```
-T ≡ (\x. x y) z ≡ (\z. z y) z
+
``````
T ≡ (\x. x y) z ≡ (\z. z y) z
```

+[Fussy note: the justification for counting `(\x. x y) z` as
+equivalent to `(\z. z y) z` is that when a lambda binds a set of
+occurrences, it doesn't matter which variable serves to carry out the
+binding. Either way, the function does the same thing and means the
+same thing. Look in the standard treatments for discussions of alpha
+equivalence for more detail.]
+
This:
T ~~> z y
@@ -24,23 +30,118 @@ Lambda expressions that have no free variables are known as **combinators**. Her
> **I** is defined to be `\x x`
-> **K** is defined to be `\x y. x`, That is, it throws away its second argument. So `K x` is a constant function from any (further) argument to `x`. ("K" for "constant".) Compare K to our definition of **true**.
+> **K** is defined to be `\x y. x`. That is, it throws away its
+ second argument. So `K x` is a constant function from any
+ (further) argument to `x`. ("K" for "constant".) Compare K
+ to our definition of `true`.
+
+> **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to K and `true` as well.
-> **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to K and true as well.
+> **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. Compare this to our definition of `false`.
-> **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. Compare this to our definition of false.
+> **B** is defined to be: `\f g x. f (g x)`. (So `B f g` is the composition `\x. f (g x)` of `f` and `g`.)
-> **ω** is defined to be: `\x. x x`
+> **C** is defined to be: `\f x y. f y x`. (So `C f` is a function like `f` except it expects its first two arguments in swapped order.)
+
+> **W** is defined to be: `\f x . f x x`. (So `W f` accepts one argument and gives it to `f` twice. What is the meaning of `W multiply`?)
+
+> **ω** (that is, lower-case omega) is defined to be: `\x. x x`
It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all.
-One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: K and I from above, and also one more, S, which behaves the same as the lambda expression `\f g x. f x (g x)`. behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (And there are different single-combinator bases you can choose.)
+One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: K and I from above, and also one more, **S**, which behaves the same as the lambda expression `\f g x. f x (g x)`. behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (And there are different single-combinator bases you can choose.)
+
+There are some well-known linguistic applications of Combinatory
+Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson.
+Szabolcsi supposed that the meanings of certain expressions could be
+insightfully expressed in the form of combinators.
+
+
+For instance, Szabolcsi argues that reflexive pronouns are argument
+duplicators.
+
+![reflexive](http://lambda.jimpryor.net/szabolcsi-reflexive.jpg)
+
+Notice that the semantic value of *himself* is exactly `W`.
+The reflexive pronoun in direct object position combines first with the transitive verb (through compositional magic we won't go into here). The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.
+
+Note that `W <~~> S(CI)`:
+
+```
S(CI) ≡
+S((\fxy.fyx)(\x.x)) ~~>
+S(\xy.(\x.x)yx) ~~>
+S(\xy.yx) ≡
+(\fgx.fx(gx))(\xy.yx) ~~>
+\gx.(\xy.yx)x(gx) ~~>
+\gx.(gx)x ≡
+W
```

+
+Ok, here comes a shift in thinking. Instead of defining combinators as equivalent to certain lambda terms,
+we can define combinators by what they do. If we have the I combinator followed by any expression X,
+I will take that expression as its argument and return that same expression as the result. In pictures,
+
+ IX ~~> X
+
+Thinking of this as a reduction rule, we can perform the following computation
+
+ II(IX) ~~> IIX ~~> IX ~~> X
+
+The reduction rule for K is also straightforward:
+
+ KXY ~~> X
+
+That is, K throws away its second argument. The reduction rule for S can be constructed by examining
+the defining lambda term:
+
+ S = \fgx.fx(gx)
+
+S takes three arguments, duplicates the third argument, and feeds one copy to the first argument and the second copy to the second argument. So:
+
+ SFGX ~~> FX(GX)
+
+If the meaning of a function is nothing more than how it behaves with respect to its arguments,
+these reduction rules capture the behavior of the combinators S, K, and I completely.
+We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of Ss and Ks:
+
+ SKKX ~~> KX(KX) ~~> X
+
+So the combinator `SKK` is equivalent to the combinator I.
