Not picking on the OA in the slightest; just thinking in terms of holistic system design. If you know what you want to happen, and you are smart enough to introspect the behavior of the tool and decide that it didnt happen, you are more than smart enough to just write it correctly in the first place.

Perhaps that is unrealistic, perhaps there is a hidden iceberg of necessary but convolutive optimizations no human could realistically or legibly write. But ok, where do you really need to engage in this kind of optimization golf? Inlined functions?

Ok, what about this targeted language feature for a future-day Zig:

1. Write an ordinary zig function 2. Write inline assembly version of that function 3. Write a "comptime assert" that first compiles to second, which only "runs" for the relevant arch. 4. What should that assert mean? That the compiler just uses your assembly version instead, but _also_ uses existing compiler machinery or an external theorem prover to verify they "behave the same up to X", for customizable values of X

That has the right feel, maybe. You are "pinning" specific, vetted, optimizations without compromising the intent, readability, or correctness of your code. And easy iteration is possible, because a failing comptime assert will just dump the assembly; you can even start with an empty manual impl.

That you can profitably loop some say 3-layer stack is likely a happy accident, where the performance loss from looping 3/4 of mystery circuit X that partially overlaps that stack is more than outweighed by the performance gain from looping 3/3 of mystery circuit Y that exactly aligns with that stack.

So, if you are willing to train from scratch, just build the looping in during training and let each circuit find its place, in disentangled stacks of various depths. Middle of transformer is:

(X₁)ᴹ ⊕ (Y₁∘Y₂)ᴺ ⊕ (Z₁∘Z₂∘Z₃)ᴾ ⊕ …

Notation: Xᵢ is a layer (of very small width) in a circuit of depth 1..i..D, ⊕ is parallel composition (which sums the width up to rest of transformer), ∘ is serial composition (stacking), and ᴹ is looping. The values of ᴹ shouldnt matter as long as they are > 1, the point is to crank them up after training.

Ablating these individual circuits will tell you whether you needed them at all, but also roughly what they were for in the first place, which would be very interesting.

But, fascinatingly, integration does in fact have a meaning. First, recall from the OP that d/dX List(X) = List(X) * List(X). You punched a hole in a list and you got two lists: the list to the left of the hole and the list to the right of the hole.

Ok, so now define CrazyList(X) to be the anti-derivative of one list: d/dX CrazyList(X) = List(X). Then notice that punching a hole in a cyclic list does not cause it to fall apart into two lists, since the list to the left and to the right are the same list. CrazyList = CyclicList! Aka a ring buffer.

There's a paper on this, apologies I can't find it right now. Maybe Alternkirch or a student of his.

The true extent of this goes far beyond anything I imagined, this is really only the tip of a vast iceberg.

I explored a similar idea once (also implemented in Python, via decorators) to help speed up some neuroscience research that involved a lot of hyperparameter sweeps. It's named after a Borges story about a man cursed to remember everything: https://github.com/taliesinb/funes

Maybe one day we'll have a global version of this, where all non-private computations are cached on a global distributed store somehow via content-based hashing.

* dynamic typing vs static typing, a continuum that JIT-ing and compiling attack from either end -- in some sense dynamically typed programs are ALSO statically typed -- with all function types are being dependent function types and all value types being sum types. After all, a term of a dependent sum, a dependent pair, is just a boxed value.

* monomorphisation vs polymorphism-via-vtables/interfaces/protocols, which trade roughly speaking instruction cache density for data cache density

* RC vs GC vs heap allocation via compiler-assisted proof of memory ownership relationships of how this is supposed to happen

* privileging the stack and instruction pointer rather than making this kind of transient program state a first-class data structure like any other, to enable implementing your own co-routines and whatever else. an analogous situation: Zig deciding that memory allocation should NOT be so privileged as to be an "invisible facility" one assumes is global.

* privileging pointers themselves as a global type constructor rather than as typeclasses. we could have pointer-using functions that transparently monomorphize in more efficient ways when you happen to know how many items you need and how they can be accessed, owned, allocated, and de-allocated. global heap pointers waste so much space.

Instead, one would have code for which it makes more or less sense to spend time optimizing in ways that privilege memory usage, execution efficiency, instruction density, clarity of denotational semantics, etc, etc, etc.

Currently, we have these weird siloed ways of doing certain kinds of privileging in certain languages with rather arbitrary boundaries for how far you can go. I hope one day we have languages that just dissolve all of this decision making and engineering into universal facilities in which the language can be anything you need it to be -- it's just a neutral substrate for expressing computation and how you want to produce machine artifacts that can be run in various ways.

Presumably a future language like this, if it ever exists, would descend from one of today's proof assistants.

We're trying to apply the insights of category theory, dependent type theory, and functional programming to deep learning. How do we best equip neural nets with strong inductive biases from these fields to help them reason in a structured way? Our upcoming ICML paper gives some flavor https://arxiv.org/abs/2402.15332 ; you can also watch https://www.youtube.com/watch?v=rie-9AEhYdY&t=387s ; but there is a lot more to say.

If you are fluent in 2 or more of { category theory, Haskell (/Idris/Agda/...), deep learning }, you'll probably have a lot of fun with us!

Check out our open positions at https://jobs.gusto.com/boards/symbolica-ai-67195a74-31b4-405...

But I do think it's fair to say they are in their infancy, and there is a missing theoretical framework to explain what is going on.

I anticipate name-free array programming will eventually be considered a historical curiosity for most purposes, and everyone will wonder how we put up without it for so long.

It's almost like when we moved from writing machine code to structured programming: instead of being content to document that certain registers corresponding to certain semantically meaningful quantities at particular times during program execution, we manipulated only named variables and left compilers to figure out how to allocate these to registers? We're at that point now with array programming.

https://nlp.seas.harvard.edu/NamedTensor

https://math.tali.link/rainbow-array-algebra/

https://arxiv.org/abs/2102.13196