Monadic Fragments of Gödel Logics: Decidability and Undecidability Results

Abstract

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 is added, while all finite-valued monadic Gödel logics remain decidable with .

SAT in Monadic Gödel Logics: A Borderline between Decidability and Undecidability

“…This result contrasts with the undecidability of the validity problem proved in [3] for all prenex monadic Gödel logics with the exception of G n and (possibly) G ↑ . 1 In general Gödel logics do not admit equivalent prenex formulas, see e.g.…”

“…Our argument, which applies also to all infinite-valued witnessed Gödel logics with △, is similar to that for the undecidability of TAUT G △ V in [3].…”

“…Proceed similarly to the decidability proof of TAUT m G △ n in [3]. Given any monadic formula P , it is indeed enough to consider interpretations with domain {1, .…”

“…The same does not hold for many-valued monadic logics in which a rather complex landscape appears and many questions are still open, see e.g. [3,9,10].…”

“…Note that in contrast with classical logic, in Gödel logics validity and satisfiability are not dual concepts. A general investigation of the (un)decidability status for the validity problem in monadic Gödel logics was carried out in [3], where it was shown that with the possible exception of Gödel logic G ↑ in which V = {1 − 1/n : n ≥ 1} ∪ {1}, validity is undecidable when V is infinite, even when restricted to prenex formulas.…”