X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=inline;f=week1.mdwn;h=b1df1ad5523599929a4241489ab9260cb49fa256;hb=40f8c745a271a15e718e2a52a94f4ef27a152d80;hp=5c880fbb7cba5289de03d371447f294cca74dbf3;hpb=f146774aed497f4e8b776bacafa82d130ae24dc1;p=lambda.git diff --git a/week1.mdwn b/week1.mdwn index 5c880fbb..b1df1ad5 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -1,33 +1,257 @@ -Here's what we did in seminar on Monday 9/13, (Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class.) +Here's what we did in seminar on Monday 9/13, -Applications -============ +Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class. This present page expands on *a lot*, and some of this material will be reviewed next week. -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.) +[Linguistic and Philosophical Applications of the Tools We'll be Studying](/applications) +========================================================================== -From linguistics ----------------- +[Explanation of the "Damn" example shown in class](/damn) -* generalized quantifiers are a special case of operating on continuations +Basics of Lambda Calculus +========================= + +See also: + +* [Chris Barker's Lambda Tutorial](http://homepages.nyu.edu/~cb125/Lambda) +* [Lambda Animator](http://thyer.name/lambda-animator/) +* [Penn lambda calculator](http://www.ling.upenn.edu/lambda/) Pedagogical software developed by Lucas Champollion, Josh Tauberer and Maribel Romero. Linguistically oriented. +* MORE + +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.) + +Here is its syntax: + +
+Variables:+ +Each variable is an expression. For any expressions M and N and variable a, the following are also expressions: + +x
,y
,z
... +
+Abstract: (λa M)
+
+
+We'll tend to write (λa M)
as just `(\a M)`, so we don't have to write out the markup code for the λ
. You can yourself write (λa M)
or `(\a M)` or `(lambda a M)`.
+
+
+Application: (M N)
+
+
+
+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 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:
+
+ ((\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 →
for one-step contraction, and the symbol ↠
for zero-or-more step reduction. Hindley and Seldin use ⊳1
and ⊳
.
+
+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 ≡
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)`.
+
+We'll regard the following two expressions:
+
+ (\x (x y))
-* (Chris: fill in other applications...)
+ (\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.
+
+* [More discussion in week 2 notes](/week2/#index1h1)
+
+
+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.)
+
+
+**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))
+
+
+**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 that as:
+
+ \x. \y. x y
+
+This:
+
+ \x. \y. (x y) x
-* expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the linked summary for more explanation and elaboration
+abbreviates:
-From philosophy
----------------
+ \x (\y ((x y) x))
-* 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.
+This on the other hand:
+
+ (\x. \y. (x y)) x
+
+abbreviates:
+
+ ((\x (\y (x y))) x)
+
+
+**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)
-* those issues may bear on Russell's Gray's Elegy argument in "On Denoting"
-* 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."
+Lambda terms represent functions
+--------------------------------
+
+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`.
+
+> `(\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`
-* 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
+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 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).
+
+
+
+Booleans and pairs
+==================
+
+Our definition of these is reviewed in [[Assignment1]].
-* the semantics for mutation will enable us to make sense of a difference between numerical and qualitative identity---for purely mathematical objects!
-* 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
+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.
+
+
+There's also a (slow, bare-bones, but perfectly adequate) version of Scheme available for online use at Scheme (functional part) | +OCaml (functional part) | +C, Java, Pasval +Scheme (imperative part) +OCaml (imperative part) |
+
untyped lambda calculus +combinatorial logic |
+||
--------------------------------------------------- Turing complete --------------------------------------------------- | +||
+ | more advanced type systems, such as polymorphic types + | + |
+ | simply-typed lambda calculus (what linguists mostly use) + | + |
-Variables:+ Mostly that's how OCaml is written too: -Each variable is an expression. For any expressions M and N and variable a, the following are also expressions: + foo 2 -x
,y
,z
... -
-Abstract: (λa M)
-
+ But a few familiar binary operators can be written infix, so:
-We'll tend to write (λa M)
as just `(\a M)`, so we don't have to write out the markup code for the λ
. You can yourself write (λa M)
or `(\a M)` or `(lambda a M)`.
+ 2 + 3
-
-Application: (M N)
-
+ You can also write them operator-leftmost, if you put them inside parentheses to help the parser understand you:
-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.
+ ( + ) 2 3
-Examples of expressions:
+ I'll mostly do this, for uniformity with Scheme and the lambda calculus.
- x
- (y x)
- (x x)
- (\x y)
- (\x x)
- (\x (\y x))
- (x (\x x))
- ((\x (x x)) (\x (x x)))
+ In OCaml and the lambda calculus, this:
-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:
+ foo 2 3
- ((\a M) N)
+ means the same as:
-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**.
+ ((foo 2) 3)
-The rule of beta-reduction permits a transition from that expression to the following:
+ 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:
- M [a:=N]
+* [[!wikipedia Currying]]
-What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
+ 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.
-What is a free occurrence?
+ 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.
-> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
+ 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.
