This year my goal is to go through one paper a week, and produce some resulting artifact to help hammer in my understanding. Today’s paper is Conal Elliott’s’s Adders and Arrows.

In my opinion, Adders and Arrows is an attempt to answer the questions “where do digital circuits come from?” and “how can we be convinced our digital circuits do what they claim?” by focusing on the concrete case of adders. What’s quite remarkable to me is that the paper takes 17 pages to build up to the “known” ripple-carry adder, which is essentially day 1 of any digital circuitry class. This feels a bit ridiculous at first blush, but immediately pays off for itself by using the same machinery to derive a carry-forward adder. Carry-forward adders come like a week later in circuitry class, and are completely unrelated to ripple-carry adders, but Elliott derives them essentially for free. He then gives a third variation on the theme, which is a ripple-carry adder in time, rather than space, and again gets it in one page.

So that’s the impressive stuff. What’s also important is that the paper doesn’t require us to trust that the corresponding circuits do what they claim — the underlying machinery passes around a big chain of equivalence proofs, which automatically get composed and witness that addition over the naturals is a model for each circuit.

Something I really liked about this paper is that it’s the first time I’ve ever gotten a glimpse of why it might be valuable to understand category theory. It’s not just good for classifying things! Turns out you can do cool things with it too. But nobody who teaches it ever says that.

The paper itself is only a draft, and it shows in several places. Notably, the core abstraction (the arrow category) is missing, and the paper ends extremely abruptly. After some spelunking, I managed to track down the implementation of the arrow category, and was pleased to realize I’d already implemented it independently.

We’ll proceed section by section.

Section 1: A model of addition

Section 1 gives us a model of the natural numbers via the Peano encoding, and also defines addition. It drives home the point that this unary encoding is only the model of our computation — its specification — and not the actual implementation. This is a common theme in Elliott’s work: “how do we even know what question we’re trying to answer?” We know by posing the dumbest possible model of the problem, and then proving any actual solution to the problem is equivalent. He stresses that the “dumbest possible” thing is an important quality — this is the only part of the problem we get no checks or balances on, so our only hope is to make it so simple that we can’t screw it up.

data Nat : Set where
zero : Nat
suc  : Nat -> Nat

_+_ : Nat -> Nat -> Nat
zero  + n = n
suc m + n = suc (m + n)

Because arbitrary categories don’t support currying, we can uncurry addition, which will come in handy later:

<+> : Nat2 -> Nat
<+> (x , y) = x + y

Section 2: Bounded numbers

Section 2 is defines the Finiteary numbers, which are natural numbers guaranteed to be smaller than a given upper bound.

data Fin : N -> Set where
zero : {n : N} -> Fin (suc n)
suc  : {n : N} -> Fin n -> Fin (suc n)

Thus, Fin 2 is the type of binary digits, and Fin 10 is that of decimal digits.

Elliott gives us an observation of Fin in terms of Nat:

toN : {n : Nat} -> Fin n -> Nat
toN zero    = zero
toN (suc f) = suc (toN f)

We’d like to find a model-preserving implementation of + over Fin (let’s call it <+F>.) But what does model-preserving meaning? As usual, the answer is “homomorphism”, and namely, that the following square must commute:

<+> . bimap toN toN = toN . <+F>

The paper says “solving this equation for <+F> yields the following recursive definition.” It’s unclear if this answer was “solved for” in the sense of being manipulated algebraically, or if it just happens to be a solution to the problem. I hope it’s the former, because I would love to see how to do that, but I suspect it’s the latter. Anyway, the definition of _+F_ is given below, plus the work I needed to do to get everything to typecheck.

n+suc : (n m : Nat) -> n + suc m == suc (n + m)
n+suc zero m = refl
n+suc (suc n) m rewrite n+suc n m = refl

weaken :  {m} {n} (y : Fin n) -> Fin (m + n)
weaken {zero} y = y
weaken {suc m} zero = zero
weaken {suc m} {suc n} (suc y) rewrite n+suc m n = suc (weaken y)

infixl 5 _+F_
_+F_ : {m n : Nat} -> Fin (suc m) -> Fin n -> Fin (m + n)
_+F_ {m} zero y = weaken y
_+F_ {suc _} (suc x) y = suc (x +F y)

