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A Mathematical Proof Too Long To Check
mikejuk writes "Mathematicians have generally gotten over their unease with computer-assisted proofs. But in the case of a new proof from researchers at the University of Liverpool, we may have crossed a line. The proof is currently contained within a 13 GB file — more space than is required to hold the entirety of Wikipedia. Its size makes it unlikely that humans will be able to check and confirm the proof. The theorem that has been proved is in connection with a long running conjecture of Paul Erdos in 1930. Discrepancy theory is about how possible it is to distribute something evenly. It occurs in lots of different forms and even has a connection with cryptography. In 1993 it was proved that an infinite series cannot have a discrepancy of 1 or less. This proved the theorem for C=1. The recent progress, which pushes C up to 2, was made possible by a clever idea of using a SAT solver — a program that finds values that make an expression true. Things went well up to length 1160, which was proved to have discrepancy 2, but at length 1161 the SAT returned the result that there was no assignment. The negative result generated an unsatisfiability certificate: the proof that a sequence of length 1161 has no subsequence with discrepancy 2 requires over 13 gigabytes of data. As the authors of the paper write: '[it]...is probably one of longest proofs of a non-trivial mathematical result ever produced. ... one may have doubts about to which degree this can be accepted as a proof of a mathematical statement.' Does this matter? Probably not — as long as other programs can check the result and the program itself has to be considered part of the proof."
it is the beginning of AI-science, not the end of human science.
Science requires testable, provable, repeatable. If a human cannot understand the proof then he cannot participate in the science. This is likely to be referred to as an "early" version of machine-exclusive science.
Yeah, I'm familiar with SAT solvers and the fact that they aren't REALLY full brute force; I oversimplified it a bit for the Slashdot crowd. Might have gone a little too far on the "lies to children" scale, mea culpa.
My point was that anyone with high school level math experience can understand the basic problem of boolean satisfiability; I was trying to draw a distinction between problems that are beyond human comprehension and those that are merely beyond human time and ability, with huge SAT instances falling into the latter category. Shouldn't have glossed over the details quite as badly as I did.