Recognizing Disguised NR(1) Instances of the Satisfiability Problem

It is well known that the satisfiability problem (SAT) for Boolean expressions in conjunctive normal form (CNF) is NP-complete, and that the problem is polynomial-time solvable if each clause contains at most two literals (2~SAT) [ 1, 4, 5, 71. The satisfiability problem is also solvable in polynomial time if none of the clauses contains more than one negated variable {3, 5, 81. We call a CNF expression k-negation restricted, NR(k), if it contains at most k negated variables per clause. It is straightforward to show that NR(2)SAT is NP-complete [3]. Since NR(l)-SAT is in P, but NR(2)SAT is NP-complete, it is of interest to determine whether a given CNF expression can be transformed into an equivalent NR(1) expression with at most a polynomial increase in the length of the expression. We call a CNF expression E a disguised NR(1) expression if it can be mapped to an equivalent NR( 1) expression E’ such that, for each variable x in E, either x H x’, or x H x”, where X denotes the negation of x and i is equivalent to x. If E’ exists it is called an apparenr NR(1) version of E (we allow E’ to be identical to E). For example,