There is that curious idea that you can think of a type in a programming language as a kind of algebraic object. Take (homogeneous) lists for example. A list of integers is either an empty list (i.e., nil) or a pair of an integer (the head of the list) and another list (the tail of the list). You can symbolically write this as
Here \(1\) is the unit type with one element, it does not matter what this element is exactly. \(A + B\) is a disjoint sum of \(A\) and \(B\). It is a tagged union type, whose values are values of \(A\) or \(B\) marked as such. \(A \cdot B\) is the product type. Its values are pairs of values of \(A\) and \(B\).
The underlying mathematical machinery includes "polynomial functors", "monads", "Lambek's theorem", etc. You can stop digging when you reach "Knaster-Tarski theorem" and "Beck's tripleability condition".
In general, we have
$$\List(x) = 1 + x \cdot \List(x)$$In Haskell this is written as
data List x = Nil | Cons x (List x)Similarly, a binary tree with values of type \(x\) at the nodes can be written as
$$\BT(x) = 1 + x\cdot \BT(x) \cdot \BT(x)$$That is, a binary tree is either empty or a triple of the root, the left sub-tree and the right sub-tree. If we are only interested in the "shapes" of the binary trees we need
$$\B = \BT(1) = 1 + \B^2$$Nothing too controversial so far. Now, apply some trivial algebra:
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \List(x) &\;=\;& 1 + x \cdot \List(x) \\ \List(x) - x \cdot \List(x) &\;=\;& 1 \\ \List(x) \cdot (1 - x) &\;=\;& 1 \\ \List(x) &\;=\;& \frac{1}{1 - x} \\ \List(x) &\;=\;& 1 + x + x^2 + x^3 + \cdots \end{array}$$Going from \(\List(x) - x \cdot \List(x)\) to \(\List(x) \cdot (1 - x)\) in the middle step arguably makes some sense: product of sets and types distributes over sum, much like it does for integers. But the rest is difficult to justify or even interpret. What is the difference of types or their infinite series?
The last equation, however, is perfectly valid: an element of \(\List(x)\) is either nil, or a singleton element of \(x\), or a pair of elements of \(x\) or a triple, etc. This is somewhat similar to the cavalier methods of eighteenth-century mathematicians like Euler and the Bernoulli brothers, who would boldly go where no series converged before and through a sequence of meaningless intermediate steps arrive to a correct result.
One can apply all kinds of usual algebraic transformations to \(\List(x) = (1 - x)^{-1}\). For example,
$$\begin{array}{r@{\;}c@{\;}l@{\quad}l@{\quad}} \List(a + b) &\;=\;& \frac{1}{1 - a - b} \\ &\;=\;& \frac{1}{1 - a}\cdot\frac{1}{1 - \frac{b}{1 - a}} \\ &\;=\;& \List(a)\cdot\List(\frac{b}{1 - a}) \\ &\;=\;& \List(a)\cdot\List(b\cdot\List(a)) \end{array}$$This corresponds to the regular expressions identity \((a|b)^* = a^*(ba^*)^*\).
Apply the same trick to binary trees:
$$\begin{array}{r@{\;}c@{\;}l@{\quad}l@{\quad}} \BT &\;=\;& 1 + \BT^2 \cdot x & \\ \BT^2 \cdot x - \BT + 1 &\;=\;& 0 & \text{(Solve the quadratic equation.)} \\ \BT &\;=\;& \frac{1\pm\sqrt{1-4\cdot x}}{2\cdot x} & \text{(Use binomial theorem)} \\ \sqrt{1-4\cdot x} &\;=\;& \sum_{k=0}^\infty \binom{\tfrac12}{k}(-4\cdot x)^k & \\ \BT(x) &\;=\;& 1 + x + 2\cdot x^2 + 5\cdot x^3 + 14\cdot x^4 + 42\cdot x^5 + \cdots & \\ &\;=\;& \sum_{n=0}^\infty C_n\cdot x^n, & \end{array}$$Where \(C_n=\frac{1}{n+1}\binom{2\cdot n}{n}\)—the \(n\)-th Catalan number, that is the number of binary trees with \(n\) nodes. To understand the last series we need to decide what \(n\cdot x\) is, where \(n\) is a non-negative integer and \(x\) is a type. Two natural interpretations are
$$n \cdot x = x + x + \cdots = \sum_0^{n - 1} x$$and
$$n \cdot x = \{0, \ldots, n - 1\} \cdot x$$which both mean the same: an element of \(n\cdot x\) is an element of \(x\) marked with one of \(n\) "branches", and this is the same as a pair \((i, t)\) where \(0 <= i < n\) and \(t\) is an element of \(x\).
