I don’t remember when I first heard the terms “coinduction” and “corecursion” but it must have been quite long ago. I had this impression that they are yet another of these difficult theoretical concepts and that I should learn about them one day. That “one day” happened recently while reading chapter 5 of “Certified Programming with Dependent Types”. It turns out that basics of coinduction are actually quite simple. In this post I’ll share with you what I already know on the subject.
Recursion in Haskell
Let’s begin with looking at Haskell because it is a good example of language not formalizing coinduction in any way. Two features of Haskell are of interest to us. First one is laziness. Thanks to Haskell being lazy we can write definitions like these (in GHCi):
ghci> let ones = 1 : ones
ghci> let fib = zipWith (+) (1:fib) (1:1:fib)
ones is – as the name implies – an infinite sequence (list) of ones.
fib is a sequence of Fibonacci numbers. Both these definitions produce infinite lists but we can use these definitions safely because laziness allows us to force a finite number of elements in the sequence:
ghci> take 5 ones
ghci> take 10 fib
Now consider this definition:
No matter how hard we try there is no way to use the definition of
inf in a safe way. It always causes an infinite loop:
ghci> (0 /= inf)
*** Exception: <<loop>>
The difference between definitions of
fib an the definition of
inf is that the former use something what is called a guarded recursion. The term guarded comes from the fact that recursive reference to self is hidden under datatype constructor (or: guarded by a constructor). The way lazy evaluation is implemented gives a guarantee that we can stop the recursion by not evaluating the recursive constructor argument. This kind of infinite recursion can also be called productive recursion, which means that although recursion is infinite each recursive call is guaranteed to produce something (in my examples either a 1 or next Fibonacci number). By contrast recursion in the definition of
inf is not guarded or productive in any way.
Haskell happily accepts the definition of
inf even though it is completely useless. When we write Haskell programs we of course don’t want them to fall into silly infinite loops but the only tool we have to prevent us from writing such code is our intelligence. Situation changes when it comes to….
Dependently-typed programming languages
These languages deeply care about termination. By “termination” I mean ensuring that a program written by the user is guaranteed to terminate for any input. I am aware of two reasons why these languages care about termination. First reason is theoretical: without termination the resulting language is inconsistent as logic. This happens because non-terminating term can prove any proposition. Consider this non-terminating Coq definition:
Fixpoint evil (A : Prop) : A := evil A.
If that definition was accepted we could use it to prove any proposition. Recall that when it comes to viewing types as proofs and programs as evidence “proving a proposition” means constructing a term of a given type.
evil would allow to construct a term inhabiting any type
Prop is a kind of logical propositions so
A is a type.) Since dependently-typed languages aim to be consistent logics they must reject non-terminating programs. Second reason for checking termination is practical: dependently typed languages admit functions in type signatures. If we allowed non-terminating functions then typechecking would also become non-terminating and again this is something we don’t want. (Note that Haskell gives you
UndecidableInstances that can cause typechecking to fall into an infinite loop).
Now, if you paid attention on your Theoretical Computer Science classes all of this should ring a bell: the halting problem! The halting problem says that the problem of determining whether a given Turing machine (read: a given computer program) will ever terminate is undecidable. So how is that possible that languages like Agda, Coq or Idris can answer that question? That’s simple: they are not Turing-complete (or at least their terminating subsets are not Turing complete). (UPDATE: but see Conor McBride’s comment below.) They prohibit user from using some constructs, probably the most important one being general recursion. Think of general recursion as any kind of recursion imaginable. Dependently typed languages require structural recursion on subterms of the arguments. That means that if a function receives an argument of an inductive data type (think: algebraic data type/generalized algebraic data type) then you can only make recursive calls on terms that are syntactic subcomponents of the argument. Consider this definition of
map in Idris:
map : (a -> b) -> List a -> List b
map f  = 
map f (x::xs) = f x :: map f xs
In the second equation we use pattern matching to deconstruct the list argument. The recursive call is made on
xs, which is structurally smaller then the original argument. This guarantees that any call to
map will terminate. There is a silent assumption here that the
List A argument passed to
map is finite, but with the rules given so far it is not possible to construct infinite list.