Something that ended up challenging me here was that Elliott uses extensional equality of functions for his commutative diagrams, but my implementation of arrow categories (coming up) requires a propositional equality. I got around the problem by postulating extensionality (which I stole from McBride), and then by defining a type-alias for extensional equality that plays well with extensionality:

postulate
extensionality : {S : Set}{T : S -> Set}
{f g : (x : S) -> T x} ->
((x : S) -> f x == g x) ->
f == g

infix 1 _=o=_
_=o=_ : {A B : Set} -> (A -> B) -> (A -> B) -> Set
_=o=_ {A} f g = (x : A) -> f x == g x

and then we can give most (it’s hard to prove things, OK?) of the proof for the commutative diagram:

toN-weaken : {m n : Nat} -> (y : Fin n) -> toN (weaken {m} y) == toN y
toN-weaken {zero} y = refl
toN-weaken {suc m} zero = refl
toN-weaken {suc m} (suc y) = ?

toN-+F :  {(m , n) : Nat2} -> toN {m + n} ∘ <+F> =o= <+> ∘ toN2 {suc m , n}
toN-+F {zero , n} (zero , y) = refl
toN-+F {suc m , n} (zero , y)
rewrite toN-+F {m , n} (zero , y)
| toN-weaken {suc m} y = refl
toN-+F {suc m , n} (suc x , y) rewrite toN-+F {m , n} (x , y) = refl

Section 3: The arrow category

Section 3 presents the fact that we can bundle up the commutative diagram with its proof into an object. Unfortunately, it only gives the barest hint about how to actually go about doing that. I’m presenting here what I think it is, but if something goes wrong in the rest of the paper, here’s where the problem is.

The paper is keen to point out that we have five things we’re bundling together:

1. A mapping from implementation input to specification input.
2. A mapping from implementation output to specification output.
3. An implementation.
4. A specification.
5. A proof that the square commutes.

Colloquially, I’d call the first two points our “observations” of the system.

The idea is, we’d like to bundle all of these things together, ideally in some sort of composable packaging. Composable usually means categorical, so we should look at our old friend SET, the category whose objects are types and arrows are functions. By itself, SET doesn’t give us any of the machinery we’d want for thinking about commutative squares. So instead, we’ll construct the arrow category over SET. Let’s call it >-SET->.

But what is the arrow category? The paper is quiet on this front, but my understanding is that it transforms the arrows in SET into objects in >-SET->. An arrow in >-SET-> is thus a pair of arrows in SET which form a commutative square. For example, consider the arrows in SET:

Fin n x Fin n        Nat x Nat
|                  |
| <+F>             | <+>
v                  v
Fin n               Nat

In >-SET->, these are just two objects, and a morphism between them is something that makes the whole square commute. In >-SET->:

bimap toNat toNat
<+F>  ------------------>  <+>
toNat

and again in SET:

bimap toNat toNat
Fin n x Fin n  ------------------> Nat x Nat
|                                |
| <+F>                           | <+>
v               toNat            v
Fin n      ------------------>    Nat

So we can consider the arrow category to be a “view” on its underlying category, like how databases present views over their data. The arrow category lets us talk about arrows directly, and ensures the commutativity of any constructions we’re able to form. As such, it’s a natural fit for our purposes of specification — we are literally unable to construct any arrows in >-SET-> which violate our specification.

Building Categories

How do we go about actually building an arrow category? First, some preliminaries to build a category:

record Category : Set where
infix 6 _~>_
field
Obj : Set
_~>_ : (A B : Obj) -> Set

id : {A : Obj} -> A ~> A
_>>_ : {A B C : Obj} -> A ~> B -> B ~> C -> A ~> C

id-l : {A B : Obj} (f : A ~> B) -> id >> f == f
id-r : {A B : Obj} (f : A ~> B) -> f >> id == f
>>-assoc : {A B C D : Obj}
(f : A ~> B)
-> (g : B ~> C)
-> (h : C ~> D)
-> f >> (g >> h) == (f >> g) >> h

And I have some syntactic sugar for dealing with arrows and composition in an arbitrary category:

infix 5 _[_,_]
_[_,_] : (C : Category) -> Obj C -> Obj C -> Set
C [ X , Y ] = _~>_ C X Y

infix 5 _[_>>_]
_[_>>_] : (C : Category)
-> {X Y Z : Obj C}
-> C [ X , Y ]
-> C [ Y , Z ]
-> C [ X , Z ]
C [ f >> g ] = _>>_ C f g

With this, we can describe an arrow in SET via SET [ A , B ], and composition via SET [ foo >> bar ].

