X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week2.mdwn;h=f7bb440907c470964678e60658a453a8aa4d79cc;hp=ce21d268c35c0a1464ccbe90b9e8f23e7f7736ab;hb=9ef074920c539db8a664e273dc17394b59b68716;hpb=ba19207955ab6675d8396fa51af98b19a708a707 diff --git a/week2.mdwn b/week2.mdwn index ce21d268..f7bb4409 100644 --- a/week2.mdwn +++ b/week2.mdwn @@ -1,3 +1,6 @@ +[[!toc]] + + Syntactic equality, reduction, convertibility ============================================= @@ -33,11 +36,11 @@ Lambda expressions that have no free variables are known as **combinators**. Her > **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**. + 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`.) @@ -62,7 +65,7 @@ duplicators. ![reflexive](http://lambda.jimpryor.net/szabolcsi-reflexive.jpg) -Notice that the semantic value of *himself* is exactly W. +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)`: @@ -100,14 +103,14 @@ S takes three arguments, duplicates the third argument, and feeds one copy to th 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 S's and K's: +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. +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. +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### @@ -118,21 +121,58 @@ We've already established that the behavior of combinatory terms can be perfectl 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). + 1. [a] a + 2. [(M N)] ([M][N]) + 3. [\a.a] I + 4. [\a.M] KM assumption: a does not occur free in M + 5. [\a.(M N)] S[\a.M][\a.N] + 6. [\a\b.M] [\a[\b.M]] + +It's easy to understand these rules based on what S, K and I do. The first rule says +that variables are mapped to themselves. +The second rule says that the way to translate an application is to translate the +first element and the second element separately. +The third rule should be obvious. +The fourth rule should also be fairly self-evident: since what a lambda term such as `\x.y` does it throw away its first argument and return `y`, that's exactly what the combinatory logic translation should do. And indeed, `Ky` is a function that throws away its argument and returns `y`. +The fifth rule deals with an abstract whose body is an application: the S combinator takes its next argument (which will fill the role of the original variable a) and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. Finally, the last rule says that if the body of an abstract is itself an abstract, translate the inner abstract first, and then do the outermost. (Since the translation of [\b.M] will not have any lambdas in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.) + +Here's an example of the translation: + + [\x\y.yx] = [\x[\y.yx]] = [\x.S[\y.y][\y.x]] = [\x.(SI)(Kx)] = S[\x.SI][\x.Kx] = S(K(SI))(S[\x.K][\x.x]) = S(K(SI))(S(KK)I) + +We can test this translation by seeing if it behaves like the original lambda term does. +The orginal lambda term lifts its first argument (think of it as reversing the order of its two arguments): + + S(K(SI))(S(KK)I) X Y = + (K(SI))X ((S(KK)I) X) Y = + SI ((KK)X (IX)) Y = + SI (KX) Y = + IY (KX)Y = + Y X + +Viola: the combinator takes any X and Y as arguments, and returns Y applied to X. + +Back to linguistic applications: one consequence of the equivalence between the lambda calculus and combinatory +logic is that anything that can be done by binding variables can just as well be done with combinators. +This has given rise to a style of semantic analysis called Variable Free Semantics (in addition to +Szabolcsi's papers, see, for instance, +Pauline Jacobson's 1999 *Linguistics and Philosophy* paper, `Towards a variable-free Semantics'). +Somewhat ironically, reading strings of combinators is so difficult that most practitioners of variable-free semantics +express there meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their +enterprise Free Variable Free Semantics. + +A philosophical application: Quine went through a phase in which he developed a variable free logic. + + + + + + + + + + +[Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of S, K, and I (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In that intermediate stage, we have `\x.I`. It's possible to avoid this, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.] 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!