-> 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.
+2. Binding suitable values to the variables `three` and `two`, and adding them.
-For instance:
+ In Scheme:
+ (let* ((three 3))
+ (let* ((two 2))
+ (+ three two)))
-> T is defined to be `(x (\x (\y (x (y z)))))`
+ 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.
-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:
+ In OCaml:
- ((\x (y x)) z)
+ let three = 3 in
+ let two = 2 in
+ ( + ) three two
-beta-reduces to:
+ In the lambda calculus:
- (y z)
+ 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.
-We'll write that like this:
+ But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in:
- ((\x (y x)) z) ~~> (y z)
+ (((\three (\two ((... three) two))) ...) ...)
+
+ In an ordinary imperatival language like C:
-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:
+ int three = 3;
+ int two = 2;
+ three + two;
- M ~~> N
+2. Mutation
-We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol →
for one-step contraction, and the symbol ↠
for zero-or-more step reduction. Hindley and Seldin use ⊳1
and ⊳
.
+ In C this looks almost the same as what we had before:
-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:
+ int x = 3;
+ x = 2;
- M <~~> N
+ Here we first initialize `x` to hold the value 3; then we mutate `x` to hold a new value.
-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 imperatival part of) Scheme, this could be done as:
-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:
+ (let ((x (box 3)))
+ (set-box! x 2))
-> T is defined to be `(M N)`.
+ 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.
-We'll regard the following two expressions:
+ In (the imperatival part of) OCaml, this could be done as:
- (\x (x y))
+ let x = ref 3 in
+ x := 2
- (\z (z y))
+ 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.
-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.
+ In the lambda calculus:
-Note that neither of those expressions are identical to:
+ 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.
- (\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.
+3. Anonymous functions
-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.
+ 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]]
-Shorthand
----------
+ 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.
-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.)
+ In the lambda calculus:
+ (\x M)
-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:
+ ---where `M` is any simple or complex expression---is anonymous. It's only when you do:
- (\x (\y (xy)))
+ ((\y N) (\x M))
-can be abbreviated as:
+ that `(\x M)` has a "name" (it's named `y` during the evaluation of `N`).
- (\x (\y. x y))
+ In Scheme, the same thing is written:
-and:
+ (lambda (x) M)
- (\x \y. (z y) z)
+ 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:
-would abbreviate:
+ (lambda (x) (+ 3 x))
- (\x \y ((z y) z))
+ In OCaml, we write our anonymous function like this:
-This on the other hand:
+ fun x -> ( + ) 3 x
- ((\x \y. (z y) z)
-would abbreviate:
+4. Supplying an argument to an anonymous function
- ((\x (\y (z y))) z)
+ 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:
-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:
+ ((lambda (x) (+ 3 x)) 2)
- (\x x y)
+ 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))`.
-as:
+ In OCaml:
- \x. x y
+ (fun x -> ( + ) 3 x) 2
-but you should include the parentheses in:
- (\x. x y) z
+5. Binding variables to values with "let"
-and:
+ Let's go back and re-consider this Scheme expression:
- z (\x. x y)
+ (let* ((three 3))
+ (let* ((two 2))
+ (+ three two)))
-Merging lambdas: an expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
+ 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:
- (\x y. M)
+ (let* ((three 3) (two 2))
+ (+ three two))
-Similarly, `(\x (\y (\z M)))` can be abbreviated as:
+ 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:
- (\x y z. M)
+ (let* ((three 3))
+ (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 `(...)`.
-Lambda terms represent functions
---------------------------------
+ It was asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:
-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:
+ (let* [(three (+ 1 2))]
+ (let* [(two 2)]
+ (+ three two)))
+
+ 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.
-(\x x) represents the identity function: given any argument M, this function
-simply returns M: ((\x x) M) ~~> M.
+ Repeating our starting point for reference:
-(\x (x x)) duplicates its argument:
-((\x (x x)) M) ~~> (M M)
+ (let* [(three 3)]
+ (let* [(two 2)]
+ (+ three two)))
-(\x (\y x)) throws away its second argument:
-(((\x (\y x)) M) N) ~~> M
+ Recall in OCaml this same computation was written:
-and so on.
+ let three = 3 in
+ let two = 2 in
+ ( + ) three two
-It is easy to see that distinct lambda expressions can represent the same
-function, considered as a mapping from input to outputs. Obviously:
+6. Binding with "let" is the same as supplying an argument to a lambda
- (\x x)
+ The preceding expression in Scheme is exactly equivalent to:
-and:
+ (((lambda (three) (lambda (two) (+ three two))) 3) 2)
- (\z z)
+ The preceding expression in OCaml is exactly equivalent to:
-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:
+ (fun three -> (fun two -> ( + ) three two)) 3 2
- (\y x. y x) (\z z)
+ Read this several times until you understand it.
- (\x. (\z z) x)
+7. Functions can also be bound to variables (and hence, cease being "anonymous").
- (\z z)
+ In Scheme:
-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 argument can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.