So we just eliminated non-termination by limiting what can be done with recursion. This means that our Haskell definitions of
fib would not be accepted in a dependently-typed language because they don’t recurse on an argument that gets smaller and as a result they construct an infinite data structure. Does that mean we are stuck with having only finite data structures? Luckily, no.
Coinduction to the rescue
Coinduction provides a way of defining and operating on infinite data structures as long as we can prove that our operations are safe, that is they are guarded and productive. In what follows I will use Coq because it seems that it has better support for coinduction than Agda or Idris (and if I’m wrong here please correct me).
Coq, Agda and Idris all require that a datatype that can contain infinite values has a special declaration. Coq uses
CoInductive keyword instead of
Inductive keyword used for standard inductive data types. In a similar fashion Idris uses
codata instead of
data, while Agda requires ∞ annotation on a coinductive constructor argument.
Let’s define a type of infinite
nat streams in Coq:
CoInductive stream : Set :=
| Cons : nat -> stream -> stream.
I could have defined a polymorphic stream but for the purpose of this post stream of nats will do. I could have also defined a
Nil constructor to allow finite coinductive streams – declaring data as coinductive means it can have infinite values, not that it must have infinite values.
Now that we have infinite streams let’s revisit our examples from Haskell:
ones is simple:
CoFixpoint ones : stream := Cons 1 ones.
We just had to use
CoFixpoint keyword to tell Coq that our definition will be corecursive and it is happily accepted even though a similar recursive definition (ie. using
Fixpoint keyword) would be rejected. Allow me to quote directly from CPDT:
whereas recursive definitions were necessary to use values of recursive inductive types effectively, here we find that we need co-recursive definitions to build values of co-inductive types effectively.
That one sentence pins down an important difference between induction and coinduction.
Now let’s define
zipWith and try our second example
CoFixpoint zipWith (f : nat -> nat -> nat) (a : stream)
(b : stream) : stream :=
match a, b with
| Cons x xs, Cons y ys => Cons (f x y) (zipWith f xs ys)
CoFixpoint fib : stream :=
zipWith plus (Cons 1 fib) (Cons 1 (Cons 1 fib)).
Unfortunately this definition is rejected by Coq due to “unguarded recursive call”. What exactly goes wrong? Coq requires that all recursive calls in a corecursive definition are:
- direct arguments to a data constructor
- not inside function arguments
Our definition of
fib violates the second condition – both recursive calls to
fib are hidden inside arguments to
zipWith function. Why does Coq enforce such a restriction? Consider this simple example:
Definition tl (s : stream) : stream :=
match s with
| Cons _ tl' => tl'
CoFixpoint bad : stream := tl (Cons 1 bad).
tl is a standard tail function that discards the first element of a stream and returns its tail. Just like our definition of
fib the definition of
bad places the corecursive call inside a function argument. I hope it is easy to see that accepting the definition of
bad would lead to non-termination – inlining definition of
tl and simplifying it leads us to:
CoFixpoint bad : stream := bad.
and that is bad. You might be thinking that the definition of
bad really has no chance of working whereas our definition of
fib could in fact be run safely without the risk of non-termination. So how do we persuade Coq that our corecursive definition of
fib is in fact valid? Unfortunately there seems to be no simple answer. What was meant to be a simple exercise in coinduction turned out to be a real research problem. This past Monday I spent well over an hour with my friend staring at the code and trying to come up with a solution. We didn’t find one but instead we found a really nice paper “Using Structural Recursion for Corecursion” by Yves Bertot and Ekaterina Komendantskaya. The paper presents a way of converting definitions like
fib to a guarded and productive form accepted by Coq. Unfortunately the converted definition looses the linear computational complexity of the original definition so the conversion method is far from perfect. I encourage to read the paper. It is not long and is written in a very accessible way. Another set of possible solutions is given in chapter 7 of CPDT but I am very far from labelling them as “accessible”.
I hope this post demonstrates that basics ideas behind coinduction are actually quite simple. For me this whole subject of coinduction looks really fascinating and I plan to dive deeper into it. I already have my eyes set on several research papers about coinduction so there’s a good chance that I’ll write more about it in future posts.