+
+Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. The most common system uses S, K, and I as defined here.
+
+###The equivalence of the untyped lambda calculus and combinatory logic###
+
+We've claimed that Combinatory Logic is equivalent to the lambda calculus. If that's so, then S, K, and I must be enough to accomplish any computational task imaginable. Actually, S and K must suffice, since we've just seen that we can simulate I using only S and K. In order to get an intuition about what it takes to be Turing complete, imagine what a text editor does:
+it transforms any arbitrary text into any other arbitrary text. The way it does this is by deleting, copying, and reordering characters. We've already seen that K deletes its second argument, so we have deletion covered. S duplicates and reorders, so we have some reason to hope that S and K are enough to define arbitrary functions.
+
+We've already established that the behavior of combinatory terms can be perfectly mimicked by lambda terms: just replace each combinator with its equivalent lambda term, i.e., replace I with `\x.x`, replace K with `\fxy.x`, and replace S with `\fgx.fx(gx)`. How about the other direction? Here is a method for converting an arbitrary lambda term into an equivalent Combinatory Logic term using only S, K, and I. Besides the intrinsic beauty of this mapping, and the importance of what it says about the nature of binding and computation, it is possible to hear an echo of computing with continuations in this conversion strategy (though you would be able to hear these echos until we've covered a considerable portion of the rest of the course).
+
+Assume that for any lambda term T, [T] is the equivalent combinatory logic term. The we can define the [.] mapping as follows:
+
+ lambda term equivalent SKI term condition
+ ----------- ------------------- ---------
+ 1. [\x.x] I
+ 2. [\x.M] K[M] x does not occur free in M
+ 3. [\x.(M N)] S[\x.M][\x.N]
+ 4. [\x\y.M] [\x[\y.M]]
+ 5. [M N] [M][N]
+
+It's easy to understand these rules based on what S, K and I do. The first rule is obvious.
+The second rule says that if a lambda abstract contains no occurrences of the variable targeted by lambda,
+what the function expressed by that lambda term does it throw away its argument and returns whatever M
+computes: it's the constant function K[M].
+The third and fourth rules say what happens when there are occurrences of the bound variable in the body.
+
+Finally, the fifth rule says what to do for an application (divide and conquer).
+
These systems are Turing complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only a single primitive operation!
-Here's more to read about combinatorial logic:
+Here's more to read about combinatorial logic.
+Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]].
+Other sources include
-* [[!wikipedia Combinatory logic]]
+* [[!wikipedia Combinatory logic]] at Wikipedia
* [Combinatory logic](http://plato.stanford.edu/entries/logic-combinatory/) at the Stanford Encyclopedia of Philosophy
* [[!wikipedia SKI combinatory calculus]]
* [[!wikipedia B,C,K,W system]]
@@ -49,7 +150,7 @@ Here's more to read about combinatorial logic:
-Evaluation strategies and Normalization
+Evaluation Strategies and Normalization
=======================================
In the assignment we asked you to reduce various expressions until it wasn't possible to reduce them any further. For two of those expressions, this was impossible to do. One of them was this:
@@ -62,7 +163,7 @@ As we saw above, each of the halves of this formula are the combinator ```
&om
This compound expression---the self-application of
````ω`

---is named Ω. It has the form of an application of an abstract (`ω`

) to an argument (which also happens to be `ω`

), so it's a redex and can be reduced. But when we reduce it, we get `ω ω`

again. So there's no stage at which this expression has been reduced to a point where it can't be reduced any further. In other words, evaluation of this expression "never terminates." (This is the standard language, however it has the unfortunate connotation that evaluation is a process or operation that is performed in time. You shouldn't think of it like that. Evaluation of this expression "never terminates" in the way that the decimal expansion of π never terminates. These are static, atemporal facts about their mathematical properties.)
-There are infinitely many formulas in the lambda calculus that have this same property. Ω is the syntactically simplest of them. In our meta-theory, it's common to assign such formula a special value, `⊥`

, pronounced "bottom." When we get to discussing types, you'll see that this value is counted as belonging to every type. To say that a formula has the bottom value means that the computation that formula represents never terminates and so doesn't evaluate to any orthodox, computed value.
