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@ -372,3 +372,9 @@ typography = do
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blockquote ? do
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@ -0,0 +1,130 @@
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---
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title: Operator Sections in Haskell: A History
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---
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Operator Sections
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-----------------
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I was explaining the Haskell notational trick of partially applying
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the *second* argument of a two-parameter function via a combination of
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back-quotes turning a named function into an operator and the operator
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section syntax. For example, you can express the function that gives
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a value modulo 10 as:
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~~~ { .haskell }
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(`mod` 10)
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~~~
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The question came up of just where the idea for operator sections came
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from. Because this is precisely the kind of useless information I
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can't help but be curious about, I resolved to find an answer. And
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after a bit of digging, I was successful.
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Haskell History
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---------------
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Our first stop on the journey through functional programming history
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is at the wonderful paper by several of the major contributors to
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Haskell, [A History of Haskell: Being Lazy With Class][HH]. Since I came
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across it, this has been my first source for the answers to questions
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of Haskell history, and once again it didn't fail me. From section 4.2:
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> The solution to this problem was to use a generalised notion of
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> *sections*, a notation that first appeared in David Wile's
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> dissertation (Wile, 1973) and was then disseminated via IFIP
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> WG2.1--among others to Bird, who adopted it in his work, and Turner,
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> who introduced it into Miranda.
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Turner, of course, refers to David Turner who is the man behind the
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languages SASL, KRC, and Miranda. Miranda was a commercial product of
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Turner's company Research Software Limited, and was the most mature
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and widely used of the family of non-strict functional languages at
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the time. The business needs of Research Software and the desire of
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the functional language community for a standard language didn't
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*quite* converge, though, so Haskell arose as a non-commercial
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alternative.
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So, Haskell got operator sections (as it got a great deal of its
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syntax) from Miranda. That's not very surprising and didn't really
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satisfy my curiosity, so I followed up on the next breadcrumb in the
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trail, Wile's dissertation.
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[HH]: http://research.microsoft.com/en-us/um/people/simonpj/papers/history-of-haskell/ "A History of Haskell: being lazy with class"
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One More Step
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-------------
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The document in question is entitled [A Generative, Nested-Sequential
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Basis for General Purpose Programming Languages][WD], and was submitted
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to Carnegie-Mellon University in November, 1973 by David Sheridan
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Wile. It describes the idea of taming the wild pointers of data
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structures via similar structuring techniques to those that were being
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applied to tame the wild control flow of GOTO-based code, and cites
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the languages BLISS, APL, and LISP as primary influences.
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When first asked about operator sections, I guessed that the influence
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had come somehow through APL, so I was gratified to see my instict
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validated. In fact, the notation presented borrows heavily from APL
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but marries it with ideas from non-strict functional programming such
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as natural representations of infinite data and the way that such data
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mediates the interaction of co-routines.
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Sure enough, on page 16 we find the following:
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> Partially instantiated functions are called "sections" in
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> mathematical literature, and we adopt the term here for
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> convenience. The nature of sections is ambiguous: they are both
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> program and data, and attempts to define them as one or the other
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> rely on a preconceived implementation.
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And on page 30, he explains further:
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> A unique primitive operation which produces primitive operators is
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> also permitted; this operation is termed "partial
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> instantiation". The "section" or "partially instantiated function"
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> was motivated in Chapter 1 as a natural mechanism for expressing
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> data structure concepts of restriction. In fact, they play a much
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> more significant role in the bsais in that many programs are
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> sequences of partially instantiated functions. In particular, we
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> allow the partial instantiation of any binary operator to produce
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> either a left- or right-unary operator.
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So, that's certainly the source of Haskell's notion of operator
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sections! But what about that reference to the mathematical
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literature? Can we trace the idea back further?
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[WD]: http://digitalcollections.library.cmu.edu/awweb/awarchive?type=file&item=362714 "A GENERATIVE, NESTED-SEQUENTIAL BASIS FOR GENERAL PURPOSE PROGRAMMING LANGUAGES"
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Recursive Functions
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-------------------
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This turns out to be [Theory of Recursive Functions and Effective
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Computability][RF], by Hartley Rogers, Jr. Rogers was a Ph.D. student
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under none other than Alonzo Church, father of the Lambda
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Calculus. This text began as a set of notes published by the MIT Math
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department in '57 and grew into its first publication as a book in '67
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during a period of huge amounts of progress in its field.
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The citation of the book points to a page in section 5.4 on projection
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theorems relating to recursively enumerable sets. Specifically, the
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definition of "section" is given in terms of a $k$-ary projection
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relation $R$ in which one of the $k$ terms, $n$, is fixed. Since binary
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operators form such a relation, fixing one of the parameters qualifies
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to be called a section of the overall relation described by the
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operator.
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[RF]: http://mitpress.mit.edu/books/theory-recursive-functions-and-effective-computability "Theory of Recursive Functions and Effective Computability | The MIT Press"
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Relation to Categorical Sections?
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---------------------------------
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I recalled having heard some other mathematical definition of the term
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section, and on a hunch I looked it up at [nLab][NL], which is a site
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discussing Category Theory. It turns out that a [section][NS] in that
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context is a right-inverse to a mapping; i.e. if you compose it on the
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right of a map f: A -> B, you get A -> B -> A, or the identity
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map. This doesn't seem to be a related concept to what Rogers described,
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but perhaps I'm missing some deeper connection.
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[NL]: http://ncatlab.org/nlab/show/HomePage "nLab: Home Page"
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[NS]: http://ncatlab.org/nlab/show/section "nLab: section"
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