---
-title: "Exponential blowup when using typeclasses with nested indices"
+title: "Exponential blowup when using unbundled typeclasses to model algebraic hierarchies"
categories: research coq
---
-When formalizing a proof in an interactive theorem prover like [Coq](https://coq.inria.fr/), one reoccuring issue is handling algebraic hierarchies.
-Such hierarchies are everywhere: Some operations are associative, while others also commute; some types have an equivalence relation, while others also have a (pre-)order or maybe even a well-ordering; and so on.
+When formalizing a proof in an interactive theorem prover like [Coq](https://coq.inria.fr/), one reoccuring issue is the handling of algebraic hierarchies.
+Such hierarchies are everywhere: some operations are associative, while others commute; some types have an equivalence relation, while others also have a (pre-)order or maybe even a well-ordering; and so on.
So the question arises: What is the best way to actually encode these hierarchies in Coq?
Coq offers two mechanism that are suited to solve this task: typeclasses and canonical structures.
Both can be instrumented in different ways to obtain a (more or less) convenient-to-use algebraic hierarchy.
-A common approach using typeclasses is the ["unbundled" approach by Bas Spitters and Eelis van der Weegen](https://arxiv.org/abs/1102.1323).
-However, as [has been observed before](https://pastel.archives-ouvertes.fr/pastel-00649586), and as learned the hard way in the Coq formalization of the [original Iris paper](https://iris-project.org/pdfs/2015-popl-iris1-final.pdf), this approach quickly leads to terms that seem to be exponential in size.
+A common approach using typeclasses is the ["unbundled" approach by Bas Spitters and Eelis van der Weegen](http://www.eelis.net/research/math-classes/mscs.pdf).
+However as we learned the hard way in the Coq formalization of the [original Iris paper](https://iris-project.org/pdfs/2015-popl-iris1-final.pdf), this approach quickly leads to terms that seem to be exponential in size.
+In this post, I will explain the cause of this exponential blowup.
+<!-- MORE -->
+
+I should note that this observation is not new, it already occurs in [François Garillot's PhD thesis](https://pastel.archives-ouvertes.fr/pastel-00649586).
+The goal of this post is to provide a more self-contained presentation, not requiring all the context set up in that thesis.
+
+## Unbundled groups
+
+A formalization of the algebraic hierarchy for groups using the unbundled approach could look as follows:
+{% highlight coq %}
+Require Import Lia ZArith.
+
+(** The operations as typeclasses for overloading, without any axioms. *)
+Class Op (A: Type) := op : A -> A -> A.
+Infix "+" := op.
+
+Class Unit (A: Type) := unit : A.
+
+Class Inverse (A: Type) := inv : A -> A.
+
+(** The algebraic classes adding the axioms, taking the operations as indices. *)
+Class Semigroup (A: Type) `{Op A} := {
+ assoc a b c : (a + b) + c = a + (b + c);
+}.
+
+Class Monoid (A: Type) `{Semigroup A, Unit A} := {
+ id_r a : a + unit = a;
+ id_l a : unit + a = a;
+}.
+
+Class Group (A: Type) `{Monoid A, Inverse A} := {
+ inv_r a : a + inv a = unit;
+ inv_l a : inv a + a = unit;
+}.
+{% endhighlight %}
+
+As usual for the unbundled approach, we end up with lots of tiny typeclasses.
+`Op`, `Unit` and `Inverse` serve as operators that can be overloaded.
+They each just take a type `A` as argument (we say they are *indexed* by the type: we can do typeclass search that, given the type, finds the instance).
+`Semigroup`, `Monoid` and `Group` supply the laws for these operators; they are indexed by the operators they involve and the typeclasses from which they "inherit".
+Taking the superclasses as indices instead of embedding them as fields simplifies typeclass search and is necessary to support diamonds in the hierarchy; see [the original paper](http://www.eelis.net/research/math-classes/mscs.pdf) for more details.
+
+We use Coq's feature for anonymous implicit generalization: `` `{Semigroup A} `` adds not only a parameter of type `Semigroup A` that we don't have to name; furthermore all typeclass arguments of `Semigroup` (in this case, `Op A`) are also implicitly added as parameter.
+This avoids lots of repetition that the unbundled style would usually incur, [but the semantics of implicit generalization are quirky and poorly documented](https://coq.discourse.group/t/documentation-of-foo/129).
+
+Next, we can show that the integers `Z` form a group:
+{% highlight coq %}
+Instance Z_op: Op Z := Z.add.
+Instance Z_unit: Unit Z := 0%Z.
+Instance Z_inv: Inverse Z := Z.opp.
+
+Instance Z_semigroup: Semigroup Z.