Building arrow categories

An arrow category is parameterizd by the category whose arrows form its objects:

module ARROW (C : Category) where
open Category
open _=>_

record ArrowObj : Set where
constructor arrow-obj
field
{alpha} : Obj A
{beta}  : Obj B
hom : C [ alpha , beta ]

open ArrowObj

record ArrowArr (X Y : ArrowObj) : Set where
constructor arrow-hom
field
f : A [ X .alpha , Y .alpha ]
g : B [ X .beta , Y .beta ]
commute : C [ f >> Y .hom ] == C [ X .hom >> g ]

Arrow : Category
Obj Arrow = ArrowObj
_~>_ Arrow = ArrowArr
-- ...

My implementation for Arrow is actually in terms of the arrow category, which is the same idea except it does some functor stuff. In the code accompanying this post, ARROW {c} is implemented as COMMA {c} ID=> ID=> where ID=> is the identity functor.

Back to section 3

For convenience, the paper implicitly defines a type synonym for constructing arrows in the arrow category. This is called _⮆_ in the paper, but I hate unicode, so mine is defined as =>>:

infix 0 _=>>_
_=>>_ :  {s1 t1 : Set}
{s2 t2 : Set}
(s1 -> s2) -> (t1 -> t2) -> Set
g =>> h = ArrowArr (arrow-obj g) (arrow-obj h)

With all of this machinery in place, we’re now ready to continue on the paper. We can construct a morphism in the arrow category corresponding to the fact that <+> is a model for <+F>, as witnessed by toN-+F:

+=>> : {(m , n) : Nat × Nat} -> toN2 { suc m , n } =>> toN { m + n }
+=>> = arrow-hom <+F> <+> $extensionality toN-+F Again, the necessity of extensionality here is a byproduct of me not having parameterized my Category by a notion of equivalence. The arrow category wants to use extensional equality, but I’ve hard-baked in propositional equality, and didn’t have time to fix it before my deadline to get this post out. Although not presented in the paper, arrow categories also have a notion of “transposition,” corresponding to flipping which arrows (in SET) lay on the horizontal and vertical axes. Because _=>>_ names two arrows, leaving the other two implicit, transposition changes the type of our arrows — making the implicit ones explicit and vice versa. We’ll need this later in section 7. transpose : {A B : CommaObj} ((comma-hom f g _) : CommaArr A B) -> f =>> g transpose {comma-obj h} {comma-obj h'} (comma-hom f g p) = comma-hom h h' (sym p) As an aside, it’s super cool that Agda can do this sort of pattern matching in a type signature. Section 4: Carry-in In order to make non-unary adders compositional, we need to support carrying-in. The play is familiar. Define what we want (the specifiation) over the naturals, write it over Fin, and then give an equivalence proof. The paper defines some type synonyms: Nat3 : Set Nat3 = Nat × Nat2 Fin3 : Nat3 -> Set Fin3 (m , n) = Fin m × Fin2 n toN3 : {(c , m , n) : Nat3} -> Fin3 (c , m , n) -> Nat3 toN3 (c , mn) = toN c , toN2 mn (where Nat2 and Fin2 are exactly what you’d expect.) It’s easy to define the specification (addN), and implementation (addF), and the proof is trivial too: addN : Nat3 -> Nat addN (c , a , b) = c + a + b addF : {(m , n) : Nat2} -> Fin3 (2 , m , n) -> Fin (m + n) addF (c , a , b) = c +F a +F b toN-addF : {mn@(m , n) : Nat2} SET [ addF >> toN ] =o= SET [ toN3 >> addN ] toN-addF {mn} (c , a , b) rewrite toN-+F {mn} (c +F a , b) | toN-+F (c , a) = refl Bundling these up into an arrow proves that addN is a model for addF: add=>>0 : {(m , n) : Nat2} -> toN3 {2 , m , n} =>> toN {m + n} add=>>0 = comma-hom addF addN$ extensionality toN-addF

