From: jim Date: Sat, 14 Feb 2015 23:22:30 +0000 (-0500) Subject: cleanup X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=ccfda3680551144d8d34e0bb262a391d9e5c9d6e cleanup --- diff --git a/topics/week3_combinatory_logic.mdwn b/topics/week3_combinatory_logic.mdwn index 4c92bfc6..df568dc3 100644 --- a/topics/week3_combinatory_logic.mdwn +++ b/topics/week3_combinatory_logic.mdwn @@ -5,10 +5,10 @@ Combinatory logic is of interest here in part because it provides a useful computational system that is equivalent to the Lambda Calculus, but different from it. In addition, Combinatory Logic has a number of applications in natural language semantics. Exploring Combinatory -Logic will involve defining a difference notion of reduction from the +Logic will involve defining a different notion of reduction from the one we have been using for the Lambda Calculus. This will provide us -with a second parallel example later when we're thinking through -such topics as evaluation strategies and recursion. +with a second parallel example when we're thinking through +topics such as evaluation strategies and recursion. Lambda expressions that have no free variables are known as **combinators**. Here are some common ones: @@ -53,7 +53,8 @@ S, K, I, B also known 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 +*primitive operations*. (That is, we refrain from defining them in terms of lambda expressions, as we did above.) +Such a language doesn't have any variables in it: not just no free variables, but no variables (or "bound positions") at all. One can do that with a very spare set of basic combinators. These days @@ -76,7 +77,7 @@ duplicators. 

 everyone   hit           himself
 S/(S!NP)   (S!NP)/NP     (S!NP)!((S!NP)/NP)
-\fAx[fx]   \y\z[HIT y z] \h\u[huu]
+\f∀x[fx]   \y\z[HIT y z] \h\u[huu]
            ---------------------------------
                   S!NP     \u[HIT u u]
 --------------------------------------------
@@ -101,8 +102,7 @@ W
###A different set of reduction rules### -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, +Instead of defining combinators in terms of antecedently understood lambda terms, we want to consider the view that takes the combinators as primitive, and understands them in terms of *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 @@ -178,7 +178,7 @@ used to establish a correpsondence between two natural language grammars, one of which is based on lambda-like abstraction, the other of which is based on Combinatory Logic like manipulations. -Assume that for any lambda term T, [T] is the equivalent Combinatory Logic term. The we can define the [.] mapping as follows: +Assume that for any lambda term T, [T] is the equivalent Combinatory Logic term. Then we can define the [.] mapping as follows: 1. [a] a 2. [(M N)] ([M][N]) @@ -187,27 +187,40 @@ Assume that for any lambda term T, [T] is the equivalent Combinatory Logic term. 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. +If the recursive unpacking of these rules ever direct you to "translate" an `S` or a `K` or an `I`, introduced at an earlier stage of translation, those symbols translate themselves. + +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. If the original lambda expression had no free variables in it, then any such translations will only be temporary. The variable will later get eliminated by the application of other rules. (If the original lambda term *does* have free variables in it, so too will the final Combinatory Logic translation. Feel free to worry about this, though you should be confident that it makes sense.) + +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. This ensures that any free occurrences of a inside M or N will end up taking on the appropriate value. 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.) -(*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 our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.) +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, `K y` 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. This ensures that any free occurrences of a inside M or N will end up taking on the appropriate value. + +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.) + +Persuade 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). -(Various, slightly differing translation schemes from Combinatory Logic to the +(Fussy note: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of our boolean false `\x. \y. y` is `[\x [\y. y]] = [\x. I] = KI`. In the intermediate stage, we have `\x. I`, which mixes primitive combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.) + +Various, slightly differing translation schemes from Combinatory Logic to the Lambda Calculus are also possible. These generate different metatheoretical correspondences between the two calculii. Consult Hindley and Seldin for -details. Also, note that the combinatorial proof theory needs to be +details. + +Also, note that the combinatorial proof theory needs to be strengthened with axioms beyond anything we've here described in order to make [M] convertible with [N] whenever the original lambda-terms M and N are convertible. But then, we've been a bit cavalier about giving the full set of -reduction rules for the Lambda Calculus in a similar way. For instance, one -issue is whether reduction rules (in either the Lambda Calculus or Combinatory -Logic) apply to embedded expressions. Generally, we