Sahlqvist Formulas in Hybrid Polyadic Modal Logics Valentin Goranko 1 Dimiter Vakarelov 2 1 Department of Mathematics, Rand Afrikaans University PO Box 524, Auckland Park 2006, Johannesburg, South Africa Email: vfg@na.rau.ac.za 2 Faculty of Mathematics and Computer Science, Sofia University blvd. James Bouchier 5, 1126 Sofia, Bulgaria Email: dvak@fmi.uni-sofia.bg Abstract Building on a new approach to polyadic modal languages and Sahlqvist formulas introduced in [10] we define Sahlqvist formulas in hybrid polyadic modal languages containing nominals and universal modality or satisfaction operators. Particularly interesting is the case of rever- sive polyadic languages, closed under all ‘inverses’ of polyadic modalities because the minimal valuations arising in the computation of the first-order equivalents of polyadic Sahlqvist for- mulae are definable in such languages and that makes the proof of first-order definability and canonicity of these formulas a simple syntactic exercise. Furthermore, the first-order de- finability of Sahlqvist formulas immediately transfers to arbitrary polyadic languages, while the direct transfer of canonicity requires a more involved proof-theoretic analysis. Keywords: hybrid polyadic modal logics, Sahlqvist formulas, nominals, universal modality, satisfaction operator, first-order definability, canonicity, completeness. 1 Introduction In [10] we propose a new treatment of polyadic modal languages as ‘purely modal polyadic languages ’ where ∨ and ∧ are treated as binary modalities (resp. a box and a diamond), boxes are composed as in PDL, and eventually every modal formula is represented as a polyadic box or a diamond. In these languages we defined a class of ‘polyadic Sahlqvist formulas ’ PSF which substantially extend the previously known class of Sahlqvist formulas (see [3]). The computation of the first-order equivalents of these formulas extends Sahlqvist-van Benthem’s algorithm to an inductive proce- dure of computing the ‘minimal’ (first-order definable) valuations of the propositional variables in the formula. Here we extend polyadic modal languages with nominals and universal modality or satisfaction operators, and define a large class of first-order definable and canoni- 1