Selecting a Fuzzy Logic Operation from the DNF-CNF Interval: How Practical are the Resulting Operations?

In classical (two-valued) logic, CNF and DNF forms of each propositional formula are equivalent to each other. In fuzzy logic, CNF and DNF forms are not equivalent, they form an interval that contains the fuzzy values of all classically equivalent propositional formulas. If we want to select a single value from this interval, then it is natural to select a linear combination of the interval's endpoints. In particular, we can do that for CNF and DNF forms of \and" and \or", thus designing natural fuzzy analogues of classical \and" and \or" operations. The problem with thus selected \and" and "or" operations is that, contrary to common sense expectations, they are not associative. In this paper, we show the largest possible value of the corresponding non-associativity is reasonably small and thus, this non-associativity does not made these operations impractical. 1 Motivation Fuzzy logic is a generalization of the classical 2-valued logic, where instead of two truth values \true" and \false" (corresponding to 1 and 0), we have the entire interval 0; 1] of truth values. To get a logic, we must extend logical operations to these values. In classical logic, every logical operation can be represented as a composition of conjunction (\and", &), disjunction (\or", _), and negation (\not", :). Therefore, to generalize an arbitrary logical operation, it is suucient to generalize these three basic logical operations. Thus, when a fuzzy logic is described, it is usually described in terms of these three operations: t-norm (\and"), t-conorm (\or"), and negation. To generalize a more complex logical operation, we can: represent this operation as a composition of \and", \or", and \not", and replace \and", \or", and \not" with, correspondingly , a t-norm, a t-conorm, and a negation operation. The main problem with this approach is that diier-ent representations of the original logical operation lead to diierent results { even if we use the simplest possible operations min, max, and 1 ? x. 2 DNF-CNF interval Since diierent classically equivalent logical formulas lead to non-equivalent fuzzy expressions, it is reasonable , instead of considering all possible classically equivalent forms, to select some special forms. In classical (2-valued) logic, two forms are most frequently used: a Conjunctive Normal Form (CNF), in which a formula is represented as disjunction of conjunctions , and a Disjunctive Normal Form (DNF), in which a formula is represented as conjunction of disjunc-tions. In view of this fact, one of …