1 Here's what we did in seminar on Monday 9/13,
3 Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class. These notes expand on *a lot*, and some of this material will be reviewed next week.
8 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.)
13 * generalized quantifiers are a special case of operating on continuations
15 * (Chris: fill in other applications...)
17 * expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the [summary](/damn) for more explanation and elaboration
22 * 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.
24 * those issues may bear on Russell's Gray's Elegy argument in "On Denoting"
26 * 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."
28 * 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
30 * the semantics for mutation will enable us to make sense of a difference between numerical and qualitative identity---for purely mathematical objects!
32 * 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
37 Basics of Lambda Calculus
38 =========================
40 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.)
45 <strong>Variables</strong>: <code>x</code>, <code>y</code>, <code>z</code>...
48 Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:
51 <strong>Abstract</strong>: <code>(λa M)</code>
54 We'll tend to write <code>(λa M)</code> as just `(\a M)`, so we don't have to write out the markup code for the <code>λ</code>. You can yourself write <code>(λa M)</code> or `(\a M)` or `(lambda a M)`.
57 <strong>Application</strong>: <code>(M N)</code>
60 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.
62 Examples of expressions:
71 ((\x (x x)) (\x (x x)))
73 The lambda calculus has an associated proof theory. For now, we can regard the
74 proof theory as having just one rule, called the rule of **beta-reduction** or
75 "beta-contraction". Suppose you have some expression of the form:
79 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**.
81 The rule of beta-reduction permits a transition from that expression to the following:
85 What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
87 What is a free occurrence?
89 > An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
91 > 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`.
93 > An occurrence of a variable is **free** if it's not bound.
98 > T is defined to be `(x (\x (\y (x (y z)))))`
100 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.
102 Here's an example of beta-reduction:
110 We'll write that like this:
112 ((\x (y x)) z) ~~> (y z)
114 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:
118 We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol <code><big><big>→</big></big></code> for one-step contraction, and the symbol <code><big><big>↠</big></big></code> for zero-or-more step reduction. Hindley and Seldin use <code><big><big><big>⊳</big></big></big><sub>1</sub></code> and <code><big><big><big>⊳</big></big></big></code>.
120 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:
124 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.)
126 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 <code>≡</code> 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:
128 > T is defined to be `(M N)`.
130 We'll regard the following two expressions:
136 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.
138 Note that neither of those expressions are identical to:
142 because here it's a free variable that's been changed. Nor are they identical to:
146 because here the second occurrence of `y` is no longer free.
148 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.
154 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.)
157 **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:
165 but you should include the parentheses in:
174 **Dot notation** Dot means "put a left paren here, and put the right
175 paren as far the right as possible without creating unbalanced
180 can be abbreviated as:
196 This on the other hand:
205 **Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
209 Similarly, `(\x (\y (\z M)))` can be abbreviated as:
214 Lambda terms represent functions
215 --------------------------------
217 All (recursively computable) functions can be represented by lambda
218 terms (the untyped lambda calculus is Turing complete). For some lambda terms, it is easy to see what function they represent:
220 > `(\x x)` represents the identity function: given any argument `M`, this function
221 simply returns `M`: `((\x x) M) ~~> M`.
223 > `(\x (x x))` duplicates its argument:
224 `((\x (x x)) M) ~~> (M M)`
226 > `(\x (\y x))` throws away its second argument:
227 `(((\x (\y x)) M) N) ~~> M`
231 It is easy to see that distinct lambda expressions can represent the same
232 function, considered as a mapping from input to outputs. Obviously:
240 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:
248 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.
250 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.
252 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).
259 Our definition of these is reviewed in [[Assignment1]].
262 It's possible to do the assignment without using a Scheme interpreter, however
263 you should take this opportunity to [get Scheme installed on your
264 computer](/how_to_get_the_programming_languages_running_on_your_computer), and
265 [get started learning Scheme](/learning_scheme). It will help you test out
266 proposed answers to the assignment.
273 Declarative/functional vs Imperatival/dynamic models of computation
274 ===================================================================
276 Many of you, like us, will have grown up thinking the paradigm of computation is a sequence of changes. Let go of that. It will take some care to separate the operative notion of "sequencing" here from other notions close to it, but once that's done, you'll see that languages that have no significant notions of sequencing or changes are Turing complete: they can perform any computation we know how to describe. In itself, that only puts them on equal footing with more mainstream, imperatival programming languages like C and Java and Python, which are also Turing complete. But further, the languages we want you to become familiar with can reasonably be understood to be more fundamental. They embody the elemental building blocks that computer scientists use when reasoning about and designing other languages.
278 Jim offered the metaphor: think of imperatival languages, which include "mutation" and "side-effects" (we'll flesh out these keywords as we proceeed), as the pâté of computation. We want to teach you about the meat and potatoes, where as it turns out there is no sequencing and no changes. There's just the evaluation or simplification of complex expressions.