The paper also makes clear that we can show that <+> is a model for addN via carryIn:

carryIn : Nat3 -> Nat2
carryIn (c , a , b) = c + a , b

addN=>> : carryIn =>> id {A = Nat}

Commentary

At this point, it’s starting to become clear what this paper is really about. The idea is that we specify super simple pieces, and then build slightly more complicated things, showing equivalence to the last piece we built. In this way, we can slowly derive complexity. Not only does it give us a fool-proof means of getting results, but it also means we can reuse the proof work. As someone whose first real project in Agda was to implement and prove the correctness of a few adders, this is a godsend. I wrote a ripple-carry adder, but was unable to use my half-adder proof to prove it correctly implements addition. And then I had to throw all of that work away when I wanted to subsequently write and prove a carry-forward adder.

Section 5: Category stuff

This section shows we went through too much effort to implement add=>>0. Really what’s going on here is we’re doing three things in a row, for the specification, implementation and proof. First, we’re reassociating the tuple, from N x (N x N) to (N x N) x N. Then we’re doing addition on the first element of the pair, and then doing addition on the resulting pair.

This is all stuff you can do in any category with all finite products. But I was too lazy to implement that in full generality, so I hard-coded it. Because arrow categories lift products, and in SET products are just the product type, it’s easy to implement categorical objects:

_×C_ : Obj Comma -> Obj Comma -> Obj Comma
_×C_ (comma-obj {A1} {B1} f) (comma-obj {A2} {B2} g) =
comma-obj {A1 × A2} {B1 × B2} λ { (x , y)  (f x , g y) }

And then we can implement first and assoc:

first :  {A B X : Obj Comma} -> Comma [ A , B ] -> Comma [ (A ×C X) , (B ×C X) ]
first {A} {B} {X} (comma-hom f g p) =
comma-hom (do-first f) (do-first g) $cong (\k (a , x) -> k a , xf x) p where do-first : {A B X : Set} -> (A -> B) -> A × X -> B × X do-first f = { (a , x) f a , x }) assoc : {A B X : Obj Comma} -> Comma [ A ×C (B ×C X) , (A ×C B) ×C X ] assoc = comma-hom reassoc reassoc ? where reassoc : {A B C : Set} -> A × (B × C) -> (A × B) × C reassoc = { (a , b , c) (a , b) , c }) where the proof is left as an exercise to the reader :) We can now implement add=>> more succinctly: add=>> : {(m , n) : Nat2} -> toN3 {2 , m , n} =>> toN {m + n} add=>> = Comma [ Comma [ assoc >> first +=>> ] >> +=>> ] While this is cool, I must admit I’m a little confused. Do first and assoc have most-general types when expressd via _=>>_? Thinking aloud here, I think not. Using the _=>>_ notation is for choosing two particular morphisms in SET, while using the more general COMMA [ X , Y ] is for any pair morphisms in SET with the right type. But I’m not confident about this. Section 6: Carrying out Carry-in is great, but what about going the other direction? Elliott starts by makig the following observation: if we fix m = n, then the type of our finitary adder is Fin2 (m , m) -> Fin (m + m), which we can rewrite the codomain as Fin (2 * m) and then reinterpret as Fin 2 x Fin m. That is to say, the type of finitary adding is to output a single digit in base m, and a bit corresponding to whether or not a carry occurred. This is a great little reminder in the value of type isomorphisms. How cool is it that we can get carry-outs for free just with some algebraic manipulations? Of course, the trick is to prove it. Start by defining two helper type synonyms: CarryIn : Nat -> Set CarryIn m = Fin3 (2 , m , m) CarryOut : Nat -> Set CarryOut m = Fin2 (2 , m) Elliott presents the following “puzzle” of a commutative diagram: addF CarryIn m --------> Fin (m + m) ^ ^ id | | ? | ? | CarryIn m --------> Fin (2 * m) ^ ^ id | | ? | ? | CarryIn m --------> CarryOut m It’s unclear how exactly one formulates these diagrams in the first place. I guess the top is pinned by addF, while the bottom corners are pinned by our desired carrying out. The middle is thus the isomorphism presented immediately before this. All of that makes sense, but I’m not convinced I could reproduce it on my own yet. So the game now is to fill in the question marks. I don’t know how to get Agda to help me with this, so I’m just going to try literally putting question marks in and seeing what happens. When doing that, we get this: puzzle1 : {m : Nat} -> Comma [ comma-obj {CarryIn m} {CarryIn m} id , comma-obj {Fin (2 * m)} {Fin (m + m)} ? ] puzzle1 = comma-hom ? addF ? Figuring out the first question-mark is simple enough, it’s an isomorphism on Fin: n+zero : (n : Nat) -> n + zero == n n+zero zero = refl n+zero (suc n) rewrite n+zero n = refl 2*m==m+m : (m : Nat) -> (2 * m) == m + m 2*m==m+m zero = refl 2*m==m+m (suc m) rewrite 2*m==m+m m | n+zero m = refl castF : {m n : Nat} -> (m == n) -> Fin m -> Fin n castF p rewrite p = id Our first hole is thus cast$ 2*m==m+m m. Interestingly, this doesn’t refine our other hole, since it’s already fully specified by the vertial components of idA and the horizontal component of addF. However, as the paper points out, we can get the second hole for free. Because cast is invertable, we can make this square commute by taking id >> addF >> cast-1. It feels a bit like cheating, but it does indeed satisfy the commutativity diagram:

