# Operator Sections in Haskell: A History

*Posted on October 17, 2014*

## Operator Sections

I was explaining the Haskell notational trick of partially applying the *second* argument of a two-parameter function via a combination of back-quotes turning a named function into an operator and the operator section syntax. For example, you can express the function that gives a value modulo 10 as:

` (`mod` 10)`

The question came up of just where the idea for operator sections came from. Because this is precisely the kind of useless information I can’t help but be curious about, I resolved to find an answer. And after a bit of digging, I was successful.

## Haskell History

Our first stop on the journey through functional programming history is at the wonderful paper by several of the major contributors to Haskell, A History of Haskell: Being Lazy With Class. Since I came across it, this has been my first source for the answers to questions of Haskell history, and once again it didn’t fail me. From section 4.2:

The solution to this problem was to use a generalised notion of

sections, a notation that first appeared in David Wile’s dissertation (Wile, 1973) and was then disseminated via IFIP WG2.1–among others to Bird, who adopted it in his work, and Turner, who introduced it into Miranda.

Turner, of course, refers to David Turner who is the man behind the languages SASL, KRC, and Miranda. Miranda was a commercial product of Turner’s company Research Software Limited, and was the most mature and widely used of the family of non-strict functional languages at the time. The business needs of Research Software and the desire of the functional language community for a standard language didn’t *quite* converge, though, so Haskell arose as a non-commercial alternative.

So, Haskell got operator sections (as it got a great deal of its syntax) from Miranda. That’s not very surprising and didn’t really satisfy my curiosity, so I followed up on the next breadcrumb in the trail, Wile’s dissertation.

## One More Step

The document in question is entitled A Generative, Nested-Sequential Basis for General Purpose Programming Languages, and was submitted to Carnegie-Mellon University in November, 1973 by David Sheridan Wile. It describes the idea of taming the wild pointers of data structures via similar structuring techniques to those that were being applied to tame the wild control flow of GOTO-based code, and cites the languages BLISS, APL, and LISP as primary influences.

When first asked about operator sections, I guessed that the influence had come somehow through APL, so I was gratified to see my instict validated. In fact, the notation presented borrows heavily from APL but marries it with ideas from non-strict functional programming such as natural representations of infinite data and the way that such data mediates the interaction of co-routines.

Sure enough, on page 16 we find the following:

Partially instantiated functions are called “sections” in mathematical literature, and we adopt the term here for convenience. The nature of sections is ambiguous: they are both program and data, and attempts to define them as one or the other rely on a preconceived implementation.

And on page 30, he explains further:

A unique primitive operation which produces primitive operators is also permitted; this operation is termed “partial instantiation”. The “section” or “partially instantiated function” was motivated in Chapter 1 as a natural mechanism for expressing data structure concepts of restriction. In fact, they play a much more significant role in the bsais in that many programs are sequences of partially instantiated functions. In particular, we allow the partial instantiation of any binary operator to produce either a left- or right-unary operator.

So, that’s certainly the source of Haskell’s notion of operator sections! But what about that reference to the mathematical literature? Can we trace the idea back further?

## Recursive Functions

This turns out to be Theory of Recursive Functions and Effective Computability, by Hartley Rogers, Jr. Rogers was a Ph.D. student under none other than Alonzo Church, father of the Lambda Calculus. This text began as a set of notes published by the MIT Math department in ’57 and grew into its first publication as a book in ’67 during a period of huge amounts of progress in its field.

The citation of the book points to a page in section 5.4 on projection theorems relating to recursively enumerable sets. Specifically, the definition of “section” is given in terms of a *k*-ary projection relation *R* in which one of the *k* terms, *n*, is fixed. Since binary operators form such a relation, fixing one of the parameters qualifies to be called a section of the overall relation described by the operator.

## Relation to Categorical Sections?

I recalled having heard some other mathematical definition of the term section, and on a hunch I looked it up at nLab, which is a site discussing Category Theory. It turns out that a section in that context is a right-inverse to a mapping; i.e. if you compose it on the right of a map f: A -> B, you get A -> B -> A, or the identity map. This doesn’t seem to be a related concept to what Rogers described, but perhaps I’m missing some deeper connection.