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Sat, 28 Aug 2021

Why is the S combinator an S?

The most important combinator in combinatory logic is the !!S!! combinator, defined simply:

$$ S x y z ⇒ (x z)(y z) $$

or in !!\lambda!!-calculus terms:

$$ S = \lambda x y z. (x z)(y z). $$

A wonderful theorem states that any !!\lambda!!-expression with no free variables can be converted into a combinator expression that contains only the combinators !!S, K,!! and !!I!!, where !!S!! is really the only interesting one of the three, !!I!! being merely the identity function, and !!K!! a constructor of constant functions:

$$ \begin{align} I x & = x \\ K x y & = x \\ \end{align} $$

In fact one can get along without !!I!! since !!S K K = I!!.

A not-too-infrequently-asked question is why the three combinators are named as they are. The !!I!! is an identity function and pretty obvious stands for “identity”.

Similarly the !!K!! constructs constant functions: !!K x!! is the combinator which ignores its argument and yields !!x!!. So it's not hard to imagine that !!K!! is short for Konstant, which is German for “constant”; no mystery there.

But why !!S!!? People typically guess that it stands for “substitution”, the idea being that if you have some application $$A\,B$$ then !!S!! allows one to substitute some term !!T!! for a free variable !!v!! in both !!A!! and !!B!! prior to the application:

$$ S\, A\, B\, T = A[v/T]\, B[v/T]. $$

Although this seems plausible, it's not correct.

Combinatory logic was introduced in a 1924 paper of Moses Schönfinkel. In it, he defines a family of combinators including the standard !!S!!, !!K!!, and !!I!!; he shows that only !!S!! and !!K!! are required. His initial set of combinators comprises the following:

$$ \begin{array}{cllrl} I & \textit{Identitätsfunktion} & \text{“identity function”}& I\,x =& x \\ C & \textit{Konstanzfunktion} & \text{“constancy function”} & C\,x\,y =& x \\ T & \textit{Vertauschungsfunktion} & \text{“swap function”} & T\,x\,y\,z=& x\,z\,y \\ Z & \textit{Zusammensetzungsfunktion} & \text{“composition function”} & Z\,x\,y\,z=& x\,(y\,z) \\ S & \textit{Verschmelzungsfunktion} & \text{“fusion function”} & S\,x\,y\,z=& x\,z\,(y\,z) \end{array} $$

(Schönfinkel also had combinators representing logical operations (one corresponding to the Sheffer stroke, which had been discovered in 1913), and to quantification, but those don't concern us right now.)

!!T!! and !!Z!! are now usually called !!C!! and !!B!!. These names probably originated in Curry's Grundlagen der kombinatorischen Logik (1930). Curry 1930 is probably also the origin of the change from !!C!! to !!K!!. I have no idea why Schönfinkel chose to abbreviate Konstanzfunktion as !!C!! instead of !!K!!. Curry notes that for !!I, K, B, C, S!! Schönfinkel has !!I, C, Z, T, S!!, but does not explain his changes. In Curry and Feys’ influential 1958 book on combinatory logic, the !!B!! and !!C!! combinators given names that are are literal translations of Schönfinkel's: “elementary permutator” and “elementary compositor”.

Returning to the !!S!! combinator, one sees that its German name in Schönfinkel's paper, Verschmelzungsfunktion, begins with the letter V, but so does Vertauschungsfunktion, so abbreviating either with V would have been ambiguous. Schönfinkel instead chose to abbreviate Verschmelzungsfunktion with S for its root schmelzen, “fusion”, and Vertauschungsfunktion with T for its root tauschen, “swap”. The word schmelzen is akin to English words “melt” and “smelt”.

The “swap” is straightforward: the !!T!! combinator swaps the order of the arguments to !!x!! in !!x\,y\,z!!: $$T\,x\,y\,z = x\,z\,y$$ but does not otherwise alter the structure of the expression.

But why is !!S!! the “melting” or “fusion” combinator? It's because Schönfinkel was interested in reducing abitrary mathematical expressions to combinators. He will sometimes have an application !!(f\, x)(g\, x)!! and he wants to ‘fuse’ the two occurrences of !!x!!. He can do this by rewriting the expression as !!S\, f\, g\, x!!. Schönfinkel says:

Der praktische Nutzen der Function !!S!! besteht ersichtlich darin, daß sie es ermöglicht, mehrmals auftresnde Veränderliche — und bis zu einem gewissen Grade auch individuelle Functionen — nur einmal auftreten zu lassen.

Clearly, the practical use of the function !!S!! will be to enable us to reduce the number of occurrences of a variable — and to some extent also of a particular function — from several to a single one.

(Translation from van Heijenoort, p. 362.)

So there you have it: the !!S!! combinator is so-named not for substitution, but because S is the first letter of schmelzen, ‘to fuse’.

References

  • Schönfinkel, M. “Über die Bausteine der mathematischen Logik” (“On the building-blocks of mathematical logic”), Mathematische Annalen (1969), p. 305–316; Berlin, Göttingen, Heidelberg.

    English translation in Van Heijenoort, Jean (ed.) From Frege to Gödel: a Source Book in Mathematical Logic, 1879–1931 (1967) pp. 355–366 Harvard University Press; Cambridge and London.

  • Curry, H.B. “Grundlagen der kombinatorischen Logik” (“Fundamentals of combinatory logic”), _American Journal of Mathematics Vol. 52, No. 3 (Jul., 1930), pp. 509-536.

  • Curry, H.B. and Robert Feys Combinatory Logic (1958) p. 152 North-Holland Publishing Company, Amsterdam.


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