want that to happen, but -making it happen requires adding explicit axioms.) +reduction rules for the Lambda Calculus in a similar way. + +For instance, one +issue we mentioned in the notes on [[Reduction Strategies|week3_reduction_strategies]] is whether reduction rules (in either the Lambda Calculus or Combinatory Logic) apply to embedded expressions. Often, we do want that to happen, but +making it happen requires adding explicit axioms. Let's check that the translation of the `false` boolean behaves as expected by feeding it two arbitrary arguments: @@ -215,9 +228,9 @@ Let's check that the translation of the `false` boolean behaves as expected by f Throws away the first argument, returns the second argument---yep, it works. -Here's a more elaborate example of the translation. The goal is to establish that combinators can reverse order, so we use the **T** combinator, where T ≡ \x\y.yx: +Here's a more elaborate example of the translation. The goal is to establish that combinators can reverse order, so we use the **T** combinator, where T ≡ \x y. y x: - [\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) + [\x y. y x] = [\x [\y. y x]] = [\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): @@ -249,12 +262,11 @@ enterprise Free Variable Free Semantics. A philosophical connection: Quine went through a phase in which he developed a variable free logic. - Quine, Willard. 1960. "Variables explained away" Proceedings of the American Philosophical Society. Volume 104: 343--347. Also in W. V. Quine. 1960. Selected Logical Papers. Random House: New - York. 227--235. +> Quine, Willard. 1960. "Variables explained away" Proceedings of the American Philosophical Society. Volume 104: 343--347. Also in W. V. Quine. 1960. Selected Logical Papers. Random House: New York. 227--235. The reason this was important to Quine is similar to the worry that using non-referring expressions such as Santa Claus might commit one to believing in -non-existant things. Quine's slogan was that "to be is to be the value of a +non-existent things. Quine's slogan was that "to be is to be the value of a variable." What this was supposed to mean is that if and only if an object could serve as the value of some variable, we are committed to recognizing the existence of that object in our ontology. Obviously, if there ARE no @@ -269,40 +281,36 @@ in two books in the 1990's. A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is from Combinatory Logic (see especially his 2012 book, Taking Scope). Steedman attempts to build -a syntax/semantics interface using a small number of combinators, including `T` ≡ `\xy.yx`, `B` ≡ `\fxy.f(xy)`, +a syntax/semantics interface using a small number of combinators, including `T` ≡ `\x y. y x`, `B` ≡ `\f x y. f (x y)`, and our friend `S`. Steedman used Smullyan's fanciful bird names for the combinators, Thrush, Bluebird, and Starling. Many of these combinatory logics, in particular, the SKI system, 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! -The combinators `K` and `S` correspond to two well-known axioms of sentential logic: ###A connection between Combinatory Logic and Sentential Logic### -One way of getting a feel for the power of the SK basis is to note -that the following two axioms +The combinators `K` and `S` correspond to two well-known axioms of sentential logic: - AK: A --> (B --> A) - AS: (A --> (B --> C)) --> ((A --> B) --> (A --> C)) + AK: A ⊃ (B ⊃ A) + AS: (A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C)) -when combined with modus ponens (from `A` and `A --> B`, conclude `B`) -are complete for the implicational fragment of intuitionistic logic. -(To get a complete proof theory for *classical* sentential logic, you -need only add one more axiom, constraining the behavior of a new connective "not".) +When these two axiom schemas are combined with the rule of modus ponens (from `A` and `A ⊃ B`, conclude `B`), the resulting proof system +is complete for the implicational fragment of intuitionistic logic. +(To get a complete proof system for *classical* sentential logic, you +need only add one more axiom schema, constraining the behavior of a new connective "not".) The way we'll favor for viewing the relationship between these axioms -and the `S` and `K` combinators is that the axioms correspond to type -schemas for the combinators. This will become more clear once we have +and the `S` and `K` combinators is that the axioms correspond to *type +schemas* for the combinators. This will become more clear once we have a theory of types in view. -Here's more to read about Combinatory Logic. -Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]]. -Other sources include - -* [[!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]] -* [Chris Barker's Iota and Jot](http://semarch.linguistics.fas.nyu.edu/barker/Iota/) -* Jeroen Fokker, "The Systematic Construction of a One-combinator Basis for Lambda-Terms" Formal Aspects of Computing 4 (1992), pp. 776-780. - +Here's more to read about Combinatory Logic. Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]]. +Other sources include: + +* [[!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]] +* [Chris Barker's Iota and Jot](http://semarch.linguistics.fas.nyu.edu/barker/Iota/) +* Jeroen Fokker, "The Systematic Construction of a One-combinator Basis for Lambda-Terms" Formal Aspects of Computing 4 (1992), pp. 776-780.