Note: This entry has been edited a couple of times (and might be edited in the future) to remove errors. Thanks to Benja Fallenstein for critique.
My
previous post about recursion drew some comments. This is a topic that frequently confuses me, and there is a lot of confusion (some of it mine) in the blog post and its comments. On reflection, it seems prudent to examine what recursion really is.
Consider the following recursive definition:
0 is an expression; if T is an expression, then
sT is an expression; nothing is an expression that doesn’t have one of these two forms. Now, it is clear that
0 is an expression. It is equally clear that
sssssK is not an expression.
In some applications, it makes sense to consider infinite expressions like “…
ssssss0“; that is, an infinite number of
ss before the final
0, and in others it doesn’t. Is that infinite utterance and expression? It does have one of the forms given in the definitions, so it is not forbidden from being an expression, but neither is there anything in the definition requiring that it is.
It is helpful to do this a bit more formally. Let us translate the self-referential definition into a set equation: The set of expressions is the set E that satisfies the equation
E =
0 ∪
se e ∈ E
But this definition is bogus! There are more than one set E that satisfies the equation, as remarked above, and thus there is no “the set E”!
We need to fix the definition in a way that makes the set E unique. The key question is, do we want to regard the infinite utterance discussed above as an expression. It is simplest to just say “the set of expressions is the
smallest set E that …” if we want it to not be an expression, and if we want it to be an expression we should say “the set of expressions is the
largest set E that …”
Let us make the following definitions:
Consider the set equation S = … S …, in which the set S is alone on the left-hand side and occurs at least once on the right-hand side. This equation is a recursive definition of the smallest set S satisfying this equation, and a corecursive definition of the largest set S satisfying this equation.
Thus, a definition like
E =
0 ∪
se e ∈ E
is not recursive because of its form; it is self-referential. By itself, it is an ambiguous and thus bogus definition. We use the words “recursive” and “corecursive” to indicate the intended method of disambiguation.
A rather nice example of this can be seen in the treatment of algebraic data types in strict (ML-style) and nonstrict (Haskell-style) functional programming languages. Consider the declaration (using Haskell syntax)
data IntList = Nil Cons Int IntList
This can be read as a set equation as follows:
IntList = Nil ∪ Cons n x n ∈ Int ∧ x ∈ IntList
Now, as a recursive definition, it is restricted to finite lists as found in ML, but as a corecursive definition, it also allows infinite lists as found in Haskell. (It unfortunately fails to address the question of bottom-terminated lists, which also can be found in Haskell.)
As it happens, the technique is not limited to set definitions; but to generalise it, we need to discuss a bit what we mean by the terms “smallest” and “largest”.
In a set context, a set is smaller than another set if it is the other’s subset. Notice how this makes the ordering partial, that is, there are sets that are not comparable (for example 1 and 2 ; neither is smaller than the other, but neither are they equal). A collection of sets does not necessarily have a “smallest” set (that is, one of the sets that is a subset of all the others). However, we can always find the lowest upper bound (”lub” or “join”) of a collection of sets as the union of the sets, and the greatest lower bound (”glb” or “meet”) as the intersection of the sets, and if the lub or glb is itself a member of the collection, it is the smallest or largest set of the collection.
Haskellists will find things instantly familiar when I say that the smallest element of a mathematical universe (if it exists) is called bottom and is denoted by ⊥; somewhat less familiar will be that the greatest element (if it exists) is called top and is denoted by ⊤.
A partial function is any function that needs not be defined everywhere. For example, f(x) = 1/x is a partial function on the reals (it is not defined for x = 0). We can form a partial ordering of (partial) functions having the same domain and codomain by specifying that f is smaller than g if f(x) = g(x) for all such x that f(x) is defined. Sometimes it will make sense to specify that f is smaller than g if f(x) is smaller than g(x) for all such x that f(x) is defined. Now, it happens that all collections of functions have a lub; and in particular, the partial function that is defined nowhere is the smallest function (the bottom function).
Now, with this definition of “ordering” of functions we can apply the definitions of recursion and corecursion to functions. For example, consider f(x) = f(x) + 1. The bottom function (defined nowhere) qualifies, since undefined + 1 is undefined; thus it is the smallest solution to the equation and thus this equation is a recursive definition of bottom. However, consider f(x) = if x = 0 then 1 else x*f(x-1): any solution to this equation must have f(0) = 1 and thus the bottom function is disqualified. In fact, the smallest solution is the factorial function, and thus the equation is a recursive definition of factorial.
In the comments to my previous entry, I was asked whether the Haskell definition “ones = 1 : ones” is recursive. Of course, the question in itself is nonsensical; whether it is a recursive definition or something else (such as corecursive) depends on what the programmer intends. I will interpret this question as whether the equation is a recursive definition of the intended result, the infinite list of all ones.
First, observe that we can order Haskell lists (including the special “undefined” list) partially as follows: l1 is smaller than l2 if l1 is undefined, if they both are empty lists or if a) the first element of l1 is smaller than the first element of l2 and b) the tail (the list with the first element removed) of l1 is smaller than the tail of l2.
The undefined list does not qualify as a solution for “ones = 1 : ones”. No finite list qualifies either. The infinite list of all ones does qualify, and it is the only solution; hence, the equation is a recursive definition of the infinite list of all ones; it is also a corecursive definition of the same list.
Finally, let us consider the question that started it all: what should we make of the definition “recursion: see recursion”. First, note that a equation x = x admits all elements of the universe as a solution; hence, the proper reading of “recursion: see recursion” as a definition is that it is ambiguous and hence bad. It is a recursive definition of the bottom element of the universe (whatever that might be for a dictionary), and a corecursive definition of the top element of the universe (whatever that might be). In any case, it does not define recursion (unless of course, the concept of recursion is the smallest or the largest concept in the universe of concepts).
