`≡`

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 +126,8 @@ 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
+
Shorthand
---------
@@ -193,8 +195,7 @@ 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 +229,7 @@ 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).
@@ -326,7 +327,7 @@ Map
Scheme (imperative part)OCaml (imperative part) -lambda calculus

+untyped lambda calculus

combinatorial logic --------------------------------------------------- Turing complete --------------------------------------------------- @@ -351,7 +352,43 @@ 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. Binding suitable values to the variables `three` and `two`, and adding them. +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". MORE + `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. + + 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. + + 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: @@ -359,21 +396,20 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo (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: + > But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in: (((\three (\two ((... three) two))) ...) ...) @@ -436,7 +472,6 @@ The following site may be useful; it lets you run a Scheme interpreter inside yo (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: @@ -703,17 +738,6 @@ 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