puzzle1 : {m : Nat}
-> Comma [ comma-obj {CarryIn m}   {CarryIn m}   id
, comma-obj {Fin (2 * m)} {Fin (m + m)} (cast $2*m==m+m m) ] puzzle1 {m} = comma-hom (SET [ addF >> cast$ sym $2*m==m+m m ]) addF ? where the proof is trivial (but I don’t know how to make it terse): SET [ SET [ addF >> cast (sym (2*m==m+m m)) ] >> cast$ 2*m==m+m m ]
== SET [ id >> addF ]

Probably there is a category of proofs, where I can just do reassoc >> second sym (sym-is-id $2*m==m+m m) >> id-r addF >> sym (id-l addF). But I don’t have that setup, and this would be annoying to do in the equational reasoning style. So it remains a hole, and you, gentle reader, can fill it in if you are keen. Also, I’d love to know how to write a proof as simple as my sketch above. So that gives us the first half of our puzzle. Now that we have the middle arrow, let’s play the same game: addF CarryIn m -------------------------------> Fin (m + m) ^ ^ id | | cast$ 2*m==m+m m
|                                           |
|          addF >> cast (sym (2*m==m+m m))  |
CarryIn m   -------------------------------> Fin (2 * m)
^                                           ^
id |                                           | ?
|                       ?                   |
CarryIn m   -----------------------------> CarryOut m

So how do we turn a CarryOut m = Fin2 (2 , m) into a Fin (2 * m)? Algebraically I think this is a bit tricky, but thankfully Data.Fin.Base has us covered:

combine :  {n k}  Fin n  Fin k  Fin (n * k)

which we can uncurry:

comb : {m n : Nat} -> Fin2 (m , n) -> Fin (m * n)
comb (f1 , f2) = combine f1 f2

and then use this to fill in the vertical arrow. Because comb is one half of an isomorphism (the other half is formed by remQuot : ∀ {n} k → Fin (n * k) → Fin n × Fin k), we can do the same trick to get the horizontal arrow for free:

puzzle2 : {m : Nat}
-> Comma [ comma-obj {CarryIn m}  {CarryIn m}   id
, comma-obj {CarryOut m} {Fin (2 * m)} comb
]
puzzle2 {m} =
comma-hom
(SET [ SET [ addF >> cast (sym (2*m==m+m m)) ] >> remQuot m ])
(SET [ addF >> cast (sym (2*m==m+m m)) ])
?

The proof is again trivial but verbose.

Let’s call the implementation arrow addCarry, because we’ll need it in section 8.

addCarry : {m : Nat} -> SET [ CarryIn m , CarryOut m ]
SET [ SET [ addF >> cast (sym (2*m==m+m m)) ] >> remQuot m ]

