can be arguments to, and results of, other functions. They can be stored in data structures. And so on. To read further: + + * [[!wikipedia Higher-order function]] + * [[!wikipedia First-class function]] - 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. + We'll begin by looking at 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. In the lambda calculus: (\x M) - is always anonymous! Here `M` stands for any expression of the language, simple or complex. It's only when you do + ---where `M` is any simple or complex expression---is anonymous. It's only when you do: ((\y N) (\x M)) @@ -432,29 +489,14 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo (lambda (x) M) - 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: + Not very different, right? For example, if `M` stands for `(+ 3 x)`, then here is an anonymous function that adds 3 to whatever argument it's given: (lambda (x) (+ 3 x)) - 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. - In OCaml, we write our anonymous function like this: - fun x -> (3 + x) - - or: - - fun x -> (( + ) 3 x) - - 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: - fun x -> ( + ) 3 x - or: - - (fun x -> (( + ) (3) (x))) - - As we saw above, parentheses can often be omitted in the lambda calculus too. But not in Scheme. Every parentheses has a specific role. 4. Supplying an argument to an anonymous function @@ -462,7 +504,7 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo ((lambda (x) (+ 3 x)) 2) - The outermost parentheses here mean "apply the function `(lambda (x) (+ 3 x))` to the argument `2`. + The outermost parentheses here mean "apply the function `(lambda (x) (+ 3 x))` to the argument `2`, or equivalently, "give the value `2` as an argument to the function `(lambda (x) (+ 3 x))`. In OCaml: @@ -474,7 +516,7 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo Let's go back and re-consider this Scheme expression: (let* ((three 3)) - (let ((two 2)) + (let* ((two 2)) (+ three two))) 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: @@ -485,23 +527,23 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo 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: (let* ((three 3)) - (let ((two 2)) + (let* ((two 2)) (+ three two))) 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 `(...)`. - Someone asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write: + It was asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write: (let* [(three (+ 1 2))] - (let [(two 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. + It was also asked 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. Repeating our starting point for reference: (let* [(three 3)] - (let [(two 2)] + (let* [(two 2)] (+ three two))) Recall in OCaml this same computation was written: @@ -543,7 +585,7 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo ((lambda (bar) (bar A)) (lambda (x) B)) - which, as we'll see, is equivalent to: + which beta-reduces to: ((lambda (x) B) A) @@ -588,7 +630,7 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo let x = A;; ... rest of the file or interactive session ... - 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.) + 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. I'm fudging a bit here, since in Scheme `(define ...)` is really shorthand for a `letrec` epression, which we'll come to in later classes.) 9. Some shorthand @@ -665,11 +707,15 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo 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. + See also: + + * [[!wikipedia Variable shadowing]] + Some more comparisons between Scheme and OCaml ---------------------------------------------- -11. Simple predefined values +* Simple predefined values Numbers in Scheme: `2`, `3` In OCaml: `2`, `3` @@ -680,18 +726,18 @@ Some more comparisons between Scheme and OCaml The eighth letter in the Latin alphabet, in Scheme: `#\h` In OCaml: `'h'` -12. Compound values +* Compound values These are values which are built up out of (zero or more) simple values. - Ordered pairs in Scheme: `'(2 . 3)` + Ordered pairs in Scheme: `'(2 . 3)` or `(cons 2 3)` In OCaml: `(2, 3)` - Lists in Scheme: `'(2 3)` + Lists in Scheme: `'(2 3)` or `(list 2 3)` In OCaml: `[2; 3]` We'll be explaining the difference between pairs and lists next week. - The empty list, in Scheme: `'()` + The empty list, in Scheme: `'()` or `(list)` In OCaml: `[]` The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"` @@ -703,17 +749,6 @@ Some more comparisons between Scheme and OCaml A shorter string, in Scheme: `""` In OCaml: `""` -13. Function application - - Binary functions in OCaml: `foo 2 3` - - Or: `( + ) 2 3` - - 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. - - 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. - - 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. What "sequencing" is and isn't