+There are infinitely many formulas in the lambda calculus that have this same property. Ω is the syntactically simplest of them. In our meta-theory, it's common to assign such formulas a special value, `⊥`

, pronounced "bottom." When we get to discussing types, you'll see that this value is counted as belonging to every type. To say that a formula has the bottom value means that the computation that formula represents never terminates and so doesn't evaluate to any orthodox, computed value.
From a "Fregean" or "weak Kleene" perspective, if any component of an expression fails to be evaluable (to an orthodox, computed value), then the whole expression should be unevaluable as well.
@@ -136,7 +237,7 @@ With normal-order evaluation (or call-by-name more generally), if we have an exp
(\x. y) (C D)
-the computation of `(C D)` won't ever have to be performed. Instead, it reduces directly to `y`. This is so even if `(C D)` is the non-evaluable `(ω ω)`

!
+the computation of `(C D)` won't ever have to be performed. Instead, `(\x. y) (C D)` reduces directly to `y`. This is so even if `(C D)` is the non-evaluable `(ω ω)`

!
Call-by-name evaluation is often called "lazy." Call-by-value evaluation is also often called "eager" or "strict". Some authors say these terms all have subtly different technical meanings, but I haven't been able to figure out what it is. Perhaps the technical meaning of "strict" is what I above called the "Fregean" or "weak Kleene" perspective: if any argument of a function is non-evaluable or non-normalizing, so too is the application of the function to that argument.
@@ -162,7 +263,8 @@ One important advantage of normal-order evaluation in particular is that it can
Indeed, it's provable that if there's *any* reduction path that delivers a value for a given expression, the normal-order evalutation strategy will terminate with that value.
-An expression is said to be in **normal form** when it's not possible to perform any more reductions. (EVEN INSIDE ABSTRACTS?) There's a sense in which you can't get anything more out of `ω ω`

, but it's not in normal form because it still has the form of a redex.
+An expression is said to be in **normal form** when it's not possible to perform any more reductions (not even inside abstracts).
+There's a sense in which you *can't get anything more out of* `ω ω`

, but it's not in normal form because it still has the form of a redex.
A computational system is said to be **confluent**, or to have the **Church-Rosser** or **diamond** property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.
@@ -178,10 +280,34 @@ The *typed* lambda calculus that linguists traditionally work with, on the other
Other more-powerful type systems we'll look at in the course will also fail to be Turing complete, though they will turn out to be pretty powerful.
+Further reading:
-##[[Lists and Numbers]]##
+* [[!wikipedia Evaluation strategy]]
+* [[!wikipedia Eager evaluation]]
+* [[!wikipedia Lazy evaluation]]
+* [[!wikipedia Strict programming language]]
+* [[!wikipedia Church-Rosser theorem]]
+* [[!wikipedia Normalization property]]
+* [[!wikipedia Turing completeness]]
-How to do with recursion with omega.
-Next week: fixed point combinators
+Decidability
+============
+
+The question whether two formulas are syntactically equal is "decidable": we can construct a computation that's guaranteed to always give us the answer.
+
+What about the question whether two formulas are convertible? Well, to answer that, we just need to reduce them to normal form, if possible, and check whether the results are syntactically equal. The crux is that "if possible." Some computations can't be reduced to normal form. Their evaluation paths never terminate. And if we just kept trying blindly to reduce them, our computation of what they're convertible to would also never terminate.
+
+So it'd be handy to have some way to check in advance whether a formula has a normal form: whether there's any evaluation path for it that terminates.
+
+Is it possible to do that? Sure, sometimes. For instance, check whether the formula is syntactically equal to Ω. If it is, it never terminates.
+
+But is there any method for doing this in general---for telling, of any given computation, whether that computation would terminate? Unfortunately, there is not. Church proved this in 1936; Turing also essentially proved it at the same time. Geoff Pullum gives a very reader-friendly outline of the proofs here:
+
+* [Scooping the Loop Snooper](http://www.cl.cam.ac.uk/teaching/0910/CompTheory/scooping.pdf), a proof of the undecidability of the halting problem in the style of Dr Seuss by Geoffrey K. Pullum
+
+
+
+##[[Lists and Numbers]]##
+