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In several explanations of coinductive definitions (for example, in the answers to What is coinduction?), we're told that while an inductive definition gives us the smallest set with a specified set of properties (the constructors), coinduction gives us the largest set such with a specified set of properties (the destructors).

I'm having trouble understanding this because the notion of "largest set" doesn't seem like it should be well-defined. Clearly I'm thinking about this wrongly, but it would be helpful to know what my mistake is.

As a simple example that I've picked up from a few introductions, let's consider infinite streams over some alphabet $\Sigma$. We consider some set $\mathcal{E}$, such that if an element of $\mathcal{E}$ is of the form $\sigma{:}s$ for $\sigma\in\Sigma$, then $s\in \mathcal{E}$.

This makes some intuitive sense, even though it's a bit informal. (What does it mean for an element of an arbitrary set to "be of the form" $\sigma{:}s$?) But then the next part of the story is that we want the largest set $\mathcal{E}$ such that this is the case.

Here's my problem: surely the set $\Sigma^\infty$ of infinite streams does obey the specified property. But then surely so do other, larger sets such as $\Sigma^\infty\cup \mathbb{R}$. An element of this set is either an infinite stream (hence it's of the form $\sigma{:}s$ and we can conclude $s\in \Sigma^\infty\cup \mathbb{R}$), or it's a real number (in which case it's not of the form $\sigma{:}s$, but that's ok because we never introduced a rule saying it has to be). So it seems I can make the set as big as I want just by adding arbitrary members to it, and $\Sigma^\infty$ isn't actually the biggest set that satisfies the given property at all.

So where's my mistake? How can I think about the "largest set" in this context in a way where it's actually well-defined?

Note: I know that coinductive definitions can be formalised as final coalgebras of polynomial functors on $\mathbf{Set}$, and I actually understand that reasonably well. But I'm coauthoring a paper aimed at an audience who won't know anything about that, so my aim is to understand the more informal motivation for coinductive definitions, so that I can explain to the reader what's going on without requiring a lot of technical background. It would be quite helpful for me to know how this informal idea of "largest set" relates to the more formal notion of a terminal coalgebra.

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    $\begingroup$ The missing piece is that every element of the set is supposed to be of the specified form ($σ:s$ or what have you). So, you cannot just union with $\mathbb R$ and get another set meeting the specification, because its elements are not of the form $σ:s$. $\endgroup$
    – Dan Doel
    Oct 14, 2022 at 16:33

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Here is a tentative self-answer.

I think it must just be that there is an additional requirement that wasn't mentioned in the tutorials I read, namely that one of the destructors must apply to each element of the set.

So in the case of infinite streams we are looking at a set $\mathcal{E}$ such that

  • every member of $\mathcal{E}$ is of the form $\sigma{:}s$ for some $\sigma$, and $s\in \mathcal{E}$.

When we ask for the largest set of this form it makes sense. (At least it does intuitively - it still seems a bit formally iffy to me, but it's hard to put my finger on why.)

It then becomes much more clear how this relates to the terminal coalgebra idea. A set where every member is of the form $\sigma{:}s$ can be modelled as a set $\mathcal{E}$ equipped with a function $\operatorname{decompose}\colon \mathcal{E}\to \mathcal{\Sigma}\times\mathcal{E}$, which we think of as splitting a stream up into its head and tail. If we think of this as a coalgebra of $\Sigma\times{-}$ then when we ask for a "larger set", we're asking for a new pair $(\mathcal{E}',\operatorname{decompose}')$ and a monomorphism $m\colon \mathcal{E}\to\mathcal{E}'$ such that $m$ is a coalgebra homomorphism. So then it's not so hard to imagine that the notions of "largest set" and "final coalgebra" are the same.

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  • $\begingroup$ Actually this doesn't make sense, does it? The unique map into the final coalgebra can be many-to-one, so the final one doesn't have to be the largest one at all. I think this idea of "largest set" might actually just be wrong. $\endgroup$
    – N. Virgo
    Oct 14, 2022 at 7:53
  • $\begingroup$ The point here is that for elements to be 'of the form $σ:s$,' means not just that there is any old function of type $\mathcal{E} → Σ×\mathcal{E}$. The point is to characterize it in such a way that bisimilarity coincides with set theoretic equality. For examples like this, some amount of encoding is going to be necessary unless non-well-founded sets are allowed. But the point is to be considering sets of traces, not carrier sets of any algebra. $\endgroup$
    – Dan Doel
    Oct 14, 2022 at 19:45
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I think the precise definition of a final coalgebra might help.

Fix a set of symbols $\Sigma$. For any set $X$ define $F(X) = \Sigma \times X$, and for any $f : X \to Y$ let $F(f) : F(X) \to F(Y)$ be the map $(s, x) \mapsto (s, f(x))$. Then $F : \mathsf{Set} \to \mathsf{Set}$ is a functor.

An $F$-coaglebra is a pair $(A, a)$ where $A$ is a set and $a : A \to F(A)$ is a map, called the structure map. Such a structure map is equivalent to having two maps $\mathsf{hd}_a : A \to \Sigma$ and $\mathsf{tl}_a : A \to A$, commonly called the head and the tail operations. The relationship with with $a$ is characterized by $a(x) = (\mathsf{hd}_a(x), \mathsf{tl}_a(x))$.

A homomorphism of $F$-coalgebras $h : (A, a) \to (B, b)$ is a map $h : A \to B$ between their underlying sets that respects the structure maps: for all $x \in A$ we require $\mathsf{hd}_b(h(x)) = \mathsf{hd}_a(x)$ and $\mathsf{tl}_b(h(x)) = h(\mathsf{tl}_a(x))$.

$F$-coalgebras and homomorphisms form a category. A terminal object in this category is the terminal $F$-coalgebra. Concretely, it is an $F$-coalgebra $(E, e)$ such that for any coalgebra $(A, a)$ there is a unique homomorphism $h : A \to E$.

Let us show that $E = \Sigma^\omega$ with $e(s_0s_1s_2s_3\cdots) = (s_0, s_1s_2s_3\cdots)$ is a final $F$-coalgebra. For an arbitrary $A$, there is a unique map $h : A \to E$ satisfying $h(x) = (\mathsf{hd}_a(x), h(\mathsf{tl}_a(x))$ for all $x \in A$. It is easy to verify that this is the only one.

The unique homomorphism $h : A \to E$ need not be injective. For example, take $A = \Sigma \times \{0,1\}$ and let $a : A \to F(A)$ be $a(s, d) = (s, (s, d))$. Then $h(s,0) = h(s, 1)$ as they are both the infinite sequence of $s$'s. It is therefore somewhat imprecise to call $E$ the “largest” $F$-coalgebra – which is why officially it is called the “terminal” one. But it is helpful to think of it as being “large” in the sense that every other $F$-coalgebra maps into it.

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