Necessary and sufficient conditions for Boolean satisfiability
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The study in this article seeks to find conditions that are necessary and sufficient for the satisfiability of a Boolean function. We will use the concept of special covering of a set introduced in [9] and study the connection of this concept with the satisfiability of Boolean functions. The basic definitions connected with the concept of special covering are given again. Generally, accepted terminology on set theory, Boolean functions, and graph theory is consistent with the terminology found in the relevant works included in the bibliography [1] [2] [3]. The newly introduced terms are not found in use by other authors and do not contradict to other terms. The bibliography also includes some articles related to the subject being studied. We show that the problem of existence of a special covering of a set is equivalent to the Boolean satisfiability problem. Thus, an important result is the proof of the existence of necessary and sufficient conditions for the existence of special covering of the set. Also we introduce the concept of proportional conjunctive normal form of a function, which is a conjunctive normal form of a function with the condition that each clause contains a negative literal or each clause contains a positive literal. We prove that Boolean function represented in conjunctive normal form is satisfable if and only if it is transformed into a function in proportional conjunctive normal form.
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