For other breakfast scholars who wish to further my study, I offer my data and code. If you are so foolhardy that you wish to explore the bounds of dark breakfast yourself, the recipe is as follows:
Author(s): Jun Chai, Javier LLorca
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I used z3 theorem prover to assess LLM output, which is a pretty decent SAT solver. I considered the LLM output successful if it determines the formula is SAT or UNSAT correctly, and for SAT case it needs to provide a valid assignment. Testing the assignment is easy, given an assignment you can add a single variable clause to the formula. If the resulting formula is still SAT, that means the assignment is valid otherwise it means that the assignment contradicts with the formula, and it is invalid.