X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week1.mdwn;h=4981c1a325d668e297ed33f74de1dfdd19937727;hp=be1b714dd9c3b6e5801d786f827b7e89c3042d19;hb=4e38540a3d46c09c41f6c4591ab9b39b401ce050;hpb=545b8774052183ad91afd0b7dc741162af7ac405 diff --git a/week1.mdwn b/week1.mdwn index be1b714d..4981c1a3 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -1,326 +1,807 @@ -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.) +Here's what we did in seminar on Monday 9/13, -Applications -============ +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 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.) +[Linguistic and Philosophical Applications of the Tools We'll be Studying](/applications) +========================================================================== -From linguistics ----------------- +[Explanation of the "Damn" example shown in class](/damn) -* generalized quantifiers are a special case of operating on continuations +Basics of Lambda Calculus +========================= -* (Chris: fill in other applications...) +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.) -* expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the linked summary for more explanation and elaboration +Here is its syntax: -From philosophy ---------------- +
+Variables:-* 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. +Each variable is an expression. For any expressions M and N and variable a, the following are also expressions: -* those issues may bear on Russell's Gray's Elegy argument in "On Denoting" +x
,y
,z
... +
+Abstract: (λa M)
+
-* 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."
+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)`.
-* 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
+
+Application: (M N)
+
-* 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
+Examples of expressions:
+ x
+ (y 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:
+ ((\a M) N)
+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**.
-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
+The rule of beta-reduction permits a transition from that expression to the following:
-1. Beta reduction
-1. Encoding pairs (and triples and ...)
-1. Encoding booleans
+ M [a:=N]
+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.
- Order matters
+For instance:
-Declarative versus imperative:
-In a pure declarative language, the order in which expressions are
-evaluated (reduced, simplified) does not affect the outcome.
+> T is defined to be `(x (\x (\y (x (y z)))))`
-(3 + 4) * (5 + 11) = 7 * (5 + 11) = 7 * 16 = 112
-(3 + 4) * (5 + 11) = (3 + 4) * 16 = 7 * 16 = 112
+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.
-In an imperative language, order makes a difference.
+To read further:
-x := 2
-x := x + 1
-x == 3
-[true]
+* [[!wikipedia Free variables and bound variables]]
-x := x + 1
-x := 2
-x == 3
-[false]
+Here's an example of beta-reduction:
+ ((\x (y x)) z)
-Declaratives: assertions of statements.
-No matter what order you assert true facts, they remain true:
+beta-reduces to:
-The value is the product of x and y.
-x is the sum of 3 and 4.
-y is the sum of 5 and 11.
-The value is 112.
+ (y z)
-Imperatives: performative utterances expressing a deontic or bouletic
-modality ("Be polite", "shut the door")
-Resource-sensitive, order sensitive:
+We'll write that like this:
-Make x == 2.
-Add one to x.
-See if x == 3.
+ ((\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:
-Untype (monotyped) lambda calculus
+ M ~~> N
-Syntax:
+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 ⊳
.
-Variables: x, x', x'', x''', ...
-(Cheat: x, y, z, x1, x2, ...)
+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:
-Each variable is a term.
-For all terms M and N and variable a, the following are also terms:
+ M <~~> N
-(M N) The application of M to N
-(\a M) The abstraction of a over M
+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.)
-Examples of terms:
+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:
-x
-(y x)
-(x x)
-(\x y)
-(\x x)
-(\x (\y x))
-(x (\x x))
-((\x (x x))(\x (x x)))
+> T is defined to be `(M N)`.
-Reduction/conversion/equality:
+We'll regard the following two expressions:
-Lambda terms express recipes for combining terms into new terms.
-The key operation in the lambda calculus is beta-conversion.
+ (\x (x y))
-((\a M) N) ~~>_beta M{a := N}
+ (\z (z y))
-The term on the left of the arrow is an application whose first
-element is a lambda abstraction. (Such an application is called a
-"redex".) The beta reduction rule says that a redex is
-beta-equivalent to a term that is constructed by replacing every
-(free) occurrence of a in M by a copy of N. For example,
+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.
-((\x x) z) ~~>_beta z
-((\x (x x)) z) ~~>_beta (z z)
-((\x x) (\y y)) ~~>_beta (\y y)
+Note that neither of those expressions are identical to:
-Beta reduction is only allowed to replace *free* occurrences of a variable.
-An occurrence of a variable a is BOUND in T if T has the form (\a N).
-If T has the form (M N), and the occurrence of a is in M, then a is
-bound in T just in case a is bound in M; if the occurrence of a is in
-N, than a is bound in T just in case a is bound in N. An occurrence
-of a variable a is FREE in a term T iff it is not bound in T.
-For instance:
+ (\x (x w))
-T = (x (\x (\y (x (y z)))))
+because here it's a free variable that's been changed. Nor are they identical to:
-The first occurrence of x in T is free. The second occurrence of x
-immediately follows a lambda, and is bound. The third occurrence of x
-occurs within a form that begins with "\x", so it is bound as well.
-Both occurrences of y are bound, and the only occurrence of z is free.
+ (\y (y y))
-Lambda terms represent functions.
-All (recursively computable) functions can be represented by lambda
-terms (the untyped lambda calculus is Turning complete).
-For some lambda terms, it is easy to see what function they represent:
+because here the second occurrence of `y` is no longer free.
-(\x x) the identity function: given any argument M, this function
-simply returns M: ((\x x) M) ~~>_beta M.
+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.
-(\x (x x)) duplicates its argument:
-((\x (x x)) M) ~~> (M M)
+* [More discussion in week 2 notes](/week2/#index1h1)
-(\x (\y x)) throws away its second argument:
-(((\x (\y x)) M) N) ~~> M
-and so on.
