X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week1.mdwn;h=24d606b2a4ced242f0a37f96e904c339e8bb19e0;hp=09ad8bd1680b95223a6944d113a23d414e5ca44b;hb=b6ecb9372e9895e6b8d49e054ed90ff69a87c247;hpb=16a6c1bb96a55803db4cd95e702030d68a650d4e diff --git a/week1.mdwn b/week1.mdwn index 09ad8bd1..24d606b2 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -1,795 +1,616 @@ -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. These notes expand on *a lot*, and some of this material will be reviewed next week. +### Getting started ### -Applications -============ +We begin with a decidable fragment of arithmetic. Our language has some **literal values**: -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.) + 0, 1, 2, 3, ... -From linguistics ----------------- +In fact we could get by with just the 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. -* generalized quantifiers are a special case of operating on continuations +We also have some predefined functions: -* (Chris: fill in other applications...) + succ, +, *, pred, - -* 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 +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`. -From philosophy ---------------- +Here's another set of functions: -* 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. + ==, <, >, <=, >=, != -* those issues may bear on Russell's Gray's Elegy argument in "On Denoting" +`==` 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. -* 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." +As mentioned in class, we represent the truth-values like this: -* 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 + 'true, 'false -* the semantics for mutation will enable us to make sense of a difference between numerical and qualitative identity---for purely mathematical objects! +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: -* 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 + 1 + 2 == 3 +evaluates to `'true`, and the expression: + 1 + 0 == 3 +evaluates to `'false`. Something else that evaluates to `'false` is the simple expression: + 'false -Declarative/functional vs Imperatival/dynamic models of computation -=================================================================== +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. -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. +The functions `succ` and `pred` come before their arguments, like this: -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. + succ 1 -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. +On the other hand, the functions `+`, `*`, `-`, `==`, and so on come in between their arguments, like this: -What the latter would amount to will become clearer as we build our way up to languages which are genuinely imperatival or dynamic. + x < y -Many of the slogans and keywords we'll encounter in discussions of these issues call for careful interpretation. They mean various different things. +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: -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. + lessthan? (x, y) -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. +or perhaps like this: -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." + lessthan? x y -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 TODO +We'll get more acquainted with the difference between these next week. For now, I'll just stick to the first form. -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.) +Another set of operations we have are: -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: + and, or, not - (\x. y) ((\x. x x) (\x. x x)) +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: -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. + x != y - x := 2 - x := x + 1 - x == 3 +will be just another way to write: -Here the comparison in the last line will evaluate to true. + not (x == y) - x := x + 1 - x := 2 - x == 3 +You see that you can use parentheses in the standard way. By the way, `<=` means ≤ or "less than or equals to", and `>=` means ≥. Just in case you haven't seen them written this way before. -Here the comparison in the last line will evaluate to false. +I've started throwing in some **variables**. We'll say variables are any expression that's written with an initial lower-case letter, then is followed by a sequence of zero or more upper- or lower-case letters, or numerals, or underscores (`_`). Then at the end you can optionally have a `?` or `!` or a sequence of `'`s, understood as "primes." Hence, all of these are legal variables: -One of our goals for this course is to get you to understand *what is* that new -sense such that only so matters in imperatival languages. + x + x1 + x_not_y + xUBERANT + x' + x'' + x? + xs -Finally, you'll see the term **dynamic** used in a variety of ways in the literature for this course: +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. -* dynamic versus static typing +We also conventionally reserve variables ending in `!` for a different special class of functions, that we will explain later in the course. -* dynamic versus lexical scoping +In fact you can think of `succ` and `pred` and `not` and the rest as also being variables; it's just that these variables have been pre-defined in our language to be bound to functions we agreed upon in advance. You can even think of `==` and `<` as being variables, too, bound to other functions. But I haven't given you parsing rules yet which would make them legal variables, because they don't start with a lower-case letter. We can make the parsing rules more liberal later. -* dynamic versus static control operators +Only a few simple expressions in our language aren't variables. These include the literal values, and also **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.) -* finally, we're used ourselves to talking about dynamic versus static semantics +The rule for symbolic atoms is that a single quote `'` followed by any single word that could be a legal variable expresses such an atom, a different atom for each different expression. +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 constitute a special subclass of symbolic atoms that we call the **booleans** or truth-values. Nothing deep hangs on them being a subclass of a larger type in this way; it just seems elegant. Some other languages make booleans their own special type, not a subclass of another type. Others make them a subclass of the numbers (yuck). We will think of them this way. -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. +Note that when writing a symbolic atom 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 expressing a symbolic atom. We use the initial `'` to make it easy for us to have a rich set of symbolic atoms, as well as a rich set of variables, without getting them mixed up. Variables never begin with `'`; symbolic atoms always do. -Map -=== +We call these things symbolic *atoms* because they aren't collections. Thus numbers are also atoms, but not symbolic ones. And functions are also atoms, but again, not symbolic ones. - - - - - - - - - - - -