+Proof. split. intros. unfold op, Z_op. lia. Qed.
+Instance Z_monoid: Monoid Z.
+Proof. split; intros; unfold op, Z_op, unit, Z_unit; lia. Qed.
+Instance Z_group: Group Z.
+Proof. split; intros; unfold op, Z_op, unit, Z_unit, inv, Z_inv; lia. Qed.
+{% endhighlight %}
+We let `lia` do most of the proving, but we have to unfold some definitions first.
+
+We can also show that a product of two {semigroups, monoids, groups} is itself a {semigroup, monoid, group}:
+{% highlight coq %}
+Section prod.
+ Context {A B: Type}.
+
+ Global Instance prod_op `{Op A, Op B}: Op (A * B) :=
+ fun '(a1, b1) '(a2, b2) => (a1 + a2, b1 + b2).
+ Global Instance prod_unit `{Unit A, Unit B}: Unit (A * B) :=
+ (unit, unit).
+ Global Instance prod_inv `{Inverse A, Inverse B}: Inverse (A * B) :=
+ fun '(a, b) => (inv a, inv b).
+
+ Global Instance prod_semigroup `{Semigroup A, Semigroup B}: Semigroup (A * B).
+ Proof.
+ split. intros [a1 b1] [a2 b2] [a3 b3]. unfold op, prod_op.
+ rewrite !assoc. reflexivity.
+ Qed.
+ Global Instance prod_monoid `{Monoid A, Monoid B}: Monoid (A * B).
+ Proof.
+ split; intros [a b]; unfold op, prod_op, unit, prod_unit;
+ rewrite ?id_l, ?id_r; reflexivity.
+ Qed.
+ Global Instance prod_group `{Group A, Group B}: Group (A * B).
+ Proof.
+ split; intros [a b]; unfold op, prod_op, unit, prod_unit, inv, prod_inv;
+ rewrite ?inv_l, ?inv_r; reflexivity.
+ Qed.
+End prod.
+{% endhighlight %}
+
+## Unexpected complexities
+
+Seems reasonable?
+Unfortunately, we have set ourselves up for some unexpectedly large terms:
+the instance of something like `Group (Z*Z*Z*Z*Z*Z)` grows with the fifth power as the number of `Z` increase!
+
+{% highlight coq %}
+Definition test: Group (Z*Z*Z*Z*Z*Z) := _.
+Set Printing All.
+Print test.
+(* Lots of output. *)
+{% endhighlight %}
+
+What is happening here?
+We can see that when we try to construct the proof terms more explicitly.
+We start by defining `Zn`, which is basically `Z^(n+1)`: a product of `n+1` times `Z`.
+{% highlight coq %}
+Fixpoint Zn (n: nat): Type :=
+ match n with
+ | O => Z
+ | S n => @prod (Zn n) Z
+ end.
+{% endhighlight %}
+Unsurprisingly, this term has size linear in `n`: the recursive case adds `@prod • Z` around the term produced for `n-1` (represented by the `•`).
+This means the term grows by a constant when `n` increases by 1, leading to linear complexity overall.
+
+Next, we define `Op (Zn n)`:
+{% highlight coq %}
+Fixpoint Zn_op (n: nat): Op (Zn n) :=
+ match n with
+ | O => Z_op
+ | S n => @prod_op (Zn n) Z (Zn_op n) Z_op
+ end.
+{% endhighlight %}
+In the recursive case, we use `prod_op` as one might expect.
+Coq could infer all the arguments here, but let us take a closer look at what it would infer: what is the size of `Zn_op` as `n` grows?
+In the recursive case, we add `@prod_op (Zn n) Z • Z_op` around the term produced for `n-1`.
+This term includes `Zn n`, which we established is linear in size.
+When `n` increases by 1, the term grows by a linear amount.
+Thus, the overall term size is quadratic in `n`!
+Ouch.
+I don't know about you, but I found this surprising when I realized it for the first time.
+
+The index basically repeats all the information that is already present in the recursive call:
+to extend a `Zn_op` by one more element, we have to repeat the full type for which we already constructed an `Op`.
+
+But, you might say, quadratic complexity is not so bad.
+Computers are fast, we can handle that.
+Unfortunately, I got bad news: things get worse as our indices start nesting.
+Let us look at the instance for `Semigroup (Zn n)`:
+{% highlight coq %}
+Fixpoint Zn_semigroup (n: nat): Semigroup (Zn n) :=
+ match n return @Semigroup (Zn n) (Zn_op n) with
+ | O => Z_semigroup
+ | S n => @prod_semigroup (Zn n) Z (Zn_op n) (Zn_semigroup n) Z_op Z_semigroup
+ end.
