Entropy of the<i>K</i>-Satisfiability Problem

Abstract

The threshold behavior of the K-satisfiability problem is studied in the framework of the statistical mechanics of random diluted systems. We find that at the transition the entropy is finite and hence that the transition itself is due to the abrupt appearance of logical contradictions in all solutions and not to the progressive decreasing of the number of these solutions down to zero. A physical interpretation is given for the different cases $K\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$, $K\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, and $K\ensuremath{\ge}3$.