The exponential explosions of `do-it-yourself' monad tutorials prompts a thought if monads, like the related continuations [Reynolds-1993], are destined to be rediscovered time and time again. One naturally wonders about the first `monad tutorial'.
An old (1994) paper on category theory monads and functional
programming included a relevant historical side-note. It turns out
that the essence of monads has been fully grasped back in 1965, by at
least one person. That person has also discovered that imperative,
control-flow--dependent computations can be embedded into a calculus by
turning control flow into data flow. That person is Peter Landin. His
1965 paper [Landin-1965] anticipated not only IO but also State
and Writer monads, call/cc, delayed evaluation and its connection with
streams.
Monads are typically equated with single-threadedness, and are therefore used as a technique for incorporating imperative features into a purely functional language. Category theory monads have little to do with single-threadedness; it is the sequencing imposed by composition that ensures single-threadedness. In a Wadler-ised monad this is a consequence of bundling the Kleisli star and flipped compose into the bind operator. There is nothing new in this connection. Peter Landin in his Algol 60 used functional composition to model semi-colon. Semi-colon can be thought of as a state transforming operator that threads the state of the machine throughout a program. The work of Peyton-Jones and Wadler has turned full circle back to Landin's earlier work as their use of Moggi's sequencing monad enables real side-effects to be incorporated into monad operations such as print. This is similar to Landin's implementation of his sharing machine where theassignandholdfunction can side-effect the store of the sharing machine because of the sequencing imposed by functional composition. Landin defined that `Imperatives are treated as null-list producing functions' [In Landin's paper,()is the syntactic representation of the empty list]. Theassignandholdimperative is subtly different in that it enables Algol's compound statements to be handled. The function takes a store location and a value as its argument, and performs the assignment to the store of the sharing machine, returning the value assigned as a result of the function. Because Landin assumed applicative order reduction, theKcombinator was used to return(), and the imperative was evaluated as a side effect by the unused argument of theK-combinator. Statements are formed by wrapping such an imperative in a lambda expression that takes()as an argument. Two consecutive Algol-60 assignments would be encoded in the lambda calculus as:Algol 60 Lambda Calculus x:= 2; ( (\() -> K () (assignandhold x 2)) . x:= -3; (\() -> K () (assignandhold x (-3))) ) ()By using a lambda with
()as its parameter,()can be thought of as the `state of the world' that is threaded throughout a program by functional composition.
()-passing trick is described in full on p.100 of Landin's paper,
with remarkable clarity:
Statements. Each statement is rendered as a 0-list- transformer, i.e. a none-adic function producing the nullist for its result. It achieves by side-effects a transformation of the current state of evaluation. ... Compound statements are considered as functional products (which we indicate informally by infixed dots).
Contrast that with a more recent explanation [Peyton-Jones-2000]:
A value of type IO a is an "action" that, when performed, may do some input/output, before delivering a value of typea. This is an admirably abstract statement, and I would not be surprised if it means almost nothing to you at the moment. So here is another, more concrete way of looking at these "actions":type IO a = World -> (a, World)This type definition says that a value of type IO a is a function that, when applied to an argument of typeWorld, delivers a newWorldtogether with a result of typea. The idea is rather program-centric: the program takes the state of the entire world as its input, and delivers a modified world as a result, modified by the effects of running the program. [p5]
Simon Peyton-Jones then goes on to explain [p14] that the functional
world-passing realization of the IO monad is indeed the implementation
of the monad in GHC. The IO monad operation (>>) is a functional
composition then -- just like it is in Landin's representation.
There are some differences: World is abstract in GHC but is represented as
the empty list () in Landin's paper. Whereas GHC threads the World
through, Landin's encoding discards and re-generates the ()
token. GHC uses a non-strict evaluation strategy whereas Landin relied on
call-by-value: although the K combinator discards the value of its
second argument, the side-effects of its evaluation are performed
nonetheless.
The differences are, however, superficial. Landin could have hid the
concrete `world' representation and the K-combinator trick by defining
assignandhold' x v = \() -> K () (assignandhold x v) and using it as
a primitive. The sequencing of compound statements is ensured by
functional composition, as data flow, and hence does not depend on the
evaluation strategy.
Strictly speaking, we ought to treat World as an abstract data type,
to prevent the compiler from optimizing the composition of two World -> World functions and reordering computations. We also need an
additional constraint that the World is to be used linearly, so the
compiler will not be tempted to replicate World-producing
computations. Landin's calculus, induced by his sharing machine,
ensures these constraints. In GHC, World is treated specially and
the duplication of redices is prevented by ad hoc rules in the optimizer.
Incidentally, call-by-name and call-by-value were introduced
ten years later, in [Plotkin-1975],
in the context of Landin's ISWIM.
Both evaluation strategies have the notion of value and both take
lambda-abstraction to be a value. The body of a function is evaluated
only when the function is applied to an argument. Landin understood
it back in 1965:
``the use of lambda, and in particular (to avoid an
irrelevant bound variable) of lambda (), to delay and possibly avoid
evaluation is exploited repeatedly in our model of Algol 60. A
function that requires an argument-list of length zero is called a
non-adic function'' [p90]. We now call that a thunk.
where notation. Peter Landin even anticipated
the debate on let vs where, saying ``The only consideration in
choosing between let and where will be the relative convenience of
writing an auxiliary definition before or after the expression it
qualifies.''
Another remark
anticipated the State and Writer monads as well.However, input/output devices can be modeled as named lists, with special, rather restricted functions associated. ... Writing is modeled by a procedure that, operates on a list, and appends a new final segment derived from other variables. (Alternatively, a purely functional approach can be contrived by including the transformed list among the results.)
In the section on Streams, Landin wrote:
This correspondence [laws of head/tail/cons] serves two related purposes. It enables us to perform operations on lists (such as generating them, mapping them, concatenating them) without using an `extensive,' item-by-item representation of the intermediately resulting lists; and it enables us to postpone the evaluation of the expressions specifying the items of a list until they are actually needed. The second of these is what interests us here.... [and footnote 6] It appears that in stream-transformers we have a functional analogue of what Conway calls `co-routines.'