We hope, though, that learning Kapulet first puts you in a position to make that shift more effortlessly, and also to more quickly see the places where there's underlying unity to Scheme, OCaml, and Haskell, despite their diverse syntaxes. (And idiosyncratic warts.)
+
+
+
### Comments
... # this is a comment in Kapulet, that goes until the end of the line
Haskell's block comments also nest.
Racket and Scheme also have block comments, though they `#| write them differently |#`.
-These block comments also nest. Another form of block comments is `#;( ... )`. Those may contain nested parentheses, and extend until the next *matching* `)`. So prefixing `#;` to a complex parenthesized expression is a way to turn the whole thing into a comment. (These two comment styles aren't part of the official Scheme standard, but they are widely implemented.)
+These block comments also nest. Another form of block comments is `#;( ... )`. Those may contain nested parentheses, and extend until the next *matching* `)`. So prefixing `#;` to a complex parenthesized expression is a way to turn the whole thing into a comment. (These two comment styles only recently became part of the official Scheme standard, but they have been widely implemented.)
+
+
+
+
+### Variables
+
+Our [[syntax for variables|topics/week1#variables]] in Kapulet is close to that in the other languages. Haskell and OCaml differ only in that they do not permit trailing `?` or `!`; however, they do permit trailing `'`s (and even permit `'`s *in the middle* of a variable too, which Kapulet does not). Scheme permits all of these characters, plus many more punctuation symbols as well, to occur anywhere in a variable. Scheme also permits variables to begin with capital letters, or to consist solely of the single character `_`; but the other languages reserve these terms for special purposes.
+
+In addition to the variables made of letters (more properly, of alphanumerics), Haskell and OCaml and Kapulet also permit some variables made exclusively of punctuation symbols, like `<` or Haskell's `>=>` and `<$>`. In Haskell, these always have infix syntax, and the variables made of letters never do. (But the former can have their infix syntax suppressed with parentheses, and the latter can be "flagged" to temporarily take on infix syntax, as we'll discuss below.)
+
+In OCaml and Kapulet, some variables made of letters also have infix syntax, such as `comp` in Kapulet or `mod` in OCaml. I haven't presented to you the complex mechanisms needed to declare this.
+
+
+
+
+### Equality and Booleans
+
+The relation that's written `==` in Kapulet is also written that way in Haskell. That symbol means something else in OCaml, having to do with mutable reference cells; to get the same notion in OCaml one writes just a single `=`. The negation of this relation is written `!=` in Kapulet, `/=` in Haskell, and `<>` in OCaml. (Again, `!=` means something else in OCaml.)
+
+FIXME 3 and Scheme Equalities
+
+The relations that are written `and`, `or`, and `not` in Kapulet are written the same way in Scheme. Note that in Scheme the first two can take zero or more arguments:
+
+ ; Scheme
+ (and)
+ (and bool1)
+ (and bool1 bool2)
+ (and bool1 bool2 bool3)
+
+As you'd expect `(and bool1)` evaluates the same as plain `bool1`; similarly with `(or bool1)`. What do you think `(and)` with no arguments should evaluate to? How about `(or)`?
+
+These relations are written in Haskell and OCaml as `&&`, `||`, and `not`. (Haskell uses `and` and `or` to express other functions, which compute the joint conjunction or disjunction of every `Bool` value in a List of such. OCaml permits `or` as an old synonym for `||`, but discourages using that spelling. OCaml also permits `&` as an old, discouraged synonym for `&&`.)
+The values that are written `'true` and `'false` in Kapulet are written in Haskell as `True` and `False`, and in OCaml as just `true` and `false`. (It'd be more consistent with OCaml's other naming policies for them to have said True and False, but they didn't.) These are written `#t` and `#f` in Scheme, but in Scheme in many contexts any value that isn't `#f` will behave as though it were `#t`, even values you might think are more "false-like", like `0` and the empty list.
+<a id=truth-like></a> Thus `(if 0 'zero 'nope)` will evaluate to `'zero`.
+Some Scheme implementations, such as Racket, permit `#true` and `#false` as synonyms for `#t` and `#f`.
-### Variable names
+Scheme also recognizes the values `'true` and `'false`, but it treats `'false` as distinct from `#f`, and thus as a "truth-like" value, like all of its other values that aren't `#f`. Kapulet essentially took Scheme's `boolean` values and collapsed them into being a subtype of its `symbol` values. FIXME also with chars.
-% Haskell's division into letter-based vs operators. Each can be "flagged" to temporarily behave as though it belonged to the other syntactic category (see below).
+ 1 2
-Although all three of these languages permits you to enclose an infix operator in parentheses to make a *section*, which no longer has infix syntax. In Kapulet, `( + )` is the same as λ `(x, y). x + y`, whereas in OCaml and Haskell it's a *curried* function, which we can write (in Kapulet syntax) as λ `x y. x + y`.
