Peter Verswyvelen
Lambda calculus is a nice theory in which every function always has one input and one output. Functions with multiple arguments can be simulated because functions are first class and hence a function can "return" a function. Multiple outputs cannot be done, one must embed these outputs in some data type, e.g. using a tuple, or one must use continuation passing style.
Both input- and output currying are completely orthogonal, for a sufficiently bent definition of orthogonal: forall f :: (a, b) -> c exists g :: a -> b -> c "likewise": forall f :: a -> (b, c) exists (g :: a -> c, h :: a -> c) This is, of course, abstract nonsense[1]: Usually, splitting a function with multiple retvals into two is just a crime against efficiency, if not clarity. I doubt any compiler could be decisive enough to get Sufficiently Smart in this context.
Now, does a similar theory exist of functions that always have one input and one output, but these inputs and outputs are *always* tuples? Or maybe this does not make any sense?
I fear it doesn't, at least if the functions are allowed to be passed and return tuples containing bottom or unit, as you'd be left with syntactically awkward versions of single-argument functions, or unary tuples, whatever. Generally speaking, functions are more fundamental than product types[2]: cons :: a -> b -> ((a -> b -> c) -> c) cons a b m = m a b car :: ((a -> b -> a) -> c) -> c) car z = z (\p q -> p) cdr :: ((a -> b -> b) -> c) -> c) cdr z = z (\p q -> q) ...as a comparison, try to define eval and apply in terms of tuples. OTOH, you might be thinking of things like applicative functors, which are product types[2] and can't (necessarily) be escaped from, and not collapsed, either, as would be the case with monads. These are things that are described in terms of a theory, not a theory themselves, though. The important notion is that inside a specific functor, one part of its type always stays the same (or it wouldn't be that functor anymore... duh.) So, to conclude: You aren't likely to find any fundamental theory restricting inputs and outputs to tuples, but you are going to find fundamental theories that enable you to say very interesting things about functions which have their inputs and outputs restricted to arbitrary types, be it sums or products. [1] Specifically, distilled from http://golem.ph.utexas.edu/category/2008/03/computer_scientists_needed_now.h... , which is a _very_ good and enlightening read [2] Stolen straight from Exercise 2.4 in the Wizard Book [3] Of a type tag and a value. Stop nitpicking. Tagged lists spoiled me. -- (c) this sig last receiving data processing entity. Inspect headers for copyright history. All rights reserved. Copying, hiring, renting, performance and/or quoting of this signature prohibited.