 Scheme (functional part) OCaml (functional part) C, Java, Pasval -Scheme (imperative part) -OCaml (imperative part) 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)))`
-Variables: `x`, `y`, `z`... -
-Abstract: `(λa M)` -
+Here there is no recursion. Rather what happens is that we *initialize* the variable `n` with the value `0`, and then so long as our sequence variable `xs` is non-empty, we *increment* that variable `n`, and *overwrite* the variable `xs` with the tail of the sequence that it is then bound to, and repeat in a loop (the `while ... do ... end` construction). This is similar to what happens in our second definition of `length`, using `aux`, but here it happens using *mutation* or *overwriting* the values of variables, and a special looping construction, whereas in the preceding definitions we achieved the same effect instead with recursion. -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)`. +We will be looking closely at mutation later in the term. For the time being, our focus will instead be on the recursive and *immutable* style of doing things---meaning no variables get overwritten. -
-Application: `(M N)` -
+It's helpful to observe that in expressions like: -Some authors reserve the term "term" for just variables and abstracts. We won't participate in that convention; we'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms. + let + x match 0; + y match x + 1; + x match x + 1; + z match 2 * x + in (y, z) -Examples of expressions: +the variable `x` has not been *overwritten* (mutated). Rather, we have *two* variables `x` and its just that the second one is *hiding* the first so long as its scope is in effect. Once its scope expires, the original variable `x` is still in place, with its orginal binding. A different example should help clarify this. What do you think this: - x - (y x) - (x x) - (\x y) - (\x x) - (\x (\y x)) - (x (\x x)) - ((\x (x x)) (\x (x x))) + let + x match 0; + (y, z) match let + x match x + 1 + in (x, 2*x) + in ([y, z], x) -The lambda calculus has an associated proof theory. For now, we can regard the proof theory as having just one rule, called the rule of **beta-reduction** or "beta-contraction". Suppose you have some expression of the form: +evaluates to? Well, consider the right-hand side of the second binding: - ((\a M) N) + let + x match x + 1 + in (x, 2*x) -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**. +This expression evaluates to `(1, 2)`, because it uses the outer binding of `x` to `0` for the right-hand side of its own binding `x match x + 1`. That gives us a new binding of `x` to `1`, which is in place when we evaluate `(x, 2*x)`. That's why the whole thing evaluates to `(1, 2)`. So now returning to the outer expression, `y` gets bound to `1` and `z` to `2`. But now what is `x` bound to in the final line,`([y, z], x)`? The binding of `x` to `1` was in place only until we got to `(x, 2*x)`. After that its scope expired, and the original binding of `x` to `0` reappears. So the final line evaluates to `([1, 2], 0)`. -The rule of beta-reduction permits a transition from that expression to the following: +This is very like what happens in ordinary predicate logic if you say: - M [a:=N] +∃ `x. F x and (` ∀ `x. G x ) and H x` -What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`. - -What is a free occurrence? - -> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`. - -> If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`. - -> An occurrence of a variable is **free** if it's not bound. - -For instance: - - -> T is defined to be `(x (\x (\y (x (y z)))))` - -The first occurrence of `x` in T is free. The `\x` we won't regard as being an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free. - -Here's an example of beta-reduction: - - ((\x (y x)) z) - -beta-reduces to: - - (y z) - -We'll write that like this: - - ((\x (y x)) z) ~~> (y z) - -Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write: - - M ~~> N - -We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol `→` for one-step contraction, and the symbol `↠` for zero-or-more step reduction. Hindley and Seldin use `⊳1` and `⊳`. - -When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **beta-convertible**. We'll write that like this: - - M <~~> N - -This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.) - -In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol `≡` for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing 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: - -> 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. - - -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.) - - -**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: - - (\x (\y. (z y) z)) - -would abbreviate: - - (\x (\y ((z y) z))) - -This on the other hand: - - (\x (\y. z y) z) - -would abbreviate: - - (\x (\y (z y)) z) - -**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) - -**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 --------------------------------- - -All (recursively computable) functions can be represented by lambda -terms (the untyped lambda calculus is Turing complete). For some lambda terms, it is easy to see what function they represent: - -> `(\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 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. - - - - - -1. Declarative vs imperatival models of computation. -2. Variety of ways in which "order can matter." -3. Variety of meanings for "dynamic." -4. Schoenfinkel, Curry, Church: a brief history -5. Functions as "first-class values" -6. "Curried" functions - -1. Beta reduction -1. Encoding pairs (and triples and ...) -1. Encoding booleans +The `x` in `F x` and in `H x` are governed by the outermost quantifier, and only the `x` in `G x` is governed by the inner quantifier. +### That's enough ### +This was a lot of material, and you may need to read it carefully and think about it, but none of it should seem profoundly different from things you're already accustomed to doing. What we worked our way up to was just the kind of recursive definitions of `factorial` and `length` that you volunteered in class, before learning any programming. +You have all the materials you need now to do this week's [[assignment|assignment1]]. Some of you may find it easy. Many of you will not. But if you understand what we've done here, and give it your time and attention, we believe you can do it. +There are also some [[advanced notes|week1 advanced notes]] extending this week's material.