X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;f=week1.mdwn;h=7a8b00aae49cd20c6f185f104ab7b24181efaa65;hb=400b008fadb498ed7e308fd30cb30d5a287936cb;hp=52080d4ac861a946477aea5a1744c57088b6b520;hpb=ff8f3749e2d882b7b7e5da789ef64cfd15109e8f;p=lambda.git diff --git a/week1.mdwn b/week1.mdwn index 52080d4a..7a8b00aa 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -1,809 +1,102 @@ -Here's what we did in seminar on Monday 9/13, +These notes will recapitulate, make more precise, and to some degree expand what we did in the last hour of our first meeting, leading up to the definitions of the `factorial` and `length` functions. -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 begin with a decidable fragment of arithmetic. Our language has some primitive literal values: -[Linguistic and Philosophical Applications of the Tools We'll be Studying](/applications) -========================================================================== + 0, 1, 2, 3, ... -[Explanation of the "Damn" example shown in class](/damn) +In fact we could get by with just the primitive literal `0` and the `succ` function, but we will make things a bit more convenient by allowing literal expressions of any natural number. We won't worry about numbers being too big for our finite computers to handle. -Basics of Lambda Calculus -========================= +We also have some predefined functions: -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... In fact, though, such types are studied, under the name "recursive type." More about these later in the seminar.) + succ, +, *, pred, - -Here is its syntax: +Again, we might be able to get by with just `succ`, and define the others in terms of it, but we'll be a bit more relaxed. Since we want to stick with natural numbers, not the whole range of integers, we'll make `pred 0` just be `0`, and `2-4` also be `0`. -
-Variables:+Here's another set of functions: -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)
-
+`==` is just what we non-programmers normally express by `=`. It's a relation that holds or not between two values. Here we'll treat it as a function that takes two values as arguments and returns a *boolean* value, that is a truth-value, as a result. The reason for using the doubled `=` symbol is that the single `=` symbol tends to get used in lots of different roles in programming, so we reserve `==` to express this meaning. I will deliberately try to minimize the uses of single `=` in this made-up language (but not eliminate it entirely), to reduce ambiguity and confusion. The `==` relation---or as we're treating it here, the `==` *function* that returns a boolean value---can at least take two numbers as arguments. Probably it makes sense for it to take other kinds of values as arguments, too. For example, it should operate on two truth-values as well. Maybe we'd want it to operate on a number and a truth-value, too? and always return false in that case? What about operating on two functions? Here we encounter the difficulty that the computer can't in general *decide* when two functions are equivalent. Let's not try to sort this all out just yet. We'll suppose that `==` can at least take two numbers as arguments, or two truth-values.
-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)`.
+As mentioned in class, we represent the truth-values like this:
-
-Application: (M N)
-
+ 'true, 'false
+These are instances of a broader class of literal values that I called *symbolic atoms*. We'll return to them shortly. The reason we write them with an initial `'` will also be explained shortly. For now, it's enough to note that the expression:
-Examples of expressions:
+ 1 + 2 == 3
- x
- (y x)
- (x x)
- (\x y)
- (\x x)
- (\x (\y x))
- (x (\x x))
- ((\x (x x)) (\x (x x)))
+evaluates to `'true`, and the expression:
-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:
+ 1 + 0 == 3
- ((\a M) N)
+evaluates to `'false`. Something else that evaluates to `'false` is the simple expression:
-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**.
+ 'false
-The rule of beta-reduction permits a transition from that expression to the following:
+That is, literal values are a limiting case of expression, that evaluate to just themselves. More complex expressions like `1 + 0` don't evaluate to themselves, but rather down to literal values.
- M [a:=N]
+The functions `succ` and `pred` come before their arguments, like this:
-What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
+ succ 1
-What is a free occurrence?
+On the other hand, the functions `+`, `*`, `-`, `==`, and so on come in between their arguments, like this:
-> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
+ x < y
-> 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`.
+Functions of this latter sort are said to have an "infix" syntax. This is just a convenience for how we write them. Our language will have to keep rigorous track of which functions have infix syntax and which don't, but we'll just rely on context and our brains to make sense of this for now. Functions with the ordinary, non-infix syntax can take two arguments, as well. If we had defined the less-than relation (boolean function) in that style, we'd write it like this instead:
-> An occurrence of a variable is **free** if it's not bound.
+ lessthan? (x, y)
-For instance:
+or perhaps like this:
+ lessthan? x y
-> T is defined to be `(x (\x (\y (x (y z)))))`
+We'll get more acquainted with the difference between these next week. For now, I'll just stick to the first form.
