Fuzzy satisfiability

This paper defines a problem that we called “fuzzy satisfiability” or “δ-satisfiability.” It describes in mathematical terms the semantics of satisfying clauses and formulas using fuzzy logic, by converting a boolean formula into an arithmetic expression via t-norm and t-conorm operators. It is shown that for any (t-norm, t-conorm) pair, the corresponding δ-satisfiability problem is NP-hard when the values of the variables are restricted to (0,1). More interesting, even when the values of the variables are in the closed interval [0,1], a large class of t-conorms exists for which the δ-satisfiability problem remains NP-hard. A simple sufficient condition is provided for t-conorms to be in this class. It is shown that the optimization versions of the problems discussed here can be formulated as special cases of nonlinear programming