Decidability of order-based modal logics

Decidability of the validity problem is established for a family of many-valued modal logics, notably modal logics based on Gödel logics, where propositional connectives are evaluated locally at worlds according to the order of values in a complete sublattice of the real unit interval, and box and diamond modalities are evaluated as infima and suprema of values in (many-valued) Kripke frames. When the sublattice is infinite and the language is sufficiently expressive, the standard semantics for such a logic lacks the finite model property. It is shown here, however, that the finite model property holds for a new equivalent semantics for the same logic. Decidability and PSPACE-completeness of the validity problem follows from this property, given certain regularity conditions on the order of the sublattice. Decidability and co-NP-completeness of the validity problem are also Preliminary results from this work were reported in the proceedings of TACL 2013 (as an extended abstract) and WoLLIC 2013 [8]. ∗Corresponding author Email addresses: [email protected] (Xavier Caicedo), [email protected] (George Metcalfe), [email protected] (Ricardo Rodrı́guez), [email protected] (Jonas Rogger) 1Supported by a Uniandes Sciences Faculty grant 2013-15. 2Supported by Swiss National Science Foundation (SNF) grant 200021 146748. Preprint submitted to Journal of Computer and System Sciences September 1, 2015 established for S5 versions of the logics that coincide with one-variable fragments of first-order many-valued logics. In particular, a first proof is provided of the decidability and co-NP-completeness for the validity problem of the one-variable fragment of first-order Gödel logic.