The final series then shows that a binary tree is either an empty tree (\(1\)) or an \(n\)-tuple of \(x\)-es associated with one of \(C_n\) shapes of binary trees with \(n\) nodes. It worked again!
Now, watch carefully. Take simple unmarked binary trees: \(\B = \BT(1) = 1 + \B^2\). In this case, we can "calculate" the solution exactly:
$$\B^2 - \B + 1 = 0$$hence
$$\B = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1}{2} \pm i\cdot\frac{3}{2} = e^{\pm i\cdot\frac{\pi}{3}}$$Both solutions are sixth-degree roots of \(1\): \(\B^6 = 1\). OK, this is still completely meaningless, right? Yes, but it also means that \(\B^7 = \B\), which --if correct-- would imply that there is a bijection between the set of all binary trees and the set of all 7-tuples of binary trees. Haha, clearly impossible... Actually, it's a more or less well-known fact.
A note is in order. All "sets" and "types" that we talked about so far are at most countably infinite so, of course, there exists a bijection between \(\B\) and \(\B^7\) simply because both of these sets are countable. What is interesting, and what the paper linked to above explicitly establishes, is that there is as they call it "a very explicit" bijection between these sets, that is a "structural" bijection that does not look arbitrarily deep into its argument trees. For example, in a functional programming language, such a bijection can be coded through pattern matching without recursion.
Here is another neat example. A rose tree is a kind of tree datatype where a node has an arbitrary list of children:
$$\R(x) = x\cdot\List(\R(x))$$Substituting \(\List(x) = (1 - x)^{-1}\) we get
$$\R(x) = \frac{x}{1 - R(x)}$$or
$$\R^2(x) - \R(x) + x = 0$$Hence \(\R = \R(1)\) is defined by the same equation as \(\B = \BT(1)\): \(\R^2 - \R + 1 = 0\). So, there must be a bijection between \(\R\) and \(\B\), and of course there is: one of the first things LISP hackers realised in the late 1950s is that an arbitrary tree can be encoded with cons cells.
Mathematically, this means that instead of "initial algebras" we treat types as "final co-algebras".
All this is, by the way, is equally applicable to infinite datatypes (that is, streams, branching transition systems, etc.)
That was pure algebra, there is nothing "analytical" here. Analysis historically started with the notion of derivative. And sure, you can define and calculate derivatives of data type constructors like \(\List\) and \(\BT\). These derivatives are generalisations of zipper-like structures: they represent "holes" in the data-type, that can be filled by an instance of the type-parameter.
