>> I wouldn't. Traditional NNs are widely called funciton approximators, or function induction. That's because they're not Turing-complete, and in this case don't even have memory. This makes them decidedly not programs, just functions.
But what the article describes is not function approximation anymore. Their system doesn't optimise a set of parameters. The search for an optimal set of parameters has been replaced entirely with a search for an optimal architecture with a single, shared parameter.
Note also that you don't need something to be Turing-complete for it to be a program. For instance, a regular automaton is a program, but it's not Turing complete. Turing completeness makes something, well, a Universal Turing Machine, which can compute any program. But a program is just a set of instructions.
> Note also that you don't need something to be Turing-complete for it to be a program. For instance, a regular automaton is a program, but it's not Turing complete. Turing completeness makes something, well, a Universal Turing Machine, which can compute any program. But a program is just a set of instructions.
A regular automaton is a program (and a function is a program), but all programs cannot be expressed as finite (regular) automatons (FAs). So FAs are a subset of programs. If we equate programs with mathematical algorithms, in general they really need the unlimited memory aspect (or at least the capability of abstracting memory, which in practice is always finite of course).
To make things clear, an example: a fixed function (or FA) cannot sort a list of numbers of variable size. You can train your function to sort any K numbers, which are the input variables. But because it has no memory, you cannot input variables sequentially (in the FA case you cannot input arbitrarily many variables). You've essentially created a sorting network[1].
Not only that, but for each input growth you need to re-train your architecture. The nice thing about program inference is that it hopefully captures the fundamental algorithm behind a program, which generalizes to arbitrary sizes. It's not that arbitrary sizes are necessary in practice (again computers are always bounded), it's that this abstraction/generalization is extremely useful, saving data and allowing large (arbitrary) variation in input.
> a Universal Turing Machine, which can compute any program
I feel there's a bit of confusion here. Turing Machines themselves can compute any program (i.e. you can represent any program as a TM) -- that follows essentially from the definition of algorithm (a set of well defined steps given unlimited paper); again languages accepted by FAs are a restricted subset of TMs (the former accepts a regular language while the latter recursively enumerable languages). There are special Turing Machines that simulate arbitrary Turing Machines -- those are called Universal Turing machines. It's a desirable characteristic of any computer, of course (the ability to input arbitrary programs).
We refer to program description languages that can describe any Turing Machine as Turing-complete. You could make a Universal Turing machine that accepts this kind of language directly or you could in principle just build a Turing machine that executes the given algorithm (for example using a gate array/FPGA to encode the state automaton, plus some large memory for the tape).
Yes, sloppy definition of Turing completenes on my part. I apologise unrservedly.
But, I still don't understand your objection. You agree that a regular automaton is a program, and that it's not Turing complete. So a program does not need to be Turing complete.
I didn't say, nor do I think, that the algorithm described in this article is Turing complete. I think it perfoms program induction. That's as far as I went. I still don't see where Turing completeness comes into it.
But what the article describes is not function approximation anymore. Their system doesn't optimise a set of parameters. The search for an optimal set of parameters has been replaced entirely with a search for an optimal architecture with a single, shared parameter.
Note also that you don't need something to be Turing-complete for it to be a program. For instance, a regular automaton is a program, but it's not Turing complete. Turing completeness makes something, well, a Universal Turing Machine, which can compute any program. But a program is just a set of instructions.