For example, wake-sleep learning and NN on a Lisp-like language [1] or synthesis of Haskell guided by refinement types [2].
[1] https://royalsocietypublishing.org/doi/full/10.1098/rsta.202....
While this technique enjoys certain advantages, i.e., it is embarrassingly parallelizable and guaranteed to enumerate distinct solutions with a bounded delay, it also somewhat unnatural. By flattening the distribution onto the integers a la Gödel numbering, it destroys locality, does not play well with incremental decoding methods (left-to-right is currently en vogue in generative language modeling), and will fail if the sample space is uncountable.
Another key step is reducing symmetries in your sample space by quotienting it somehow (e.g., by α-equivalence). The author seems to be invoking some kind of equivalence relation by “superposition”, but the technical details here are a little fuzzy.
This problem is also closely related to model counting in the CSP literature, so a practical speedup could lead to improvements on a lot of interesting downstream benchmarks.
In general, the problem of program induction from input-output examples is not well-posed, so specialized solvers that can make stronger assumptions will usually have an advantage on domain-specific benchmarks. Most existing program synthesizers do not satisfy all of these desiderata (e.g., soundness, completeness, naturalness, incrementality).
Right! A great way to do this is to learn a program by using it to prove the training examples while it is being learned. A very cool ability that some systems of the new wave of Inductive Logic Programming can pull off, but probably nothing else can far as I can tell.
That's Inductive Functional Programming (IFP), a kind of Inductive Programming that also includes Inductive Logic Programming (ILP). The canonical example of IFP is Magic Haskeller:
https://nautilus.cs.miyazaki-u.ac.jp/~skata/MagicHaskeller.h...
As an example of a modern ILP I suggest Popper:
https://github.com/logic-and-learning-lab/Popper/
Or Louise (mine):
https://github.com/stassa/louise
One of the DreamCoder papers describes Inductive Programming as a form of weakly supervised learning, in the sense that such systems learn to generate programs not from examples of programs, but from examples of the target programs' beuav908rs, i.e. their inputs and outputs. By contrast LLMs or slightly older neural program synthesis systems are trained on examples that consist of pairs of (programming-task, program-solving-the-task).
Another way to see the difference between Inductive Programming systems and conventional machine learning systems used for program synthesis is that Inductive Programming systems learn by solving problems rather than from observing solutions.
The advantage is that, in this way, we can learn programs that we don't know how to write (because we don't have to generate examples of such programs) whereas with conventional machine learning we can only generate programs like the ones the system's been trained on before.
Another advantage is that it's much easier to generate examples. For instance, if I want to learn a program that reverses a list, I give some examples of lists and their reverse, e.g. reverse([a,b,c],[c,b,a]) and reverse([1,2,3],[3,2,1]) whereas e.g. an LLM must be trained on explicit examples of list-reversing programs; like, their source code.
IFP and ILP systems are also very sample efficient, so they only need a handful of examples, often just one, whereas neural net-based systems may need millions (no exaggeration- can give a ref if needed).
The disadvantage is that learning a program usually (but not always - see Louise, above) implies searching a very large combinatorial space and that can get very expensive, very, very fast. But, there are ways around that.
It is a very interesting field which I hope makes a comeback once systems become neurosymbolic.
I'm one of the last purely symbolic hold-outs in ILP I guess. But if you're interested in recent neurosymbolic work using MIL this is some recent work from my colleagues:
Abductive Knowledge Induction from Raw Data
What does this mean?
IMHO, this might eventually become mainstream. For example, see all the work that merges theorem provers with RL and NN.