For example, suppose you have a list \((x_0, \ldots, x_k, \ldots, x_{n - 1})\) of type \(\List(x)\) then a hole is obtained by cutting out the element \(x_k\) that is \((x_0, \ldots, x_{k-1}, \blacksquare, x_{k+1}, \ldots, x_{n - 1})\) and is fully determined by the part of the list to the left of the hole \((x_0, \ldots, x_{k - 1})\) and the part to the right \((x_{k+1}, \ldots, x_{n - 1})\). Both of these can be arbitrary lists (including empty), so a hole is determined by a pair of lists and we would expect \(\d \List(x) = \List^2(x)\). But Euler would have told us this immediately, because \(\List(x) = (1 - x)^{-1}\) hence
$$\d \List(x) = \d (1 - x)^{-1} = (1 - x)^{-2} = \List^2(x)$$In general the derivative of a type constructor (of a "function" analytically speaking) \(T(x)\) comes with a "very explicit" surjective function
$$\plug : \d T(x) \cdot x \to T(x)$$injective on each of its arguments, and has the usual properties of a derivative, among others:
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \d (A + B) &\;=\;& \d A + \d B \\ \d (A \cdot B) &\;=\;& \d A \cdot B + A \cdot \d B \\ \d_x A(B(x)) &\;=\;& \d_B A(B) \cdot \d_x B(x) \\ \d x &\;=\;& 1 \\ \d x^n &\;=\;& n\cdot x^{n-1} \\ \d n &\;=\;& 0 \end{array}$$For binary trees we get
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \d \BT(x) &\;=\;& \d (1 + x\cdot \BT^2(x)) \\ &\;=\;& 0 + \BT^2(x) + 2\cdot x\cdot \BT(x) \cdot \d \BT(x) \\ &\;=\;& \BT^2(x) / (1 - 2\cdot x\cdot \BT(x)) \\ &\;=\;& \BT^2(x) \cdot \List(2\cdot x\cdot \BT(x)) \\ &\;=\;& \BT^2(x) \cdot \Zipper_2(x) \end{array}$$Here \(\Zipper_2(x) = \List(2\cdot x\cdot \BT(x))\) is the classical Huet's zipper for binary trees. We have an additional \(\BT^2(x)\) multiplier, because Huet's zipper plugs an entire sub-tree, that is, it's a \(\BT(x)\)-sized hole, whereas our derivative is an \(x\)-sized hole.
For \(n\)-ary trees \(\T = 1 + x\cdot \T^n(x)\) we similarly have
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \d \T(x) &\;=\;& \d (1 + x\cdot \T^n(x)) \\ &\;=\;& 0 + \T^n(x) + n\cdot x\cdot \T^{n-1}(x) \cdot \d \T(x) \\ &\;=\;& \T^n(x) / (1 - n\cdot x\cdot \T^{n -1}(x)) \\ &\;=\;& \T^n(x) \cdot \List(n\cdot x\cdot \T^{n - 1}(x)) \\ &\;=\;& \T^n(x) \cdot \Zipper_n(x) \end{array}$$At this point we have basic type constructors for arrays (\(x^n\)), lists and variously shaped trees, which we can combine and compute their derivatives. We can also try constructors with multiple type parameters (e.g., trees with different types at the leaves and internal nodes) and verify that the usual partial-differentiation rules apply. But your inner Euler must be jumping up and down: much more is possible!
Indeed it is. Recall the "graded" representation of \(\List(x)\):
$$\List(x) = 1 + x + x^2 + x^3 + \cdots$$This formula says that there is one (i.e, \(|x|^0\)) list of length \(0\), one list (of length \(1\)) for each element of \(x\), one list for each ordered pair of elements of \(x\), for each ordered triple, etc. What if we forget the order of \(n\)-tuples? Then, by identifying each of the possible \(n!\) permutations, we would get, instead of lists with \(n\) elements, multisets (bags) with \(n\) elements.
$$\Bag(x) = \frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^k}{k!} + \cdots = e^x$$[0]
Formally we move from polynomial functors to "analytic functors" introduced by A. Joyal.
This counting argument, taken literally, does not hold water. For one thing, it implies that every term in \(e^x\) series is an integer (hint: there are no \(n!\) \(n\)-tuples belonging to the same class as \((t, t, \ldots, t)\)). But it drives the intuition in the right direction and \(\Bag(x)\) does have the same properties as the exponent. Before we demonstrate this, let's look at another example.
A list of \(n\) elements is one of \(|x|^n\) \(n\)-tuples of elements of \(x\). A bag with \(n\) elements is one of such tuples, considered up to a permutation. A set with \(n\) elements is an \(n\)-tuple of different elements considered up to a permutation. Hence, where for lists and bags we have \(x^n\), sets have \(x\cdot(x - 1)\cdot\cdots\cdot(x - n + 1)\) by the standard combinatorial argument: there are \(x\) options for the first element, \(x - 1\) options for the second element and so on.
