If a "problem we care about" is not stated as a formal language, does it mean it does not exist in the hierarchy of formal languages? Or is it just as yet unclassified?
It means that there are two problems: one, to formalize the problem as stated while capturing all relevant details, and two, solving the resulting formal problem. Until you solve problem one, you can't use formal methods to say anything about the problem (it's not even clear a priori that a problem is even solvable).
Unfortunately, the task of a formalizing an informal problem is itself an informal problem that we don't know how to formalize, so we can't say much about it. So overall, we can't say much about how hard the general problem "given a problem statement from a human, solve that problem" is, whether any particular system (including a human!) can solve it and how long that might take with what resources.
My 2 cents: Since LLMs (Large Language Models) operate as at least a subset of Turing machines (which recognize recursively enumerable languages), the chain of thought (CoT) approach could be equivalent to or even more expressive than that subset. In fact, CoT could perfectly be a Turing machine.
If we leave CoT aside for a moment, it's worth exploring the work discussed in the paper "Neural Networks and the Chomsky Hierarchy"[1], which analyzes how neural networks (including LLMs) map onto different levels of the Chomsky hierarchy, with a particular focus on their ability to recognize formal languages across varying complexity.
tsimionescu|1 year ago
Unfortunately, the task of a formalizing an informal problem is itself an informal problem that we don't know how to formalize, so we can't say much about it. So overall, we can't say much about how hard the general problem "given a problem statement from a human, solve that problem" is, whether any particular system (including a human!) can solve it and how long that might take with what resources.
viraptor|1 year ago
I couldn't find details about this - do you know of a paper or some resource which digs into that idea?
wslh|1 year ago
If we leave CoT aside for a moment, it's worth exploring the work discussed in the paper "Neural Networks and the Chomsky Hierarchy"[1], which analyzes how neural networks (including LLMs) map onto different levels of the Chomsky hierarchy, with a particular focus on their ability to recognize formal languages across varying complexity.
[1] https://ar5iv.labs.arxiv.org/html/2207.02098v1
flir|1 year ago
Are we going to need an infinite number of LLMs, arranged on a tape?