+{% endhighlight %}
+How big is this term?
+Well, the term added as `n` increases by 1 is `@prod_semigroup (Zn n) Z (Zn_op n) • Z_op Z_semigroup`.
+This includes `Zn_op n`, which is quadratic -- so `Zn_semigroup` is cubic!
+
+`Zn_semigroup` has an `Op (Zn n)` as index, and again the full nested term describing the operator of both semigroups that we take the product of is getting repeated each time we add just one more `Z`.
+So, while `Op (Zn n)` has an index that makes it grow quadratically, `Semigroup (Zn n)` *has an index that has an index*, and grows cubically.
+To compute the complexity of our terms, we have to figure out *how nested* our indices are: with no indices (such as `Zn n` itself), the term is linear; for each extra nested index the power increases by 1.[^1]
+If we look at `Group`, we see that it has a `Monoid` index which has a `Semigroup` index which has an `Op` index which has a `Type` index.
+That's 4 nested indices, and hence the terms have size `O(n^5)`.
+More generally speaking, the term size is *exponential* in the height of our hierarchy.
+That hurts.
+
+[^1]: Note that the *number* of indices does not matter, like the fact that `Monoid` has both an `Op` and a `Unit` index. That just affects the constant factor. It's the nesting of indices that gets us here.
+
+## Conclusion
+
+We have seen that the number of nested indices of a typeclass has exponential impact on the size of proof terms that arise when deriving typeclass instances for compound types.
+This effect has grinded the original Iris coq development back in 2015 to a halt: compiling the correctness proof of the logic itself took around 3.5 hours and needed more than the 4GB of RAM than my laptop had back then.
+
+In my view, this basically rules out the unbundled approach for any serious formalization effort.
+But what are the alternatives?
+
+Instead of making superclasses indices, we could make them fields, as in:
+{% highlight coq %}
+Class Monoid (A: Type) `{Op A, Unit A} := {
+ monoid_semigroup :> Semigroup A;
+ id_r a : a + unit = a;
+ id_l a : unit + a = a;
+}.
+{% endhighlight %}
+
+This limits the "index depth" to 1, and thus limits term complexity to `O(n^2)`.
+However, this scheme does not support diamonds in the hierarchy as instances will be duplicated.
+Also, the instance we implicitly added by writing `:>`, namely that `Monoid A -> Semigroup A`, means we'll check all `Monoid` instances any time we just want a `Semigroup`.
+This leads to exponential search times when performing backtracking typeclass search: a `Semigroup (A*A)` can be obtained either by searching for a `Monoid (A*A)` or via `prod_semigroup`, and in case of backtracking Coq will go through a combinatorial explosion when trying all possible ways to derive `Semigroup (A*A*A*...)`.[^2]
+
+[^2]: In genaral, such combinatorial blowups happen very easily with backtracking, and they are hard to detect in advance. This is one reason why I believe that backtracking-per-default is a bad choice for proof search.
+
+Maybe there is a way to make the unbundled approach viable with some help from the proof assistant.
+Some way to eliminate indices would help (similar to primitive projections, but also covering operations such as `prod_op`).
+Or maybe the proof assistant could exploit the huge amounts of sharing that are present in these large terms.
+If all subterms are properly shared, I think the overall term size remains linear.
+Coq deduplicates term representation in memory, but (to my knowledge) does not exploit sharing when traversing the term (for unification or type-checking or any other purpose).
+Defeating the exponential complexity here would require exploiting the sharing in *every* pass over the term.
+
+In Iris, we instead went with an adaptation of [this approach based on canonical structures](https://hal.inria.fr/hal-00816703v1/document).
+This fully bundled approach avoids indices entirely and thus maintains a linear term size.
+On the other hand, it comes with a lot of boilerplate code, and mixing canonical structures with typeclasses can be tricky to debug. (Unfortunately we cannot avoid typeclasses entirely as we need to have `Proper` instances in our algebraic classes.)
+When composing types, Iris users have to write things like `prodC (listC natC) (optionC fracC)` instead of `(list nat) * (option frac)`, which is an additional barrier of entry to an already complicated development.
+Also, I have to say I don't really have an intuitive grasp of canonical structure resolution, so when something does not work I don't even know how to start debugging.
+On the other hand, the correctness proof of the logic *and* the general proof mode infrastructure *and* a few examples verified in the logic take less than five minutes to compile in our latest Coq formalization.
+That's quite some speedup!
+
+All in all, I think the question of how to best represent algebraic hierarchies in a proof assistant is still wide open.
+I am curious which ideas people will come up with!
+
+#### Footnotes
projects / web.git / commitdiff