+<a id=pre-curried></a>
+But all three of these languages permits you to enclose an infix operator in parentheses to make a *section*, which no longer has infix syntax. In Kapulet, `( + )` is the same as λ `(x, y). x + y`, whereas in OCaml and Haskell it's a *curried* function, which we can write (in Kapulet syntax) as λ `x y. x + y`. We'll discuss [[sections|#sections]] and [[curried functions|#curried]] below.
-Kapulet and OCaml have some operators spelled with letters also taking infix syntax, such as `comp` in Kapulet or `mod` in OCaml. In Haskell, this is never the case: variables that are made of letters are only treated as function terms being applied to arguments *when they're at the start* of a list of expressions; and variables that are made of punctuation symbols, and not enclosed in parentheses, will only be treated as infix operators. However, Haskell permits you to temporarily "flag" a letter-made function term to behave like an infix operator, by enclosing it in `` ` `` marks. Thus in Haskell you can write:
+Kapulet and OCaml have some variables made of (or spelled with) letters also taking infix syntax, such as `comp` in Kapulet or `mod` in OCaml. In Haskell, this is never the case: variables that are made of letters are only treated as function terms being applied to arguments *when they're at the start* of a list of expressions; and variables that are made of punctuation symbols, and not enclosed in parentheses, will only be treated as infix operators. However, Haskell permits you to temporarily "flag" a function term made of letters to behave like an infix operator, by enclosing it in `` ` `` marks. Thus in Haskell you can write:
3 `mod` 2
But without the `` ` ``, you'd have to write: `mod 3 2`.
-Scheme has no infix operators. It ruthlessly demands that all functions to be applied to arguments come at the start of a list of expressions, regardless of whether those functions are spelled using letters, punctuation symbols, or a mix of the two. Thus in Scheme one always writes:
+Scheme has no infix operators. It ruthlessly demands that all functions to be applied to arguments come at the start of a list of expressions, regardless of whether the functions are specified by variables made of letters, punctuation symbols, or a mix of the two, or even if the functions are computed by evaluating more complex expressions. Thus in Scheme one always writes:
(+ 3 2)
((+) 3 2)
-what that would mean is that `+` is first being applied to *zero* arguments, which is different from not applying it all. (In Kapulet, OCaml, and Haskell, one would write that `f` is being applied to "zero arguments" like this: `f ()`.) Scheme helpfully defines the result of applying `+` to zero arguments to be `0`. So `((+) 3 2)` would evaluate to whatever `(0 3 2)` does, and that's an error, because `0` is not a function.
+what that would mean is that `+` is first being applied to *zero* arguments, which is different from not applying it all. (In Kapulet, OCaml, and Haskell, one would write that `f` is being applied to "zero arguments" like this: `f ()`. We will discuss FIXME) Scheme helpfully defines the result of applying `+` to zero arguments to be `0`. So `((+) 3 2)` would evaluate to whatever `(0 3 2)` does, and that's an error, because `0` is not a function.
-Note that `(0 3 2)`, although it *is*, qua expression, a list of numbers, does not evaluate to a list. To get an expression that *evaluates to* that list, you'd have to use `(list 0 3 2)` or `'(0 3 2)`. (Notice the initial `'`.)
+Note that `(0 3 2)`, although it *is*, qua expression, a list of numbers, does not evaluate to a list. To get an expression that *evaluates to* that list, you'd have to use `(list 0 3 2)` or `'(0 3 2)`. (Notice the initial `'`.) More on this [[below|#writing-scheme-lists]].
In Scheme, you can also write `(+ 3 2 10)`, and so on. You only have to write `(+ (+ 3 2) 10)` if you really want to.
-% Parentheses have other roles in Scheme, too.
+Parentheses have many other roles in Scheme, too; they're a ubiquitous part of the syntax, and don't always express function application. You might sometimes feel they are overused.
+You may sometimes see `[ ... ]` being used in Scheme, instead of `( ... )`. This is just a stylistic variant; they work exactly the same. The official Scheme standard doesn't permit this usage, but most Scheme implementations do. It can help keep track of which closing `]` or `)` goes with which opening `[` or `(`. The opening and closing symbols always have to correspond.
+
+<a id=curried></a>
In Scheme, the default style for defining functions is as taking several arguments simultaneously, that is the *uncurried* style. In OCaml and Haskell, the default style is to define them *curried*. Curried functions can easily be partially applied:
(* OCaml *)
will result in `5`.
-In Scheme, the default would be to define `add` like this:
+In Scheme, the common idiom would be to define `add` like this:
(define add (lambda (x y) (+ x y)))
-Then you cannot say `(add 2)`, because `add` will be expecting two arguments, but you only supplied one. You can however define curried functions in Scheme, it's just more laborious:
+(We'll explain `define` [[below#define]].) After this, you cannot say `(add 2)`, because `add` will be expecting two arguments, but you only supplied one. You can however define curried functions in Scheme, it's just more laborious:
(define curried_add (lambda (x) (lambda (y) (+ x y))))
(define add2 (curried_add 2))
OCaml and Haskell also permit defining functions in uncurried form:
(* OCaml *)
- let add = fun (x, y) -> x + y in
+ let add = fun (x, y) -> x + y (* uncurried*) in
let add2 = fun add 2 in ...
