Critical Behavior at the k - Satis ability

The satissability of random Boolean formulae with precisely k variables per clause is a popular testbed for the performance of search algorithms in artiicial intelligence and computer science. For k = 2, formulae are almost aways satissable when the ratio of clauses to variables is less than 1; for ratios larger than 1, the formulae are almost never satissable. We present data showing a similar threshold behavior for higher values of k. We also show how nite-size scaling, a method from statistical physics, can be used to characterize size dependent eeects near the threshold. Finally, we commment on the relationship between thresholds and computational complexity. acknowledges the hospitality of the Salk Institute, where a portion of the work was carried out, with support from the McDonnell-Pew Foundation.