Section 7: Vertical composition

Finally, we can use vertical composition to combine our two puzzles:

puzzle : {m : Nat} -> id =>> (cast (2*m==m+m m) ∘ comb {2} {m})
puzzle = transpose $Comma [ transpose puzzle2 >> transpose puzzle1 ] using our transpose machinery from earlier. Vertical composition composes along the axis of specification — if the implementation of one arrow matches the specification another, we can combine them into one. Section 8: Moving away from unary representations Unary sucks. Let’s generalize our adding machinery to any arbitrary type. First we’ll make types corresponding to CarryIn and CarryOut: DIn : Set -> Set DIn t = Bool × t × t DOut : Set -> Set DOut t = t × Bool I’m going to go rogue for a while, and try to do this section without referencing the paper. We want to make a morphism in the arrow category corresponding to using addCarry as the specification for addition over DIn and DOut. Let’s play the same puzzle game, and set up a commutative diagram. At the top is our specification, addCarry : CarryIn m -> CarryOut m. That then pins our top two objects, and obviously our bottom two are DIn t and DOut t: addCarry CarryIn m ---------> CarryOut m ^ ^ bval x (μ x μ) | | μ x bval | addD | DIn t -------------> DOut t where bval : Bool -> Fin 2. Here, μ plays the part of toNat, and addD is addition over D t-indexed numbers. We can package this up into a record indexed by μ: record Adder {t : Set} {m : Nat} (μ : t -> Fin m) : Set where constructor _-|_ field addD : DIn t -> DOut t is-adder : SET [ addD >> bimap μ bval ] == SET [ bimap bval (bimap μ μ) >> addCarry ] and trivially construct the desired commutative diagram from an Adder: Adder=>> : {t : Set} {m : Nat} {μ : t -> Fin m} -> Adder μ -> bimap bval (bimap μ μ) =>> bimap μ bval Adder=>> (addD -| proof) = comma-hom addD addCarry proof So let’s implement a full-adder. This is a “well-known” result, but I didn’t know it offhand. I’m sure I could have sussed this out on my own, but instead I just found it on Wikipedia: and : Bool -> Bool -> Bool and true b = b and false b = false or : Bool -> Bool -> Bool or true b = true or false b = b xor : Bool -> Bool -> Bool xor true true = false xor true false = true xor false true = true xor false false = false full-add : DIn Bool -> DOut Bool full-add (cin , a , b) = let o = xor a b in xor o cin , or (and a b) (and o cin) We can construct an Adder for full-add with observation bval by case-bashing our way through the proof: BitAdder : Adder bval BitAdder = full-add -| extensionality \ { (false , false , false) -> refl ; (false , false , true) -> refl ; (false , true , false) -> refl ; (false , true , true) -> refl ; (true , false , false) -> refl ; (true , false , true) -> refl ; (true , true , false) -> refl ; (true , true , true) -> refl } The next step is obviously to figure out how to compose Adders — ideally to construct adders for vectors. But I don’t know how to do this offhand. So it’s time to look back at the paper. OK, right. So we have an obvious tensor over μ, which is just to lift two of them over a pair: tensorμ : {tm tn : Set} -> {m n : Nat} -> (tm -> Fin m) -> (tn -> Fin n) -> (tm × tn -> Fin (n * m)) tensorμ μm μn (tm , tn) = comb$ μn tn , μm tm

Likewise, we have one over the adding functions themselves, pushing the carry-out of one into the carry-in of the next:

: {m n : Set}
-> (DIn m -> DOut m)
-> (DIn n -> DOut n)
-> (DIn (m × n) -> DOut (m × n))
let (m , cin') = addm $cin , ma , mb (n , cout) = addn$ cin' , na , nb
in (m , n) , cout

Allegedly these form a true adder as witnessed by Adder (tensorμ μm μn), but the proof isn’t included in the paper, and I had a hard time tracking it down in the source code. So rather than taking this by fiat, let’s see if we can convince ourselves.

As a first attempt of convincing myself, I tried to construct adder22 : Adder (tensorμ bval bval) which is a tensor of two full-adders. I constructed a case bash for the proof, and Agda complained! After some sleuthing, I had missed a swap somewhere in the paper, and thus had my carry bit in the wrong place in full-adder.

After sorting that out, the case bash works on adder22. So that’s a good sanity check that this works as promised. But, why? Presuably I should be able to run my commutative diagram all the way to its specification to debug what’s going on.