Here the last displayed line will fail, because `add` expects as its argument a tuple of two numbers.
-Kapulet essentially works like OCaml and Haskell; though for pedagogical reasons I started out by introducing uncurried definitions, rather than the *curried* definitions those other languages predominantly use.
+Kapulet essentially works like OCaml and Haskell; though for pedagogical reasons we started out by introducing uncurried definitions, rather than the *curried* definitions those other languages predominantly use.
-As we mentioned, in Kapulet, OCaml, and Haskell, there is a shorthand that enables you to write things like:
+<a id=sections></a>
+[[As we mentioned in the course notes|topics/week1_advanced_notes#sections]], in Kapulet, OCaml, and Haskell, there is a shorthand that enables you to write things like:
# Kapulet
let
ten_minus match lambda x. 10 - x;
- and_ys match lambda x. x & ys
+ and_ys match lambda x. x & ys;
+ plus match lambda (x, y). x + y
in (ten_minus, and_ys)
like this:
# Kapulet
let
ten_minus match (10 - );
- and_ys match ( & ys)
+ and_ys match ( & ys);
+ plus match ( + )
in (ten_minus, and_ys)
-There are just minor differences between these languages. First, OCaml doesn't have the `( + 10)` or `(10 + )` forms, but only the `( + )`. Second, as a special case, OCaml doesn't permit you to do this with its list-consing operator `::`. You have to write `fun x xs -> x :: xs`, not `( :: )`. Whereas in Kapulet `( & )`, `(x & )`, and `( & xs)` are all sections using its sequence-consing operator `&`; and in Haskell, `( : )`, `(x : )`, and `( : xs)` are the same.
+There are just minor differences between these languages. First, OCaml doesn't have the `( + 10)` or `(10 + )` forms, but only the `( + )`.
+
+Second, as a special case, OCaml doesn't permit you to do this with its list cons-ing operator `::`. You have to write `fun x xs -> x :: xs`, not `( :: )`. Whereas in Kapulet `( & )`, `(x & )`, and `( & xs)` are all sections using its sequence cons-ing operator `&`; and in Haskell, `( : )`, `(x : )`, and `( : xs)` are the same.
-Thirdly, in Kapulet, `( - 10)` also expresses λ `x. x - 10` (consistently with `(10 - )`), but Haskell (and OCaml) treat this form differently, and interpret it as meaning the integer `- 10`. Here's how to express some things in Kapulet:
+Third, as [[mentioned above|#pre-curried]], OCaml's and Haskell's `( + )` and the like evaluate to *curried* functions.
+
+Fourth, in Kapulet, `( - 10)` expresses λ `x. x - 10` (consistently with `(10 - )`), but Haskell (and OCaml) treat this specific form differently, and interpret it as meaning the integer `- 10`. Here's how to express some things in Kapulet:
# Kapulet
(0 - 2)
( - ) (5, 3)
-and here are their translations into Haskell:
+and here are their translations into natural Haskell:
-- Haskell
- ( -2 )
+ ( -2 ) -- (0 - 2) also works
(subtract 2) -- subtract 2 10 == 8
negate -- (0 - ) also works
( - ) 5 3
OCaml expresses `(0 - )` or `negate` as `~-`. You can write `3 * (0 - 2)` in OCaml as either `3 * ( -2 )` or as `3 * ~-2`.
-I know all these languages fairly well, and I still find this last issue difficult to keep track of. You may be starting to understand why I spoke of "warts."
-
-
-### Equality and Booleans
-
-The relation that's written `==` in Kapulet is also written that way in Haskell. That symbol means something else in OCaml, having to do with mutable reference cells; to get the same notion in OCaml one writes just a single `=`. The negation of this notion is written `!=` in Kapulet, `/=` in Haskell, and `<>` in OCaml. (Again, `!=` means something else in OCaml.)
+I know all these languages fairly well, and I still find this fourth issue difficult to keep track of. You may be starting to understand why I spoke of "warts."
-The relations that are written `and`, `or`, and `not` are written in Haskell and OCaml as `&&`, `||`, and `not`. (Haskell uses `and` and `or` to express functions that form the conjunction or disjunction of every `Bool` value in a List of such. OCaml permits `or` as an old synonym for `||`, but discourages using that spelling. OCaml also permits `&` as an old, discouraged synonym for `&&`.)
-
-Scheme %%
-
-The values that are written `'true` and `'false` in Kapulet are written in Haskell as `True` and `False`, and in OCaml as just `true` and `false`. They're written `#t` and `#f` in Scheme, but in Scheme in many contexts any value that isn't `#f` will behave as though it were `#t`, even values you might think are more "false-ish", like `0` and the empty list. Thus `(if 0 'yes 'no)` will evaluate to `'yes`.
