In the case of DeepSeek-R1, they used a series of heuristic reward functions that were built for different data types. The paper mentions the use of sandboxed environments to execute generated code against a suite of tests, for example, to evaluate it for correctness. The reward functions also evaluated syntax and formatting.

In general, the use of externally verifiable sources of truth (like simulators) is referred to as "grounding" and there has been quite a bit of research around it over the years, if you're interested in digging deeper. I've always found it super compelling as a research direction.

I think it just means that you can objectively score an answer as being correct or not. (e.g. if the generated program passes some tests; a discovered proof is valid, etc).

As in there's an objective truth that can be determined by a computer. E.g. whether code compiles, whether a unit test passes, whether the answer given to a mathematical question like 3+5 is correct. Many other fields have no objective truth (like art or creative writing), or objective truth requires measurement of the physical world (although if the world can be simulated accurately enough for the problem class at hand, then sufficient training data can still be generated by a computer).

The other replies have said what was meant, but I don’t think they’ve explicitly addressed whether or not that is the sense used in the idea of NP.

I would say… it is at least somewhat similar.

A problem in NP might be of the form “For this value of X, does there exist a Y such that q(X,Y)?” for some predicate q and value X, and where when the answer is “yes”, the answer of “yes” can be verified by being given a value Y, and evaluating q(X,Y). (Specifically in the case of 3SAT, X would be a 3CNF formula, Y would be an assignment of values to the variables in the formula, and q(X,Y) would be “the formula X when evaluated with variable assignments Y, results in 'true’.”.)

This is sort of like the task of “Given requirements X that can be checked automatically, produce code Y which satisfies those requirements”, except that in this case the question is specifically asking for Y, not just asking whether such a Y exists, but.. well, often in practice when one wants a solution to a problem in NP, one actually wants the witness, not just whether there exists such a Y, right?

So, I would say there is a substantial similarity, but also a difference.

For some reasoning data (e.g. you talking out loud as you figure something out, mistakes and all) to be useful for RL training, the conclusion to your reasoning needs to be correct/verified, else that's not the kind of reasoning you want to learn!

Some types of reasoning output, such as solving a math problem or writing a computer program can be automatically verified (e.g. respectively by a symbolic solver, or by compiling and running the program), but in the general case it's hard for a computer to verify whether a chain of reasoning is correct and arrived at a valid answer or not, although LLM-as-judge should work some of the time.

They mean that the solutions can be verified to be correct in a binary sense. E.g. a coding solution passes all the unit tests vs writing poetry.

There's a big difference. The membership of these classes is determined in the worst case - so if there is no polynomial time solution in the worst case then it's NP.

For this problem we don't care if it's possible that sometimes there are things that aren't verifiable, or the answers aren't exact, we just need training signal.