Decidability Preservation and Complexity Bounds for Combined Logics

Abstract

Transfer theorems for combined logics provide essential tools and insight for reasoning about complex logical systems. In this paper, we present the first sufficient criterion (contextual extensibility) for decidability to be preserved through combination of propositional logics, and we study the complexity upper bounds induced by the method. In order to assess the scope and usability of our criterion, we illustrate its use in re-obtaining two standard important (though partial) results of the area: the preservation of decidability for disjoint combinations of logics, and the preservation of decidability for fusions of modal logics. Due to the very abstract nature and generality of the idea underlying contextual extensibility, we further explore its applicability beyond propositional logics. Namely, we explore the particular case of 2-deductive systems, and as a byproduct, we obtain the preservation of decidability for disjoint combinations of equational logics and discuss the relationship of this result and of our criterion with several related results with meaningful applications in satisfiability modulo theories.