-
-Scheme recognizes the values `'true` and `'false`, but it treats `'false` as distinct from `#f`, and thus as a "true-ish" value, like all of its other values that aren't `#f`.
[1, 2, 3]
-Haskell is very similar, except that it calls these Lists, and its cons-ing operator is written `:`. OCaml also calls them `list`s, and its cons-operator is written `::`. (OCaml *also* uses `:`, but it uses it to deal with types, and Haskell in turn also uses `::`, but that's what *it* uses to deal with types. Grr.)
+Haskell is very similar, except that it calls these Lists, and its cons-ing operator is written `:`. OCaml also calls them `list`s, and its cons-operator is written `::`. (OCaml *also* uses Haskell's symbol `:`, but it uses it to deal with types; and Haskell in turn also uses OCaml's symbol `::`, but that's what *it* uses to deal with types. Grr.)
Kapulet writes the operator that concatenates or appends sequences as `&&`. Thus:
[1; 2] @ [3; 4; 5] ;;
"over" ^ "due" ;;
-evaluate to `[1; 2; 3; 4; 5]` and `"overdue"`. Note that OCaml separates its `list` items with semicolons not commas. If you write `[1, 2, 3]` in OCaml, it will think that's a one-element list whose first element is a triple, that is, what you'd write in Haskell as `[(1, 2, 3)]`.
+evaluate to `[1; 2; 3; 4; 5]` and `"overdue"`. Note that OCaml separates its `list` elements with semicolons not commas. If you write `[1, 2, 3]` in OCaml, it will think that's a one-element list whose first element is a triple, that is, what you'd write in Haskell as `[(1, 2, 3)]`.
Here are some list functions in Kapulet:
dropwhile
reverse
# new functions
- concat # converts [[10, 20], [30], [], [40, 50]]
- # to [10, 20, 30, 40, 50] (only merging a single layer of []s)
+ join # converts [[10, 20], [30], [], [40, 50]]
+ # to [10, 20, 30, 40, 50] (but only "joining" a single layer of []s)
(mem) # infix syntax, 2 mem [1, 2, 3] == 'true
- nth # nth [10, 20, 30] 1 == 20, because the first element
- # is at position 0; fails if index is out of bounds
+ nth # nth [10, 20, 30] 1 == 20, because 10 occupies position 0
+ # fails if the index is out of bounds
all? p xs # all? odd? [1, 3, 5] == 'true
any? p xs # any? even? [1, 3, 5] == 'false
filter
Data.List.partition
map
- zipWith {- zip handles the special case where f is the function that forms ordered pairs
+ zipWith {- zip handles the special case of zipWith where f is the function that forms ordered pairs
both zipWith and zip stop with the shortest list -}
- unzip -- doesn't take an f argument, assumes (\(x, y) -> (x, y))
+ unzip {- unlike unmap2, doesn't take an explicit f argument
+ just assumes it's (\(x, y) -> (x, y)) -}
takeWhile
dropWhile
reverse
- concat
+ concat -- corresponding to join
elem -- not infix syntax, but often written as: 2 `elem` [1, 2, 3]
- (!!) -- infix syntax: [10, 20, 30] !! 1 == 20; fails if index is out of bounds
+ (!!) -- infix syntax: [10, 20, 30] !! 1 == 20
+ -- fails if the index is out of bounds
all p xs
any p xs
Here they are in OCaml:
- length
+ List.length
(@) (* or List.append *)
(* no function corresponding to empty? *)
List.tl (* compare List.hd, which fails on [] *)
List.split (* like Haskell's unzip, doesn't take an f argument *)
(* no function corresponding to takewhile or dropwhile *)
List.rev
- List.concat (* also List.flatten, which still only merges a single layer of []s *)
+ List.concat (* also List.flatten, which still only "joins" a single layer of []s *)
List.mem (* not infix syntax *)
- List.nth (* List.nth [10; 20; 30] 1 = 20; fails if index is out of bounds *)
+ List.nth (* List.nth [10; 20; 30] 1 = 20; fails if the index is out of bounds *)
List.for_all p xs
List.exists p xs
-How does all this look in Scheme? Well, Scheme has a notion they call a (proper) `list`, and also a notion they call a `vector`. There are also what Scheme calls "improper" `list`s, with `(cons 1 'nonlist)` or `'(1 . nonlist)`, where `'nonlist` is any non-list (here it's a `symbol`) being a special case. Let's ignore the improper `list`s. Scheme's (proper) `list`s and `vector`s each has a claim to correspond to Kapulet's sequences, Haskell's Lists, and OCaml's `list`s. Each is also somewhat different. The dominant differences are:
+<a id=scheme-lists></a>
+How does all this look in Scheme? Well, Scheme has a notion they call a (proper) `list`, and also a notion they call a `vector`. There are also what Scheme calls "improper" `list`s, with `(cons 1 'nonlist)` or `'(1 . nonlist)`, where `'nonlist` is any non-list (here it's a `symbol`) being a limiting case. Let's ignore the improper `list`s. Scheme's (proper) `list`s and `vector`s each have a claim to correspond to Kapulet's sequences / Haskell's Lists / OCaml's `list`s. But they also differ from those. The main differences are:
1. these structures in Scheme can contain heterogenously-typed elements, including further `list`s and `vector`s in some positions but not in others
2. in the official Scheme standard, `list`s and `vector`s are both *mutable* containers, that is, one and the same persisting `list` structure can have different
elements at different stages in a program's evaluation
-Many Scheme implementations also provide immutable versions of `list`s and `vector`s, more closely approximating the sequences/lists in Kapulet, Haskell, and OCaml. With some configurations, Racket even makes the immutable versions the defaults. But none of these are yet part of the official Scheme standard. Also, difference 1 is present in all Scheme implementations. This makes Scheme's `list`s and `vector`s in some ways more akin to *tuples* in the other languages (to "proper" tuples in Kapulet).
