Many hard examples for resolution
Journal of the ACM (JACM)
On the satisfiability and maximum satisfiability of random 3-CNF formulas
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
A machine program for theorem-proving
Communications of the ACM
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Where the really hard problems are
IJCAI'91 Proceedings of the 12th international joint conference on Artificial intelligence - Volume 1
Heuristic average-case analysis of the backtrack resolution of random 3-satisfiability instances
Theoretical Computer Science
Annals of Mathematics and Artificial Intelligence
MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability
Artificial Intelligence
Zchaff2004: an efficient SAT solver
SAT'04 Proceedings of the 7th international conference on Theory and Applications of Satisfiability Testing
Abstraction refinement for bounded model checking
CAV'05 Proceedings of the 17th international conference on Computer Aided Verification
A boolean model for enumerating minimal siphons and traps in petri nets
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
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Determining whether a propositional theory is satisfiable is a prototypical example of an NP-complete problem. Further, a large number of problems that occur in knowledge representation, learning, planning, and other areas of AI are essentially satisfiability problems. This paper reports on a series of experiments to determine the location of the crossover point -- the point at which half the randomly generated propositional theories with a given number of variables and given number of clauses are satisfiable -- and to assess the relationship of the crossover point to the difficulty of determining satisfiability. We have found empirically that, for 3-SAT, the number of clauses at the crossover point is a linear function of the number of variables. This result is of theoretical interest since it is not clear why such a linear relationship should exist, but it is also of practical interest since recent experiments [Mitchell et al. 92; Cheeseman et al. 91] indicate that the most computationally difficult problems tend to be found near the crossover point. We have also found that for random 3-SAT problems below the crossover point, the average time complexity of satisfiability problems seems empirically to grow linearly with problem size. At and above the crossover point the complexity seems to grow exponentially, but the rate of growth seems to be greatest near the crossover point.