Structure of random r-SAT below the pure literal threshold

It is well known that there is a sharp density threshold for a random r-SAT formula to be satisfiable, and a similar, smaller, threshold for it to be satisfied by the pure literal rule. Also, above the satisfiability threshold, where a random formula is with high probability (whp) unsatisfiable, the unsatisfiability is whp due to a large "minimal unsatisfiable subformula" (MUF). By contrast, we show that for the (rare) unsatisfiable formulae below the pure literal threshold, the unsatisfiability is whp due to a unique MUF with smallest possible "excess", failing this whp due to a unique MUF with the next larger excess, and so forth. In the same regime, we give a precise asymptotic expansion for the probability that a formula is unsatisfiable, and efficient algorithms for satisfying a formula or proving its unsatisfiability. It remains open what happens between the pure literal threshold and the satisfiability threshold. We prove analogous results for the k-core and ...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Alexander D. Scott, Gregory B. Sorkin
• Keywords: CORR 2010, Education, Pure Literal, Pure Literal Threshold, Satisfiability Threshold