+Many Scheme implementations also provide immutable versions of `list`s and `vector`s, more closely approximating the sequences/lists in Kapulet, Haskell, and OCaml. With some configurations, Racket even makes the immutable versions the defaults. But none of these are yet part of the official Scheme standard. Also, difference 1 is present in all Scheme implementations. This makes Scheme's `list`s and `vector`s in some ways more akin to *tuples* in the other languages (to "proper" tuples in Kapulet) (see [[below|#tuples]]).
+<a id=writing-scheme-lists></a>
There are also some differences in how `list`s are specified in Scheme versus the other languages. In Scheme, one writes the empty list like this:
(list)
null? ; corresponds to Kapulet's empty?, Haskell's null
car ; corresponds to Haskell's head
cdr ; corresponds to Kapulet's and Haskell's tail
- (list-tail xs k) ; corresponds to Kapulet's drop (k, xs); fails if out-of-bounds
+ (list-tail xs k) ; corresponds to Kapulet's drop (k, xs)
+ ; fails if the list has length < k
; no official function corresponding to take or split or filter or partition
map ; corresponds to Kapulet's map and map2
; can take one or more list arguments
; no official function corresponding to unmap2 or takewhile or dropwhile
reverse
- ; no official function corresponding to concat
- member ; corresponds to Kapulet's (mem) and Haskell's elem
+ ; no official function corresponding to join/concat
+ member ; corresponds to Kapulet's (mem) and Haskell's elem FIXME: eqv? version
(list-ref xs k) ; corresponds to Kapulet's `nth xs k`
+ ; fails if the index k is out of bounds
; no official function corresponding to all or any
All of the functions listed as missing from the official Scheme standard can be found in various add-on libraries, or you could define them yourself if you had to.
+<a id=tuples></a>
+FIXME tuples
-
-<!--
-
-symbol=?
-
-characters: #\c #\xff #\space #\newline
-
-
-
+LATER
### Other functions
Same in all: `succ`, `pred`, `fst`, `snd`.
Kapulet's `dup` isn't predefined in Haskell but can be easily expressed as `\x -> (x, x)`.
--->
+
+
+
+### Case, Cond, and If ... then ...
+
+The complex expression that's written like this in Kapulet:
+
+ # Kapulet
+ case some_expression of
+ 0 then result0;
+ 1 then result1;
+ x then resultx
+ end
+
+is written very similarly in Haskell:
+
+ -- Haskell
+ case some_expression {
+ 0 -> result0;
+ 1 -> result1;
+ x -> resultx
+ }
+
+<a id=haskell-whitespace></a>
+Unlike the other languages we're discussing, Haskell pays special attention to the whitespace/indentation of what you write. This permits you to omit the `{`, `;`, and `}`s in the above, if you've got the indentation right. And that's how you will often see Haskell code displayed. On this website, though, I propose to always include the `{`s and so on when displaying Haskell code, because the indentation rules aren't 100% intuitive. It's easy to read properly-indented Haskell code, but until you've learned and practiced the specific rules, it's not always easy to write it.
