* 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
+
+
+
Declarative/functional vs Imperatival/dynamic models of computation
===================================================================
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.
-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 pate 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.
+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.
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.
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":
- true and true = true
- true and * = *
- true and false = false
- * and true = *
- * and * = *
- * and false = *
- false and true = false
- false and * = false
- false and false = false
+true and true = true
-And then we'd notice that `* and false` has a different intepretation than `false and *`. (The same phenomenon is already present with the mateial conditional in bivalent logics; but seeing that a non-symmetric semantics for `and` is available even for functional languages is instructive.)
+And then we'd notice that `* and false` has a different intepretation than <code>false and *</code>. (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.)
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:
(fun x -> (( + ) (3) (x)))
- As we'll see below, parentheses can often be omitted in the lambda calculus too. But not in Scheme. Every parentheses has a specific role.
+ 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
(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:
+ 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:
(let* ((three 3) (two 2))
(+ three two))
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 below.)
+ 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.)
9. Some shorthand
which just means:
- (let [(bar (lambda (x) B))] ... rest of the file or interactive session ...)
+ (let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...)
which just means:
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.
+ <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.
The OCaml expression just means:
(fun x -> ((fun x -> x) 2) 3)
- and there's no more change of state going on here than there is in:
+ and there's no more mutation going on there than there is in:
- <blockquote>
- &exists;x. (F x and &exists;x (not (F x)))
- </blockquote>
+ <pre>
+ <code>∀x. (F x or ∀x (not (F x)))</code>
+ </pre>
+
+ 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.
Some more comparisons between Scheme and OCaml
11. Simple predefined values
- Numbers in Scheme: 2, 3
- In OCaml: 2, 3
+ Numbers in Scheme: `2`, `3`
+ In OCaml: `2`, `3`
- Booleans in Scheme: #t, #f
- In OCaml: true, false
+ Booleans in Scheme: `#t`, `#f`
+ In OCaml: `true`, `false`
- The eighth letter in the Latin alphabet, in Scheme: #\h
- In OCaml: 'h'
+ The eighth letter in the Latin alphabet, in Scheme: `#\h`
+ In OCaml: `'h'`
12. Compound values
These are values which are built up out of (zero or more) simple values.
- Ordered pairs in Scheme: '(2 . 3)
- In OCaml: (2, 3)
+ Ordered pairs in Scheme: `'(2 . 3)`
+ In OCaml: `(2, 3)`
- Lists in Scheme: '(2 3)
- In OCaml: [2; 3]
+ Lists in Scheme: `'(2 3)`
+ In OCaml: `[2; 3]`
We'll be explaining the difference between pairs and lists next week.
- The empty list, in Scheme: '()
- In OCaml: []
+ The empty list, in Scheme: `'()`
+ In OCaml: `[]`
- The string consisting just of the eighth letter of the Latin alphabet, in Scheme: "h"
- In OCaml: "h"
+ 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 longer string, in Scheme: `"horse"`
+ In OCaml: `"horse"`
- A shorter string, in Scheme: ""
- In OCaml: ""
+ A shorter string, in Scheme: `""`
+ In OCaml: `""`
13. Function application
- Binary functions in OCaml: foo 2 3
+ Binary functions in OCaml: `foo 2 3`
- Or: ( + ) 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.
+ 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.
+ 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
+------------------------------
+
+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)
-Computation = sequencing changes?
+Or even:
- Different notions of sequencing
+ (define foo A)
+ (define foo B)
+ (foo 2)
- Concatanation / syntactic complexity is not sequencing
+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.
- Shadowing is not mutating
+Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:
- Define isn't mutating
+ (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"))
-
-(let [(three 3) (two 2)] (+ 3 2))
+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:
+
+ begin A; B; C end
+
+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.
+
+
1. Declarative vs imperatival models of computation.
2. Variety of ways in which "order can matter."
3. Variety of meanings for "dynamic."