That is,
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \Set(x) &\;=\;& 1 + \frac{x}{1!} + \frac{x\cdot(x - 1)}{2!} + \frac{x\cdot(x - 1)\cdot(x - 2)}{3!} + \cdots + \frac{x\cdot(x - 1)\cdot\cdots\cdot(x - k + 1)}{k!} + \cdots \\ &\;=\;& (1 + 1)^x \;\;\;\;\text{(By binomial)} \\ &\;=\;& 2^x \end{array}$$OK, we have \(\Bag(x) = e^x\) and \(\Set(x) = 2^x\). The latter makes sense from the cardinality point of view: there are \(2^x\) subsets of set \(x\) (even when \(x\) is infinite). Moreover, both \(\Set\) and \(\Bag\) satisfy the basic exponential identity \(\alpha^{x + y} = \alpha^x\cdot\alpha^y\). Indeed a bag of integers and Booleans can be uniquely and unambiguously separated into the bag of integers and the bag of Booleans, that can be poured back together to produce the original bag, and the same for sets. This does not work for lists: you can separate a list of integers and Booleans into two lists, but there is no way to restore the original list from the parts. This provides a roundabout argument, if you ever need one, that function \((1 - x)^{-1}\) does not have form \(\alpha^x\) for any \(\alpha\).
Next, \(\alpha^x\) is the type of functions from \(x\) to \(\alpha\), \(\alpha^x = [x \to \alpha]\). For sets this means that an element of \(\Set(x)\), say \(U\), can be thought of as the characteristic function \(U : x \to \{0, 1\}\)
$$U(t) = \begin{cases} 1, & t \in U,\\ 0, & t \notin U. \end{cases}$$For \(\Bag(x)\) we want to identify a bag \(V\) with a function \(V : x \to e\). What is \(e\)? By an easy sleight of hand we get:
$$e = e^1 = \Bag(1) = \{ 0, 1, \ldots \} = \nat$$(Because for the one-element unit type \(1\), a bag from \(\Bag(1)\) is fully determined by the (non-negative) number of times the only element of \(1\) is present in the bag.)
Hence, we have \(\Bag(x) = \nat^x = [x \to \nat]\), which looks surprisingly reasonable: a bag of \(x\)-es can be identified with the function that for each element of \(x\) gives the number of occurrences of this element in the bag. (Finite) bags are technically "free commutative monoids" generated by \(x\).
Moreover, the most famous property of \(e^x\) is that it is its own derivative, so we would expect
$$\d \Bag(x) = \Bag(x)$$And it is: if you take a bag and make a hole in it, by removing one of its elements, you get nothing more and nothing less than another bag. The "plug" function \(\plug : \Bag(x)\cdot x \to \Bag(x)\) is just multiset union:
$$\plug (V, t) = V \cup \{t\},$$which is clearly "very explicit" and possesses the required properties of injectivity and surjectivity. This works neither for lists, trees and arrays (because you have to specify where the new element is to be added) nor for sets (because if you add an element already in the set, the plug-map is not injective).
The central rôle of \(e^x\) in analysis indicates that perhaps it is multiset, rather than list, that is the most important datastructure.
If we try to calculate the derivative of \(\Set(x)\) we would get \(\d\Set(x) = \d(2^x) = \d e^{\ln(2)\cdot x} = \ln(2)\cdot 2^x = \ln(2)\cdot\Set(x)\), of which we cannot make much sense (yet!), but we can calculate
$$\Bag(\d\Set(x)) = e^{\d\Set(x)} = e^{\ln(2)\cdot\Set(x)} = 2^{Set(x)} = \Set(\Set(x))$$Nice. Whatever the derivative of \(\Set(x)\) is, a bag of them is a set of sets of \(x\).
Thinking about a type constructor \(T(x) = \sum_{i>0} a_i\cdot x^i\) as an non-empty (\(i > 0\)) container, we can consider its pointed version, where a particular element in the container is marked:
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \Punct(T(x)) &\;=\;& \sum_{i>0} i\cdot a_i\cdot x^i \\ &\;=\;& x\cdot \sum_{i>0} i\cdot a_i\cdot x^{i-1} \\ &\;=\;& x\cdot \sum_{i>0} a_i\cdot \d x^i \\ &\;=\;& x\cdot \d \sum_{i>0} a_i\cdot x^i \\ &\;=\;& x\cdot \d T(x) \end{array}$$This has clear intuitive explanation: if you cut a hole in a container and kept the removed element aside, you, in effect, marked the removed element.
When I first studied these functions long time ago, they used to be called \(\sh\) and \(\ch\), pronounced roughly "shine" and "coshine". The inexorable march of progress...