+
+This is written only a little bit differently in OCaml:
+
+ (* OCaml *)
+ match some_expression with
+ 0 -> result0 |
+ 1 -> result1 |
+ x -> resultx
+
+Note there is no closing `end` or `}`. You can enclose the whole expression in parentheses if you want to, and when embedding it in some larger expressions (like another `match` expression), you may need to. Sometimes the `|` dividers are written at the start of a line, and you are allowed to include an extra one before the first line, so you could also see this written as:
+
+ (* OCaml *)
+ match some_expression with
+ | 0 -> result0
+ | 1 -> result1
+ | x -> resultx
+
+The syntax for [[guards|topics/week1_advanced_notes#guards]] and [[as-patterns|topics/week1_advanced_notes#as-patterns]] also only varies slightly between these languages:
+
+ # Kapulet
+ case some_expression of
+ pat1 when guard then result1;
+ pat1 when different_guard then result2;
+ ((complex_pat) as var, pat4) then result3
+ end
+
+<a id=haskell-guards></a>
+
+ -- Haskell
+ case some_expression {
+ pat1 | guard -> result1;
+ | different_guard -> result2;
+ (var@(complex_pat), pat4) -> result3
+ }
+
+ (* OCaml *)
+ match some_expression with
+ pat1 when guard -> result1 |
+ pat1 when different_guard -> result2 |
+ ((complex_pat) as var, pat4 -> result3
+
+
+The official Scheme standard only provides for a limited version of this. There is a `case` construction, available since at least "version 5" of the Scheme standard (r5rs), but it only accepts literal values as patterns, not any complex patterns containing them or any patterns containing variables. Here is how it looks:
+
+ ; Scheme
+ (case some_expression
+ ((0) 'result0)
+ ((1) 'result1)
+ ((2 3 5) 'smallprime)
+ (else 'toobig))
+
+The results can be complex expressions; I just used bare symbols here for illustration. Note that the literal patterns in the first two clauses are surrounded by an extra pair of parentheses than you might expect. The reason is shown in the third clause, which begins `(2 3 5)`. This does not mean to match a list containing the values `2` `3` and `5`. Instead it means to match the simple value `2` *or* the simple value `3` *or* the simple value `5`. The final `else` clause is optional. FIXME no match?
+
+The patterns here can be any literal value (what the Scheme standards call a "datum"). Numbers are permitted, as are boolean literals (`#t` and `#f`) and symbolic atoms (`'alpha` and the like, though inside a pattern position in a `case`-construction, you omit the initial `'`). You can also use the list literal `'()` (again, omit the initial `'` when writing it as a pattern). Some implementations of Scheme allow more complex list patterns, matching literal lists like `'(alpha 0 () #t)`; others don't.
+
+There are various add-on libraries to Scheme that will permit you to pattern-match in more ambitious ways, approximating what you can do in Kapulet, OCaml, and Haskell. We will explain some of these later in the course, after we've introduced you to the notion of *datatypes*.
+
+What programmers using standard Scheme tend to do instead is to use *predicates* that query the type and/or structure of an unknown value, and then take separate evaluation paths depending on the result. This can be done with an `if ... then ... else ...` construction, or with Scheme's more general `cond` construction. In Scheme, these two are equivalent:
+
+ ; Scheme
+ (if test1 'result1 ; else what follows:
+ (if test2 'result2 ; else what follows:
+ (if test3 'result3 'somethingelse)))
+
+ (cond
+ (test1 'result1)
+ (test2 'result2)
+ (test3 'result3)
+ (else 'somethingelse))
+
+The tests tend to use predicates like `null?` (are you the empty list?), `pair?` (are you a non-empty list, whether proper or improper?), `list?` (are you a proper list, whether empty or not?), `symbol?`, `boolean?`, `number?`, `zero?` (you get the idea). The *Little Schemer* books use their own predicates they call `atom?` (are you a non-list?) and `lat?` (are you a list all of whose members are atoms?)
+
+You can also use more complex tests you write on the spot, or your own antecedently-defined functions:
+
+ ; Scheme...in case the parens left any doubt
+ (define smallprime? (lambda (x) (if (= x 2) #t (if (= x 3) #t (if (= x 5) #t #f)))))
+
+ (cond
+ ((= x 0) 'infant)
+ ((smallprime? x) 'myfavorite)
+ ((and (> x 10) (< x 20)) 'teenaged)
+ (else 'unknown))
+
+Remember that in Scheme, an expression doesn't have to evaluate to `#t` to be treated as "truth-like". *Every* value other than `#f` is treated as truth-like. As I [[said before|#truth-like]] `(if 0 'zero 'nope)` evaluates to `'zero`.
+
+You may sometimes see Scheme `cond` constructions written with this kind of clause:
+
+ (cond
+ ...
+ (test-expression => function-value)
+ ...)
+
+That's the same as the following:
+
+ (cond
+ ...
+ (test-expression (function-value test-expression))
+ ...)
+
+Except that it only evaluates the test-expression once.
+
+The clauses in Scheme's `cond` expressions can contain *multiple* expressions after the test. This only becomes useful when you're working with mutable values and side-effects, which we've not gotten to yet. The `if` expressions only take a single expression for the "then" branch and a single expression for the "else" branch. You can turn a complex series of expressions, which may involve side-effects, into a single expression by wrapping it in a `(begin ...)` construction. The `(begin ...)` construction as a whole evaluates to whatever the last expression it contains does.
+
+Scheme standards after r5rs also provide two further conditional constructions, which are for the situations where you want to perform a meaningful action only on the "then" branch, or only on the "else" branch:
+
+ (when test-expression
+ result-expression1...)
+
+ (unless test-expression
+ result-expression2...)
