First-order satisfiability in Gödel logics: An NP-complete fragment

Defined over sets of truth values V which are closed subsets of [0,1] containing both 0 and 1, Gödel logics are prominent examples of many-valued logics. We investigate a first-order fragment of extended with Δ, that is powerful enough to formalize important properties of fuzzy rule-based systems. The satisfiability problem in this fragment is shown to be NP-complete for all , also in the presence of an additional, involutive negation. In contrast to the one-variable case, in the fragment considered, only two infinite-valued Gödel logics extended with Δ differ w.r.t. satisfiability. Only one of them enjoys the finite model property.