280 Now, when you ask the Scheme interpreter to simplify an expression for you, that's a kind of dynamic interaction between you and the interpreter. You may wonder then why these languages should not also be understood imperatively. The difference is that in a purely declarative or functional language, there are no dynamic effects in the language itself. It's just a static semantic fact about the language that one expression reduces to another. You may have verified that fact through your dynamic interactions with the Scheme interpreter, but that's different from saying that there are dynamic effects in the language itself.
282 What the latter would amount to will become clearer as we build our way up to languages which are genuinely imperatival or dynamic.
284 Many of the slogans and keywords we'll encounter in discussions of these issues call for careful interpretation. They mean various different things.
286 For example, you'll encounter the claim that declarative languages are distinguished by their **referential transparency.** What's meant by this is not always exactly the same, and as a cluster, it's related to but not the same as this means for philosophers and linguists.
288 The notion of **function** that we'll be working with will be one that, by default, sometimes counts as non-identical functions that map all their inputs to the very same outputs. For example, two functions from jumbled decks of cards to sorted decks of cards may use different algorithms and hence be different functions.
290 It's possible to enhance the lambda calculus so that functions do get identified when they map all the same inputs to the same outputs. This is called making the calculus **extensional**. Church called languages which didn't do this **intensional**. If you try to understand that kind of "intensionality" in terms of functions from worlds to extensions (an idea also associated with Church), you may hurt yourself. So too if you try to understand it in terms of mental stereotypes, another notion sometimes designated by "intension."
292 It's often said that dynamic systems are distinguished because they are the ones in which **order matters**. However, there are many ways in which order can matter. If we have a trivalent boolean system, for example---easily had in a purely functional calculus---we might choose to give a truth-table like this for "and":
296 true and false = false
300 false and true = false
302 false and false = false
304 And then we'd notice that `* and false` has a different intepretation than `false and *`. (The same phenomenon is already present with the material conditional in bivalent logics; but seeing that a non-symmetric semantics for `and` is available even for functional languages is instructive.)
306 Another way in which order can matter that's present even in functional languages is that the interpretation of some complex expressions can depend on the order in which sub-expressions are evaluated. Evaluated in one order, the computations might never terminate (and so semantically we interpret them as having "the bottom value"---we'll discuss this). Evaluated in another order, they might have a perfectly mundane value. Here's an example, though we'll reserve discussion of it until later:
308 (\x. y) ((\x. x x) (\x. x x))
310 Again, these facts are all part of the metatheory of purely functional languages. But *there is* a different sense of "order matters" such that it's only in imperatival languages that order so matters.
316 Here the comparison in the last line will evaluate to true.
322 Here the comparison in the last line will evaluate to false.
324 One of our goals for this course is to get you to understand *what is* that new
325 sense such that only so matters in imperatival languages.
327 Finally, you'll see the term **dynamic** used in a variety of ways in the literature for this course:
329 * dynamic versus static typing
331 * dynamic versus lexical scoping
333 * dynamic versus static control operators
335 * finally, we're used ourselves to talking about dynamic versus static semantics
337 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.
344 <td width=30%>Scheme (functional part)</td>
345 <td width=30%>OCaml (functional part)</td>
346 <td width=30%>C, Java, Pasval<br>
347 Scheme (imperative part)<br>
348 OCaml (imperative part)</td>
350 <td width=30%>lambda calculus<br>
351 combinatorial logic</td>
353 <td colspan=3 align=center>--------------------------------------------------- Turing complete ---------------------------------------------------</td>
356 <td width=30%>more advanced type systems, such as polymorphic types
360 <td width=30%>simply-typed lambda calculus (what linguists mostly use)
368 Here's how it looks to say the same thing in various of these languages.
370 1. Binding suitable values to the variables `three` and `two`, and adding them.
384 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:
390 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.
392 But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in:
394 (((\three (\two ((... three) two))) ...) ...)
396 In an ordinary imperatival language like C:
404 In C this looks almost the same as what we had before:
409 Here we first initialize `x` to hold the value 3; then we mutate `x` to hold a new value.
411 In (the imperatival part of) Scheme, this could be done as:
416 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.
418 In (the imperatival part of) OCaml, this could be done as:
423 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.
425 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.
431 3. Anonymous functions
433 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.
435 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.
437 In the lambda calculus:
441 is always anonymous! Here `M` stands for any expression of the language, simple or complex. It's only when you do
445 that `(\x M)` has a "name" (it's named `y` during the evaluation of `N`).
447 In Scheme, the same thing is written:
451 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:
455 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.
457 In OCaml, we write our anonymous function like this:
465 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:
471 (fun x -> (( + ) (3) (x)))
473 As we saw above, parentheses can often be omitted in the lambda calculus too. But not in Scheme. Every parentheses has a specific role.