+
+If the test-expression evaluates to `#f`, then the `when` expression evaluates to a special "void" value; mutatis mutandis for the `unless` expression. This is analogous to `()` in OCaml, Haskell, and Kapulet. FIXME
+
+In the last three languages, the expressions in the then-branch and the else-branch of a conditional have to have the same type. You can't say `if test-expression then 0 else []`. Also, they expect the test-expression to evaluate specifically to a boolean value, not merely to `'false` versus "anything else". They are stricter about types here than Scheme is.
+
+In the special case where an else-branch evaluate to `()` (and thus so too must the then-branch), and the else-branch does so using no complex expression but merely the literal `()`, then OCaml permits you to omit that else-branch. So in OCaml you can write this:
+
+ if test_expression then then_result
+
+instead of
+
+ if test_expression then then_result else ()
+
+This is similar to Scheme's `when`-construction. Kapulet and Haskell have no analogue.
+
+
+
+
+### Lambda expressions
+
+In Kapulet you write λ-expressions (sometimes called "anonymous functions") with a prefix of either λ or the spelled-out `lambda`. That's followed by one or more patterns, separated by spaces, then a period, then a single expression which makes up the body of the function. When there are multiple patterns, the function expressed is *curried*, thus:
+
+ lambda (x, y) z. result
+
+means the same as:
+
+ lambda (x, y). (lambda z. result)
+
+The parentheses could have been omitted around `lambda z. result`; they're just there to focus your attention.
+
+Haskell and OCaml are very similar to this, they just use some slightly different notation. In Haskell you'd write:
+
+ -- Haskell
+ \(x, y) z -> result
+
+and in OCaml you'd write:
+
+ (* OCaml *)
+ fun (x, y) z -> result
+
+You may sometimes see λ-expressions in OCaml written using `function` instead of `fun`. These overlap somewhat in their usage. The difference is that `function` only allocates a position for *one* argument pattern, so can't straightforwardly define curried functions. (You can however embed `function` expressions inside other `function` expressions.) On the other hand, `function` can take multiple *variant* patterns for that single position. Thus with `function` you can say:
+
+ (* OCaml *)
+ function [] -> result1 |
+ x::xs -> result2
+
+whereas with `fun` you'd have to write:
+
+ (* OCaml *)
+ fun ys -> match ys with
+ [] -> result1 |
+ x::xs -> result2
+
+In Scheme, lambda expressions are written like this:
+
+ ; Scheme
+ (lambda (vars...) body-expressions...)
+
+Scheme only permits simple variables as its argument patterns, and the lambda-expression can be defined to take zero or more arguments:
+
+ ; Scheme
+ (lambda () ...)
+ (lambda (x) ...)
+ (lambda (x y) ...)
+ (lambda (x y z) ...)
+
+We will discuss functions that "take zero arguments" a few weeks into the semester.
+
+There is special syntax for defining functions that may take *varying* numbers of arguments (recall `and` and `+`), where Scheme binds a single variable to a list containing all of the received arguments (or all of the arguments after the *n*th...). I won't explain that syntax here.
+
+
+
+
+### Let, Letrec, and Define
+
+Kapulet has the syntax:
+
+ # Kapulet
+ let
+ pat1 match expr1;
+ pat2 match expr2;
+ pat3 match expr3
+ in result
+
+which is equivalent to:
+
+ # Kapulet
+ let
+ pat1 match expr1
+ in let
+ pat2 match expr2
+ in let
+ pat3 match expr3
+ in result
+
+There is also a corresponding `letrec` form. In `let`, the bindings in `pat1` are in effect for the evaluation of all of `expr2`, `expr3`, and `result` (but not any further, if this is part of a more complex expression); similarly for the bindings in `pat2` and `pat3`. In `letrec`, all of the bindings on the left-hand side are in effect for all of the right-hand side expressions, as well as for the result.
+
+OCaml only has the second, more verbose form of this, and writes it a bit differently:
+
+ (* OCaml *)
+ let
+ pat1 = expr1
+ in let
+ pat2 = expr2
+ in let
+ pat3 = expr3
+ in result
+
+If you want to define some mutually recursive functions with `letrec`, OCaml uses a special syntax for that, using `letrec ...` <code><em>and</em></code> `... in ...`:
+
+ (* OCaml *)
+ letrec
+ even = fun x -> if x = 0 then true else odd x
+ and
+ odd = fun x -> if x = 0 then false else even x
+ in ...
+
+Haskell has both of the syntactic forms that Kapulet does, though like OCaml, it uses `=` rather than `match`. And it wraps all the binding clauses with `{ ... }` (see [[earlier remarks|#haskell-whitespace]] about Haskell and whitespace/indentation):
+
+ -- Haskell
+ let {
+ pat1 = expr1;
+ pat2 = expr2;
+ pat3 = expr3
+ } in result
+
+Also, in Haskell `let` always means `letrec`. There is no term in Haskell that means what simple `let` does in Kapulet and OCaml.
