Monadic fragments of Gödel logics: decidability and undecidability results

Authors:
Matthias Baaz;Agata Ciabattoni;Christian G. Fermüller
Affiliations:
Technische Universität Wien, Vienna, Austria;Technische Universität Wien, Vienna, Austria;Technische Universität Wien, Vienna, Austria
Venue:
LPAR'07 Proceedings of the 14th international conference on Logic for programming, artificial intelligence and reasoning
Year:
2007

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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 infinitevalued 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.