A few hours later…

I came up with the following, which can run a commutative diagram:

arrowOf : {A B : CommaObj} -> CommaArr A B -> CommaObj × CommaObj
arrowOf {A} {B} _ = A , B

debug : {A B C D : Set} -> {f : A -> B} -> {g : C -> D} -> f =>> g -> A -> D
debug arr x =
let (_ , comma-obj y) = arrowOf arr
(comma-hom f _ _) = arr
in y (f x)

Of course, the diagrams we get from Adder=>> only get us as far as addCarry. In order to get all the way to Nats, we need to vertically compose a bunch of other diagrams too. In order, they’re puzzle, addF=>> and addN=>>. The actual diagram I came up with was this attrocious thing:

: Cat2.CommaArr.f
(Comma [
>> transpose addN=>> ]) =>> Cat2.CommaArr.g
(Comma [
>> transpose puzzle ]
Adder=>>N = transpose $Comma [ Comma [ Comma [ transpose (Adder=>> adder22) >> transpose puzzle ] >> transpose addF=>> ] >> transpose addN=>> ] and finally, I can evaluate the thing: debug' : Nat debug' = debug Adder=>>N (false , (true , false) , (true , false)) Nice. OK, so back to answering the question. Each of the Bool x Bool tuples is a little-endian vector, which get added together, plus the carry. In the process of doing all of this work, I inadvertantly figured out how the tensoring works. What’s more interesting is tensoring together two different adders. For example, the trivial-add (section 8.3): data One : Set where one : One oneval : One -> Fin 1 oneval one = zero trivial-add : DIn One -> DOut One trivial-add (b , _ , _) = one , b If we construct tensorAdder trivial-add BitAdder, we get an adder whose vectors are One x Bool. This is an extremely interesting representation, as it means our number system doesn’t need to have the same base for each digit. In fact, that’s where the compositionality comes from. We’re pretty comfortable assigning 2^i to each digit position, but this representation makes it clear that there’s nothing stopping us from choosing arbitrary numbers. What’s really doing the work here is our old friend comb : {m n : Nat} -> Fin2 (m , n) -> Fin (n * m). Expanding the type synonym makes it a little clearer comb : {m n : Nat} -> Fin m x Fin n -> Fin (n * m) — this thing is literally multiplying two finite numbers into one! Looking at tensorAdd under this new lens makes it clearer too. Recall: tensorAdd addm addn (cin , (ma , na) , (mb , nb)) = let (m , cin') = addm$ cin  , ma , mb
(n , cout) = addn \$ cin' , na , nb
in (m , n) , cout

Here we’re pointwise adding the digits, where m is the least significant of the two digits. Our carry-in goes into the m, and its carry-out goes into n. OK, so this thing is just doing adder-chaining.

Section 8.4 talks about extending this to vectors, but the trick is just to fold them via tensorAdd. The paper uses a right-fold. I’m curious about what happens if you do a left fold, but might circle back around to that question since I only have a few more hours to get this post out and I want to spend some time in Mexico while I’m here. Upon deeper thought, I don’t think anythig would change — we’d still get a ripple carry adder. Worth playing around with though.

Section 9: Speculation

Section 9 is the most exciting part of the paper in my eyes. It defines speculation, which Elliott uses to implement a carry-ahead adder. I think the paper cheats a bit in this section — and makes it clear that we might have been cheating a bit earlier too. But first some preliminaries. The paper defines speculate:

speculate : {A C : Set} -> (Bool × A -> C) -> (Bool × A -> C)
speculate f (b , a) = if b then f (true , a) else f (false , a)

This looks like it should be a no-op, and indeed, there’s a trivial proof that speculate f =o= f. The trick then is to lift speculate over an adder:

spec
: {t : Set} {m : Nat} {μ : t -> Fin m}

and the claim is that if we now do the same vector fold over spec a instead of a, we will get a carry-ahead adder! Sorcery! Magic!

But also… wat? Is that actually true?

I think here is where the paper is playing fast and loose. In SET, speculate a is exactly a. But the paper shows us a circuit diagram for this speculated fold, and it does indeed show the right behavior. So what’s going on?

What’s going on is that the paper isn’t actually working in SET. Well, it is. Sorta. In fact, I think there’s some fancy-pants compiling-to-categories going on here. In the same way that xor presented above is a SET-equivalent version of the xor operation in the CIRCUIT category (but is not actually xor in CIRCUIT), if_then_else_ is actually the SET version of an equivalent operation in CIRCUIT. In CIRCUIT, if_then_else_ is actually implemented as inlining both its true and false branches, and switching between them by anding their outputs against the conditional.