+
+<a id=five-lets></a>
+Scheme has *four (or five)* syntactic forms here, including `let`, `let*`, `letrec`, and `letrec*`. The difference between the last two [is subtle](http://stackoverflow.com/questions/13078165) and only arises in the presence of continuations; you can just use `letrec` for ordinary purposes. I won't try to explain the difference between `let` and `let*` here, except to say this:
+
+1. When there's only a single pattern-binding clause, as in `(let ((var expression)) result)`, `let` and `let*` work the same.
+2. When there are multiple pattern-binding clauses, as in `(let ((var1 expression1) (var2 expression2)) result)`, then they work somewhat differently and `let*` is probably the one that works like you're expecting.
+
+The `let*` form is the one that corresponds to `let` in Kapulet. I recommend you get in the habit of just always using `let*` (or `letrec`) in Scheme, instead of `let`.
+
+When you're at the "toplevel" of your program, or of a library/module/compilation-unit (the terminology differs), there is also another syntactic form possible. In Kapulet, you'd write:
+
+ # Kapulet
+ let
+ pat1 match expr1;
+ ...
+ end
+ ... # rest of program or library
+
+Notice that this form ends with `end`, not with `in result`. The above is roughly equivalent to:
+
+ # Kapulet
+ let
+ pat1 match expr1;
+ ...
+ in ... # rest of program or library
+
+That is, the bindings initiated by the clauses of the `let`-construction remain in effect until the end of the program or library. They can of course be "hidden" by subsequent bindings to new variables spelled the same way. The program:
+
+ # Kapulet
+ let
+ x match 0
+ end
+ let
+ x match 1
+ end
+ x
+
+evaluates to `1`, just like:
+
+ # Kapulet
+ let
+ x match 0
+ in let
+ x match 1
+ in x
+
+does. There's a similar form for `letrec`.
+
+OCaml can do the same:
+
+ let
+ x = 0;;
+ let
+ x = 1;;
+ x
+
+The double-semicolons are hints to OCaml's "toplevel interpreter" that a syntactic unit has finished. In some contexts they're not needed, but it does no harm to include them if you're not sure.
+
+Haskell's "toplevel interpreter" (ghci) permits a syntactic form that looks superficially quite like these:
+
+ let x = 2
+ x
+
+but under the covers something quite different is happening. (Specifically, you're working "inside the IO Monad", except that in this special context, expressions like `x` that don't evaluate to monadic values are permitted and evaluated. We don't expect that you will understand yet what any of this means.) If you're writing *in a file* that you want Haskell to interpret or compile, on the other hand, you have to do something a bit different (which you can't easily also do at the toplevel in ghci). [[Recall|topics/week1_advanced_notes#funct-declarations]] the shortcut by which we permitted:
+
+ # Kapulet
+ let
+ f match lambda pat1. body1;
+ g match lambda pat2 pat3. body2
+ in ...
+
+to be written more concisely as:
+
+ # Kapulet
+ let
+ f pat1 = body1;
+ g pat2 pat3 = body2
+ in ...
+
+OCaml and Haskell permit that same shorthand. And what Haskell permits at the toplevel of *files* are just the bare binding clauses of such expressions, that is, without the surrounding `let` and `in`. That is, a Haskell file can look like this:
+
+ -- Haskell file.hs
+ f pat1 = body1
+
+ g pat2 pat3 = body2
+ ...
+
+Note there are no semicolons here. These are called "toplevel declarations" of the functions `f` and `g`. A single function can have multiple declarations (within a single scoping context), using different patterns:
+
+ -- Haskell file.hs
+ f [] = 0
+ f (x:xs) = 1 + f xs
+
+defines `f` as a function that returns the length of a single List argument. (You can also do that *inside* Haskell's `let`-constructions, too.) This is what corresponds *in Haskell files* to `let ... end` in Kapulet.
+
+<a id=define></a>
+Scheme has a version of `letrec ... end`, which it writes as `define`. Thus in Scheme this:
+
+ ; Scheme
+ (define var1 expr1)
+ ... ; rest of program
+
+evaluates the same as this:
+
+ ; Scheme
+ (letrec ((var1 expr1))
+ ... ; rest of program
+ )
+
+This is what we can call Scheme's [[fifth|#five-lets]] form of the `let` family.
+
+Some versions of Scheme permit you also to include `define` inside some (but not all) complex expressions. Thus you can write:
+
+ (lambda (x)
+ (define var1 expr1)
+ ...)
+
+instead of:
+
+ (lambda (x)
+ (letrec ((var1 expr1))
+ ...))
+
+There is no analogue to this in the other languages.
+
+
(This page is being worked on...)
+
+FIXME
+
+symbol=?
+
+characters: #\c #\xff #\space #\newline
+
+
+
+
+
+### Further Installments ...
+
+We will expand these comparisons (on separate web pages) as we introduce additional ideas in the course, such as types and monads and continuations.
+
+
+
+
+
+
FIXME