X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week1.mdwn;h=5ea3f421e69234c88e0fd974a486ad61e580fae0;hp=a5e697a5026751b9fb6b261f657cd2f91f69c99a;hb=573a8b36ce653c84c2aecb2b81ef99128cb41d13;hpb=cf87edfd9d4cb5026287b39b190797ef72237f19 diff --git a/week1.mdwn b/week1.mdwn index a5e697a5..5ea3f421 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -1,43 +1,16 @@ Here's what we did in seminar on Monday 9/13, -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. - -Applications -============ - -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.) - -From linguistics ----------------- - -* generalized quantifiers are a special case of operating on continuations - -* (Chris: fill in other applications...) - -* 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 - -From philosophy ---------------- - -* 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" - -* 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." - -* 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 - -* the semantics for mutation will enable us to make sense of a difference between numerical and qualitative identity---for purely mathematical objects! - -* 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 - +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. +[Linguistic and Philosophical Applications of the Tools We'll be Studying](/applications) +========================================================================== +[Explanation of the "Damn" example shown in class](/damn) Basics of Lambda Calculus ========================= -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.) +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.) Here is its syntax: @@ -57,20 +30,21 @@ We'll tend to write `(λa M)` as just `(\a M)`, so we don't hav Application: `(M N)` -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. Examples of expressions: x (y x) (x x) - (_x y) - (_x x) - (_x (_y x)) - (x (_x x)) - ((_x (x x)) (_x (x x))) + (\x y) + (\x x) + (\x (\y x)) + (x (\x x)) + ((\x (x x)) (\x (x 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: +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: ((\a M) N) @@ -95,7 +69,11 @@ 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. +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. + +To read further: + +* [[!wikipedia Free variables and bound variables]] Here's an example of beta-reduction: @@ -121,7 +99,7 @@ When M and N are such that there's some P that M reduces to by zero or more step 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: +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)`. @@ -145,6 +123,8 @@ 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 --------- @@ -152,47 +132,53 @@ 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: +**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 (\y (x y))) + (\x (x y)) -can be abbreviated as: +as: + + \x (x y) + +but you should include the parentheses in: - (\x (\y. x y)) + (\x (x y)) z and: - (\x (\y. (z y) z)) + z (\x (x y)) -would abbreviate: - (\x (\y ((z y) z))) +**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: -This on the other hand: + \x (\y (x y)) - (\x (\y. z y) z) +can be abbreviated as: -would abbreviate: + \x (\y. x y) - (\x (\y (z y)) z) +and that as: -**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. \y. x y - (\x. x y) +This: -as: + \x. \y. (x y) x - \x. x y +abbreviates: -but you should include the parentheses in: + \x (\y ((x y) x)) - (\x. x y) z +This on the other hand: -and: + (\x. \y. (x y)) x + +abbreviates: + + ((\x (\y (x y))) x) - z (\x. x y) **Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as: @@ -206,8 +192,7 @@ Similarly, `(\x (\y (\z M)))` can be abbreviated as: 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: +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`. @@ -241,7 +226,7 @@ yet when applied to any argument M, all of these will always return M. So they h 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). +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). @@ -258,7 +243,7 @@ computer](/how_to_get_the_programming_languages_running_on_your_computer), and proposed answers to the assignment. - +There's also a (slow, bare-bones, but perfectly adequate) version of Scheme available for online use at . @@ -293,7 +278,7 @@ It's often said that dynamic systems are distinguished because they are the ones false and * = false false and false = false -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 `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 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: @@ -320,7 +305,7 @@ Finally, you'll see the term **dynamic** used in a variety of ways in the litera * dynamic versus static typing -* dynamic versus lexical scoping +* dynamic versus lexical [[!wikipedia Scope (programming) desc="scoping"]] * dynamic versus static control operators @@ -328,6 +313,16 @@ Finally, you'll see the term **dynamic** used in a variety of ways in the litera 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. +To read further about the relation between declarative or functional programming, on the one hand, and imperatival programming on the other, you can begin here: + +* [[!wikipedia Declarative programming]] +* [[!wikipedia Functional programming]] +* [[!wikipedia Purely functional]] +* [[!wikipedia Referential transparency (computer science)]] +* [[!wikipedia Imperative programming]] +* [[!wikipedia Side effect (computer science) desc="Side effects"]] + + Map === @@ -335,11 +330,11 @@ Map Scheme (functional part) OCaml (functional part) -C, Java, Pasval
+C, Java, Python
Scheme (imperative part)
OCaml (imperative part) -lambda calculus
+untyped lambda calculus
```-	`∀x. (F x or ∀x (not (F x)))`
``````∀x. (F x or ∀x (not (F x)))