So, the carry-ahead adder is not nearly as simple as it’s presented in this paper. There’s a huge amount of work going on behind the scenes:

1. defining the CIRCUIT category
2. implementing if_then_else_ in CIRCUIT
3. showing that the arrow category lifts if_then_else_

Furthermore, I’m not exactly sure how this all works. Like, when we define speculate as if b then f (true , a) else f (false , a), are we literally inlining f with the conditional fixed, and simplfying the resulting circuit? I mean, we could just fix true by tying it to HIGH, but the diagrams included in the paper don’t appear to do that. If so, who is responsible for simplfying? Does it matter? This is a big hole in the paper, and in my opinion, greatly diminishes its impact, since it’s the claim I was most excited about.

To the paper’s credit, the vector fold and speculative fold turn into nice combinators, and give us a little language for building adders, which is extremely cool.

Section 10: Reusing circuitry over time

Ripple-carry adders are slow but use few gates. Carry-ahead adders are much faster, but use asymptotically more gates. Clearly there is a tradeoff here between latency and manufacturing cost. Section 10 gives us another point in the design space where we just build one full-adder, and loop it into itself. This also sounds exciting, but I’m a bit wary after section 9.

And as presented, I don’t think I trust the paper to deliver on this front. There is some finagling, but at it’s core, we are given a looping construct:

loop
: {A B S : Set}
-> {n : Nat}
-> (S × A -> B × S)
-> S × Vec A n
-> Vec B n × S
loop f (s , nil) = nil , s
loop f (s , cons a v) =
let b , s' = f (s , a)
in bimap (cons b) id (loop f (s' , v))

Thinking about what this would mean in CIRCUIT makes it unclear how we would go about implementing such a thing in real hardware — especially so if the embedding sticks f into the hardware and then loops over it over time. You’re going to need some sort of ring buffer to read off the outputs and stick them in the resulting vector. You’re going to need timing signals to know when your ring buffer is consistent. There’s clearly a lot going on in this section that is left unsaid, and there aren’t even any pretty pictures to help us reverse engineer the missing bits.

So I’m going to leave it there.

Conclusion

Adders and Arrows was a fun paper to go through. It forced me to up my category game, and I got a much better sense of what the arrow category does, and how it can be useful. Furthermore, just going through a non-trivial project aggressively improved my Agda competencies, and I’m excited to do more along these lines.

The paper itself is a bit too terse for my liking. I would have liked a section saying “here’s what we’re going to do, and here’s how we’re going to go about it,” rather than just being thrown in and trying to deduce this for myself. In particular, it took me an embarassing amount of time to realize how to get natural numbers out of my adder arrows, and why the first 6 sections were worth having done.

Technically, I found the ergonomics of working with arrow-category arrows very challenging. Two of the SET morphisms show up in the type, but the other two show up as values, and there is no easy way to see which diagrams can be vertically composed. My Adder=>>N arrow abve shows the pain of trying to give a type to such a thing.

I had two major points of complaint about this paper. The first is that the source code isn’t very accessible. It exists in a repo, but is scattered around a bunch of modules and whenever I wanted to find something I resorted to just looking at each — being unable to make rhyme or reason of how things were organized. Worse, a huge chunk of the underlying machinery is in a separate repo, one which is significantly more advanced in its Agda usage than I am. A proliferation of weird unicode symbols that aren’t the ones that show up in the PDF make this especially challenging to navigate.

My other major complaint is that sections 9 and 10 were extremely underwhelming, though. If the paper does what it promises, it doesn’t actually show how. There is a lot going on behind the scenes that aren’t even alluded to in the paper. Granted, the version I’m reading is a draft, so hopefully this will be cleared up.

I don’t yet have a major takeaway from this paper, other than that arrow categories are cool for specifying problems and proving that your implementations adhere to those specifications. But as implemented, for my given adeptness at Agda, they are too hard to use. Composition is tricky to wrap ones head around given the type signatures used in this paper, but hopefully that’s an aesthetic problem more than a fundamental issue. In particular, tranpose needs to have type CommaArr A B C D -> CommaArr C D A B — this would make vertical and horizontal composition much easier to think about.

All in all, powering through this paper has given me some new tools for thought, and helped me see how category theory might be useful to mere mortals.

My implementation of this code is available on Github.