-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.
+Another set of operations we have are:
-To read further:
+ and, or, not
-* [[!wikipedia Free variables and bound variables]]
+The first two of these are infix functions that expect two boolean arguments, and gives a boolean result. The third is a function that expects only one boolean argument. Our earlier function `!=` means "doesn't equal", and:
-Here's an example of beta-reduction:
+ x != y
- ((\x (y x)) z)
+will in general be just another way to write:
-beta-reduces to:
+ not (x == y)
- (y z)
+You see that you can use parentheses in the standard way.
-We'll write that like this:
+I've started throwing in some variables. We'll say variables are any expression that starts with a lower-case letter, then is followed by a sequence of 0 or more upper- or lower-case letters, or underscores (`_`). Then at the end you can optionally have a `?` or `!` or a sequence of `'`s, understood as "prime" symbols. Hence, all of these are legal variables:
- ((\x (y x)) z) ~~> (y z)
+ x
+ x1
+ x_not_y
+ xUBERANT
+ x'
+ x''
+ x?
+ xs
-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:
+We'll follow a *convention* of using variables with short names and a final `s` to represent collections like sequences (to be discussed below). But this is just a convention to help us remember what we're up to, not a strict rule of the language. We'll also follow a convention of only using variables ending in `?` to represent functions that return a boolean value. Thus, for example, `zero?` will be a function that expects a single number argument and returns a boolean corresponding to whether that number is `0`. `odd?` will be a function that expects a single number argument and returns a boolean corresponding to whether than number is odd. Above, I suggested we might use `lessthan?` to represent a function that expects *two* number arguments, and again returns a boolean result.
- M ~~> N
+We also conventionally reserve variables ending in `!` for a different special class of functions, that we will explain later in the course.
-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 fact you can think of `succ` and `pred` and `not` and all the rest as also being variables; it's just that these variables have been pre-defined in our language to be bound to special functions we designated in advance. You can even think of `==` and `<` as being variables, too, bound to other functions. But I haven't given you rules yet which would make them legal variables, because they don't start with a lower-case letter. We can make the rules more liberal later.
-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:
+Only a few things in our language aren't variables. These include the **keywords** like `let` and `case` and so on that we'll discuss below. You can't use `let` as a variable, else the syntax of our language would become too hard to mechanically parse. (And probably too hard for our meager brains to parse, too.)
- M <~~> N
+The rule for symbolic atoms is that a single quote `'` followed by any single word that could be a legal variable is a symbolic atom. Thus `'false` is a symbolic atom, but so too are `'x` and `'succ`. For the time being, I'll restrict myself to only talking about the symbolic atoms `'true` and `'false`. These are a special subgroup of symbolic atoms that we call the *booleans* or *truth-values*. Nothing deep hangs on these being a subclass of a larger category in this way; it just seems elegant. Other languages sometimes make booleans their own special type, not a subclass of any other limited type. Others make them a subclass of the numbers (yuck). We will think of them this way.
-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.)
+Note that in symbolic atoms there is no closing `'`, just a `'` at the beginning. That's enough to make the whole word, up to the next space (or whatever) count as naming a symbolic atom.
-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))
-
- (\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
-
-abbreviates:
-
- \x (\y ((x y) x))
-
-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)
-
-
-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`
-
-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]].
-
-
-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) - | - |
∀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:
-
- 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.
+We call these things symbolic *atoms* because they aren't collections. Thus numbers are also atoms, just not symbolic ones. And functions are also atoms, but again, not symbolic ones.
+Functions are another class of values we'll have in our language. They aren't "literal" values, though. Numbers and symbolic atoms are simple expressions in the language that evaluate to themselves. That's what we mean by calling them "literals." Functions aren't expressions in the language at all; they have to be generated from the evaluation of more complex expressions.
+(By the way, I really am serious about thinking of *the numbers themselves* as being expressions in this language; rather than some "numerals" that aren't themselves numbers. We can talk about this down the road. For now, don't worry about it too much.)
+I said we wanted to be starting with a fragment of arithmetic, so we'll keep the function values off-stage for the moment, and also all the symbolic atoms except for `'true` and `'false`. So we've got numbers, truth-values, and some functions and relations (that is, boolean functions) defined on them. We also help ourselves to a notion of bounded quantification, as in ∀`x < M.` φ, where `M` and φ are (simple or complex) expressions that evaluate to a number and a boolean, respectively.
+*More to come*