As the number of clauses increases, so should the number of variables (randomly chosen). That is, the alpha that your result mentions can well fluctuate between over-and-underconstrained values even with a googol clauses.
For any particular 3-SAT instance, you can calculate α. If the random algorithm that generated it doesn’t favor any particular clause types, then the distribution of outputs with c clauses and v variables should be uniform among all such 3-SAT instances, and therefore we can use α to figure out if the formula is (most likely) satisfiable.
Furthermore, unless you deliberately try to generate random instances with a particular value of α, I imagine under most models the value of α will be (with high probability) well on one side of the threshold, and so the probability that your formula is satisfiable will be either very nearly 1 or very nearly 0, depending on the random model.
Yes, you can calculate alpha, then depending on its value predict whether the formula is satisfiable with a high probability. Is that still intuiting an answer?
The question is not whether—faced with a typically very, very large (if randomly chosen) number of clauses—you can implement smarter algorithms, the question is whether human-level intuitions will still help you with randomly constructed truth statements.
Not whether they can be resolved, or whether there are neat tricks for certain subclasses.
Of course if the initial use of “intuition” that sparked this debate is used synonymously with “heuristic”, the point is moot. I was referring to the subset that is more typically referred to as human intuitions.
For any particular 3-SAT instance, you can calculate α. If the random algorithm that generated it doesn’t favor any particular clause types, then the distribution of outputs with c clauses and v variables should be uniform among all such 3-SAT instances, and therefore we can use α to figure out if the formula is (most likely) satisfiable.
Furthermore, unless you deliberately try to generate random instances with a particular value of α, I imagine under most models the value of α will be (with high probability) well on one side of the threshold, and so the probability that your formula is satisfiable will be either very nearly 1 or very nearly 0, depending on the random model.
Yes, you can calculate alpha, then depending on its value predict whether the formula is satisfiable with a high probability. Is that still intuiting an answer?
The question is not whether—faced with a typically very, very large (if randomly chosen) number of clauses—you can implement smarter algorithms, the question is whether human-level intuitions will still help you with randomly constructed truth statements.
Not whether they can be resolved, or whether there are neat tricks for certain subclasses.
Of course if the initial use of “intuition” that sparked this debate is used synonymously with “heuristic”, the point is moot. I was referring to the subset that is more typically referred to as human intuitions.
(I should’ve said k-sat anyways.)