475 4. Supplying an argument to an anonymous function
477 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:
479 ((lambda (x) (+ 3 x)) 2)
481 The outermost parentheses here mean "apply the function `(lambda (x) (+ 3 x))` to the argument `2`.
485 (fun x -> ( + ) 3 x) 2
488 5. Binding variables to values with "let"
490 Let's go back and re-consider this Scheme expression:
496 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:
498 (let* ((three 3) (two 2))
501 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:
507 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 `(...)`.
509 Someone asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:
511 (let* [(three (+ 1 2))]
515 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.
517 Repeating our starting point for reference:
523 Recall in OCaml this same computation was written:
529 6. Binding with "let" is the same as supplying an argument to a lambda
531 The preceding expression in Scheme is exactly equivalent to:
533 (((lambda (three) (lambda (two) (+ three two))) 3) 2)
535 The preceding expression in OCaml is exactly equivalent to:
537 (fun three -> (fun two -> ( + ) three two)) 3 2
539 Read this several times until you understand it.
541 7. Functions can also be bound to variables (and hence, cease being "anonymous").
545 (let* [(bar (lambda (x) B))] M)
547 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)`.
551 let bar = fun x -> B in
556 (let* [(bar (lambda (x) B))] (bar A))
558 as we've said, means the same as:
560 ((lambda (bar) (bar A)) (lambda (x) B))
562 which, as we'll see, is equivalent to:
566 and that means the same as:
570 in other words: evaluate `B` with `x` assigned to the value `A`.
572 Similarly, this in OCaml:
574 let bar = fun x -> B in
581 and that means the same as:
586 8. Pushing a "let"-binding from now until the end
588 What if you want to do something like this, in Scheme?
590 (let* [(x A)] ... for the rest of the file or interactive session ...)
595 ... for the rest of the file or interactive session ...
597 Scheme and OCaml have syntactic shorthands for doing this. In Scheme it's written like this:
600 ... rest of the file or interactive session ...
602 In OCaml it's written like this:
605 ... rest of the file or interactive session ...
607 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.)
611 OCaml permits you to abbreviate:
613 let bar = fun x -> B in
621 It also permits you to abbreviate:
623 let bar = fun x -> B;;
629 Similarly, Scheme permits you to abbreviate:
631 (define bar (lambda (x) B))
637 and this is the form you'll most often see Scheme definitions written in.
639 However, conceptually you should think backwards through the abbreviations and equivalences we've just presented.
645 (define bar (lambda (x) B))
649 (let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...)
653 (lambda (bar) ... rest of the file or interactive session ...) (lambda (x) B)
655 or in other words, interpret the rest of the file or interactive session with `bar` assigned the function `(lambda (x) B)`.
660 You can override a binding with a more inner binding to the same variable. For instance the following expression in OCaml:
666 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:
671 <em>but it's not the same!</em> In the latter case we have mutation, in the former case we don't. You will learn to recognize the difference as we proceed.
673 The OCaml expression just means:
675 (fun x -> ((fun x -> x) 2) 3)
677 and there's no more mutation going on there than there is in:
679 <pre><code>∀x. (F x or ∀x (not (F x)))
682 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.
685 Some more comparisons between Scheme and OCaml
686 ----------------------------------------------
688 11. Simple predefined values
690 Numbers in Scheme: `2`, `3`
693 Booleans in Scheme: `#t`, `#f`
694 In OCaml: `true`, `false`
696 The eighth letter in the Latin alphabet, in Scheme: `#\h`
701 These are values which are built up out of (zero or more) simple values.
703 Ordered pairs in Scheme: `'(2 . 3)`
706 Lists in Scheme: `'(2 3)`
708 We'll be explaining the difference between pairs and lists next week.
710 The empty list, in Scheme: `'()`
713 The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`
716 A longer string, in Scheme: `"horse"`
719 A shorter string, in Scheme: `""`
722 13. Function application
724 Binary functions in OCaml: `foo 2 3`
728 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.
730 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.
732 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.
735 What "sequencing" is and isn't
736 ------------------------------
738 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.
740 Neither do they have any useful notion of sequencing. But what this would be takes some care to identify.
742 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,
744 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.
746 Third, the kinds of bindings we see in:
757 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.
759 Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:
763 In OCaml it looks like this:
771 In the presence of imperatival elements, sequencing order is very relevant. For example, these will behave differently:
773 (begin (print "under") (print "water"))
775 (begin (print "water") (print "under"))
779 begin x := 3; x := 2; x end
781 begin x := 2; x := 3; x end
783 However, if A and B are purely functional, non-imperatival expressions, then:
787 just evaluates to C (so long as A and B evaluate to something at all). So:
791 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.
793 We'll discuss this more as the seminar proceeds.