Monadic fragments of Gödel logics: Decidability and undecidability results

Abstract

The monadic fragments of first-order Gödel logics are investigated. It is shown that all finite-valued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinite-valued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of an important sub-case, that is well motivated also from an application oriented point of view, is proven. A tight bound for the cardinality of finite models that have to be checked to guarantee validity is extracted from the proof. Moreover, monadic G↑, like all other infinite-valued logics, is shown to be undecidable if the projection operator Δ is added, while all finite-valued monadic Gödel logics remain decidable with Δ. © Springer-Verlag Berlin Heidelberg 2007.