Number of models and satisfiability of sets of clauses

AbstractWe present a way of calculating the number of models of propositional formulas represented by sets of clauses. The complexity of such a procedure is O(ψnk), where k is the length of clauses and n is the number of variables in the clauses. The value of ψ2 is approximately 1.619, value of ψ3 is approximately 1.840 and the value of ψk approaches 2 when k is large. Further we apply the theory on satisfiability problems, especially on the 3-SAT problems. The complexity of the 3-SAT problems is O(ψn), where n is the number of variables in the clauses. The value of ψ is approximately 1.571 which is better than the results in Schiermeyer (1993) and Monien and Schiermeyer (1985)

An approximation algorithm for #k-SAT

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An Approximation Algorithm for #k-SAT

We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k >= 3 within a running time that is not only non-trivial, but also significantly better than that of the currently fastest exact algorithms for the problem. More precisely, our algorithm is a randomized approximation scheme whose running time depends polynomially on the error tolerance and is mildly exponential in the number n of variables of the input formula. For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For 4-CNF input the bound increases to O(1.6155^n). We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way. Relaxing the requirements on the quality of the approximation, on k-CNF input we obtain significantly reduced running times in comparison to the above bounds