Necessary and Sufficient Condition for Satisfiability of a Boolean Formula in CNF and its Implications on P versus NP problem
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In this paper, a necessary and sufficient condition for satisfiability of a boolean formula, in CNF, has been determined. It has been found that the maximum cardinality of satisfiable boolean formula increases exponentially, with increase in number of variables. Due to which, any algorithm require exponential time, in worst case scenario, depending upon the number of variables in a boolean formula, to check satisfiability of the given boolean formula. Which proves the non-existence of a polynomial time algorithm for satisfiability problem. As satisfiability is a NP-complete problem, and non-existence of a polynomial time algorithm to solve satisfiability proves exclusion of satisfiability from class P. Which implies P is not equal to NP. Further, the necessary and sufficient condition can be used to optimize existing algorithms, in some cases, the unsatisfiability of a given boolean function can be determined in polynomial time. For this purpose, a novel function has been defined, that can be used to determine cardinality of a given boolean formula, and occurances of a literal in the given formula, in polynomial time.
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