+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:
-It is easy to see that distinct lambda terms can represent the same
-function. For instance, (\x x) and (\y y) both express the same
-function, namely, the identity function.
+ \x (x y)
------------------------------------------
-Dot notation: dot means "put a left paren here, and put the right
+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(xy))) = \x\y.xy, and \x\y.(z y) x =
-(\x(\y((z y) z))), but (\x\y.(z y)) x = ((\x(\y(z y))) x).
+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
-Church figured out how to encode integers and arithmetic operations
-using lambda terms. Here are the basics:
+ Numbers in Scheme: `2`, `3`
+ In OCaml: `2`, `3`
-0 = \f\x.fx
-1 = \f\x.f(fx)
-2 = \f\x.f(f(fx))
-3 = \f\x.f(f(f(fx)))
-...
+ Booleans in Scheme: `#t`, `#f`
+ In OCaml: `true`, `false`
-Adding two integers involves applying a special function + such that
-(+ 1) 2 = 3. Here is a term that works for +:
+ The eighth letter in the Latin alphabet, in Scheme: `#\h`
+ In OCaml: `'h'`
-+ = \m\n\f\x.m(f((n f) x))
+* Compound values
-So (+ 0) 0 =
-(((\m\n\f\x.m(f((n f) x))) ;+
- \f\x.fx) ;0
- \f\x.fx) ;0
+ These are values which are built up out of (zero or more) simple values.
-~~>_beta targeting m for beta conversion
+ Ordered pairs in Scheme: `'(2 . 3)` or `(cons 2 3)`
+ In OCaml: `(2, 3)`
-((\n\f\x.[\f\x.fx](f((n f) x)))
- \f\x.fx)
+ 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.
-\f\x.[\f\x.fx](f(([\f\x.fx] f) x))
+ The empty list, in Scheme: `'()` or `(list)`
+ In OCaml: `[]`
-\f\x.[\f\x.fx](f(fx))
+ The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`
+ In OCaml: `"h"`
-\f\x.\x.[f(fx)]x
+ A longer string, in Scheme: `"horse"`
+ In OCaml: `"horse"`
-\f\x.f(fx)
+ A shorter string, in Scheme: `""`
+ In OCaml: `""`
-----------------------------------------------------
+What "sequencing" is and isn't
+------------------------------
-A concrete example: "damn" side effects
+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.
-1. Sentences have truth conditions.
-2. If "John read the book" is true, then
- John read something,
- Someone read the book,
- John did something to the book,
- etc.
-3. If "John read the damn book",
- all the same entailments follow.
- To a first approximation, "damn" does not affect at-issue truth
- conditions.
-4. "Damn" does contribute information about the attitude of the speaker
- towards some aspect of the situation described by the sentence.
+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.
------------------------------------------
-Old notes, no longer operative:
+Third, the kinds of bindings we see in:
-1. Theoretical computer science is beautiful.
+ (define foo A)
+ (foo 2)
- Google search for "anagram": Did you mean "nag a ram"?
- Google search for "recursion": Did you mean "recursion"?
+Or even:
- Y = \f.(\x.f (x x)) (\x.f (x x))
+ (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.
-1. Understanding the meaning(use) of programming languages
- helps understanding the meaning(use) of natural langauges
+Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:
- 1. Richard Montague. 1970. Universal Grammar. _Theoria_ 34:375--98.
- "There is in my opinion no important theoretical difference
- between natural languages and the artificial languages of
- logicians; indeed, I consider it possible to comprehend the
- syntax and semantics of both kinds of languages within a
- single natural and mathematically precise theory."
+ (begin A B C)
- 2. Similarities:
+In OCaml it looks like this:
- Function/argument structure:
- f(x)
- kill(it)
- pronominal binding:
- x := x + 1
- John is his own worst enemy
- Quantification:
- foreach x in [1..10] print x
- Print every number from 1 to 10
+ begin A; B; C end
- 3. Possible differences:
+Or this:
- Parentheses:
- 3 * (4 + 7)
- ?It was four plus seven that John computed 3 multiplied by
- (compare: John computed 3 multiplied by four plus seven)
- Ambiguity:
- 3 * 4 + 7
- Time flies like and arrow, fruit flies like a banana.
- Vagueness:
- 3 * 4
- A cloud near the mountain
- Unbounded numbers of distinct pronouns:
- f(x1) + f(x2) + f(x3) + ...
- He saw her put it in ...
- [In ASL, dividing up the signing space...]
-
-
-2. Standard methods in linguistics are limited.
+ (A; B; C)
- 1. First-order predicate calculus
+In the presence of imperatival elements, sequencing order is very relevant. For example, these will behave differently:
- Invented for reasoning about mathematics (Frege's quantification)
+ (begin (print "under") (print "water"))
+
+ (begin (print "water") (print "under"))
- Alethic, order insensitive: phi & psi == psi & phi
- But: John left and Mary left too /= Mary left too and John left
+And so too these:
- 2. Simply-typed lambda calculus
+ begin x := 3; x := 2; x end
- Can't express the Y combinator
+ begin x := 2; x := 3; x end
+However, if A and B are purely functional, non-imperatival expressions, then:
-3. Meaning is computation.
+ begin A; B; C end
- 1. Semantics is programming
+just evaluates to C (so long as A and B evaluate to something at all). So:
- 2. Good programming is good semantics
+ begin A; B; C end
- 1. Example
+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.
- 1. Programming technique
+We'll discuss this more as the seminar proceeds.
- Exceptions
- throw (raise)
- catch (handle)
- 2. Application to linguistics
- presupposition
- expressives
- Develop application:
- fn application
- divide by zero
- test and repair
- raise and handle
- fn application
- presupposition failure
- build into meaning of innocent predicates?
- expressives
- throw
- handle
- resume computation
-