The Maximum Length of Prime Implicates for Instances of 3-SAT

Schrag and Crawford ( 1996) present strong experimental evidence that the occurrence of prime implicates of varying lengths in random instances of 3-SAT exhibits behaviour similar to the wellknown phase transition phenomenon associated with satisfiability. Thus, as the ratio of number of clauses (m) to number of propositional variables (n) increases, random instances of 3-SAT progress from formulae which are generally satisfiable through to formulae which are generally not satisfiable, with an apparent sharp threshold being crossed when m/n 4.2. For instances of 3SAT, Schrag and Crawford ( 1996) examine with what probability the longest prime implicate has length k (for k > 0) -unsatisfiable formulae correspond to those having only a prime implicate of length O-demonstrating that similar behaviour arises. It is observed by Schrag and Crawford ( 1996) that experiments failed to identify any instance of 3-SAT over nine propositional variables having a prime implicate of length 7 or greater, and it is conjectured that no such instances are possible. In this note we present a combinatorial argument establishing that no 3-SAT instance on n variables can have a prime implicate whose length exceeds max{ [n/21 + 1, L2n/3j }, validating this conjecture for the case n = 9. We further show that these bounds are the best possible. An easy corollary of the latter constructions is that for all k > 3, instances of k-SAT on n variables can be formed, that have prime implicates of length n o(n). @ 1997 Elsevier Science B.V.