Fuzzy normal forms

Abstract

Linguistic expressions of a natural language are transformed at least three times before they are expressed in computational formulas. Between the linguistic and the computational expressions, there are the meta-linguistic and propositional expressions under various interpretations and assumptions. Fuzzy normal forms are canonical expressions derived at the propositional level with particular assumptions. The fuzzy normal forms can be generated from fuzzy truth tables with a normal form derivation algorithm and an orthodox interpretation of conjunction, disjunction and complementation operations. It is shown that these fuzzy normal forms display a unique structure for certain general classes of crisp operators over fuzzy sets. The fuzzy normal forms, generated by crisp operators over fuzzy sets that correspond to max-min-standard-complement De Morgan triple, are shown to be equivalent to the classical normal forms of two-valued logic, i.e., disjunctive and conjunctive normal forms known as DNF and CNF, with a particular property, i.e., the crisp containment FDNF (3) ⊆ FCNF (3). Whereas the fuzzy normal forms, generated by crisp operators that correspond to non-idempotent t-norms, t-conorms and standard complement De Morgan triples do not have, in general, a similar crisp containment property, i.e., FDNF (2)  FCNF (2) . Neither do the fuzzy normal forms, generated by crisp operators over fuzzy sets that correspond to non-idempotent and non-commutative and non-associative conjunction, disjunction and standard complement De Morgan triples have a crisp containment property, i.e., FDNF (1)  FCNF (1) . However, they still provide bounds on the combination of concepts.