The next logical step is to try to recall as many Taylor expansions as one can to see what types they correspond to. Start with the easy ones: hyperbolic functions
$$\cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$$and
$$\sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$$\(\cosh(x)\) is the type of bags of \(x\) with an even number of elements and \(\sinh(x)\) is the type of bags of \(x\) with an odd number of elements. Of course, \(\cosh\) and \(\sinh\) happen to be an even and an odd function respectively. So to answer the question in the title: \(\cosh(\List(\Bool))\) is the type of bags of even number of lists of Booleans. It is easy to check that all usual hyperbolic trigonometry identities hold.
We can also write down a general function type \([ x \to (1 + y) ]\):
$$[ x \to (1 + y) ] = (1 + y)^x = 1 + y\cdot x + y^2\cdot\frac{x\cdot(x - 1)}{2!} + y^3\cdot\frac{x\cdot(x-1)\cdot(x-2)}{3!} + \cdots$$Combinatorial interpretation(s) (there are at least two of them!) are left to the curious reader.
We have by now seen the series with the terms of the form \(n_k\cdot x^k\) (\(n_k \in \nat\)) corresponding to various tree-list types, the series with the terms \(n_k\cdot\frac{x^k}{k!}\) corresponding to function-exponent types. What about "logarithm"-like series with terms of the form \(n_k\cdot\frac{x^k}{k}\)? Similarly to the exponential case, we try to interpret them as types where groups of \(n\) \(n\)-tuples are identified. The obvious candidates for such groups are all possible rotations of an \(n\)-tuple. The learned name for \(n\)-tuples identified up to a rotation is "an (aperiodic) necklace", we will call it a ring. The type of non-empty rings of elements of \(x\) is given by
$$\Ring(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots$$Compare with:
$$\begin{array}{r@{\;}c@{\;}l@{\quad}} \ln(1 + x) &\;=\;& x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \\ \ln(1 - x) &\;=\;& - x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots \\ - \ln(1 - x) &\;=\;& x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots \\ &\;=\;& \Ring(x) \end{array}$$Now comes the cool part:
$$\Bag(\Ring(x)) = e^{\Ring(x)} = e^{- \ln(1 - x)} = (1 - x)^{-1} = \List(x)$$We are led to believe that for any type there is a "very explicit" bijection between bags of rings and lists. That is, it is possible to decompose any list to a multiset of rings (containing the same elements, because nothing about \(x\) is given) in such a way that the original list can be unambiguously recovered from the multiset.
Sounds pretty far-fetched? It is called Chen–Fox–Lyndon theorem, by the way. This theorem has a curious history. The most accessible exposition is in De Bruijn, Klarner, Multisets of aperiodic cycles (1982). The "Lyndon words" term is after Lyndon, On Burnside's Problem (1954). The original result is in Ширшов, Подалгебры свободных лиевых алгебр (1953), where it is used to estimate dimensions of sub-algebras of free Lie algebras. See also I.57 (p. 85) in Analytic Combinatorics by Flajolet and Sedgewick.
If you are still not convinced that \(\Ring(x) = -\ln(1 - x)\), consider
$$\d\Ring(x) = -\d\ln(1 - x) = (1 - x)^{-1} = \List(x)$$If you cut a hole in a ring, you get a list!
We still have no interpretation for alternating series like \(\ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + \cdots\). Fortunately, our good old friend \(\List(x) = (1 - x)^{-1}\) gives us
$$\List(2) = -1$$which, together with the inclusion–exclusion principle, can be used to concoct plausibly looking explanations of types like \(\sin(x) = x - x^3/3! + x^5/5! + \cdots\), etc.
Of course, (almost) all here can be made formal and rigorous. Types under \(+\) and \(\cdot\) form a semiring. The additive structure can be extended to an abelian group by the Grothendieck group construction. The field of fractions can be built. Puiseux and Levi-Civita series in this field provide "analysis" without the baggage of limits and topology.
But they definitely had more fun back in the eighteenth century. Can we, instead of talking about "fibre bundles over \(\Ring(\lambda t.t\cdot x)\)", dream of Möbius strips of streams?