+ (let* [(bar (lambda (x) B))] M)
-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.
+ 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)`.
-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 seminar (and further discussion is best pursued in person).
+ Similarly, in OCaml:
+ let bar = fun x -> B in
+ M
+ This in Scheme:
+ (let* [(bar (lambda (x) B))] (bar A))
+ as we've said, means the same as:
-Booleans and pairs
-==================
+ ((lambda (bar) (bar A)) (lambda (x) B))
-Our definition of these is reviewed in [[Assignment1]].
+ which beta-reduces to:
+
+ ((lambda (x) B) A)
+
+ and that means the same as:
+
+ (let* [(x A)] B)
+
+ in other words: evaluate `B` with `x` assigned to the value `A`.
+
+ Similarly, this in OCaml:
+
+ let bar = fun x -> B in
+ bar A
+
+ is equivalent to:
+
+ (fun x -> B) A
+
+ and that means the same as:
+
+ let x = A in
+ B
+
+8. Pushing a "let"-binding from now until the end
+
+ What if you want to do something like this, in Scheme?
+
+ (let* [(x A)] ... for the rest of the file or interactive session ...)
+
+ or this, in OCaml:
+
+ let x = A in
+ ... for the rest of the file or interactive session ...
+
+ 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:
+
+ 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. 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
+
+ OCaml permits you to abbreviate:
+
+ let bar = fun x -> B in
+ M
+
+ as:
+
+ let bar x = B in
+ M
+
+ It also permits you to abbreviate:
+
+ let bar = fun x -> B;;
+
+ as:
+
+ let bar x = B;;
+
+ Similarly, Scheme permits you to abbreviate:
+
+ (define bar (lambda (x) B))
+
+ as:
+
+ (define (bar x) B)
+
+ and this is the form you'll most often see Scheme definitions written in.
+
+ However, conceptually you should think backwards through the abbreviations and equivalences we've just presented.
+
+ (define (bar x) B)
+
+ just means:
+
+ (define bar (lambda (x) B))
+
+ which just means:
+
+ (let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...)
+
+ which just means:
+
+ (lambda (bar) ... rest of the file or interactive session ...) (lambda (x) B)
+
+ or in other words, interpret the rest of the file or interactive session with `bar` assigned the function `(lambda (x) B)`.
+
+
+10. Shadowing
+
+ You can override a binding with a more inner binding to the same variable. For instance the following expression in OCaml:
+
+ let x = 3 in
+ let x = 2 in
+ x
+
+ 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:
+
+ int x = 3;
+ x = 2;
+
+ but it's not the same! In the latter case we have mutation, in the former case we don't. You will learn to recognize the difference as we proceed.
+
+ The OCaml expression just means:
+
+ (fun x -> ((fun x -> x) 2) 3)
+
+ and there's no more mutation going on there than there is in:
+
+ ∀x. (F x or ∀x (not (F x)))
+
+
+ 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
+----------------------------------------------
+
+* Simple predefined values
+
+ Numbers in Scheme: `2`, `3`
+ In OCaml: `2`, `3`
+
+ Booleans in Scheme: `#t`, `#f`
+ In OCaml: `true`, `false`
+
+ The eighth letter in the Latin alphabet, in Scheme: `#\h`
+ In OCaml: `'h'`
+
+* Compound values
+
+ These are values which are built up out of (zero or more) simple values.
+
+ Ordered pairs in Scheme: `'(2 . 3)` or `(cons 2 3)`
+ In OCaml: `(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: `'()` or `(list)`
+ In OCaml: `[]`
+
+ The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`
+ In OCaml: `"h"`
+
+ A longer string, in Scheme: `"horse"`
+ In OCaml: `"horse"`
+
+ A shorter string, in Scheme: `""`
+ In OCaml: `""`
+
+
+
+What "sequencing" is and isn't
+------------------------------
+
+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.
+
+Neither do they have any useful notion of sequencing. But what this would be takes some care to identify.
+
+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,
+
+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.
+
+Third, the kinds of bindings we see in:
+
+ (define foo A)
+ (foo 2)
+
+Or even:
+
+ (define foo A)
+ (define foo B)
+ (foo 2)
+
+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.
+
+Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:
+
+ (begin A B C)
+
+In OCaml it looks like this:
+
+ begin A; B; C end
+
+Or this:
+
+ (A; B; C)
+
+In the presence of imperatival elements, sequencing order is very relevant. For example, these will behave differently:
+
+ (begin (print "under") (print "water"))
+
+ (begin (print "water") (print "under"))
+
+And so too these:
+ begin x := 3; x := 2; x end
+ begin x := 2; x := 3; x end
+However, if A and B are purely functional, non-imperatival expressions, then:
+ begin A; B; C end
+just evaluates to C (so long as A and B evaluate to something at all). So:
-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
+ begin A; B; C end
-1. Beta reduction
-1. Encoding pairs (and triples and ...)
-1. Encoding booleans
+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.
+We'll discuss this more as the seminar proceeds.