Deciding theoremhood in fibred logics without shared connectives

Fibring is a powerful mechanism for combining logics, and an essential tool for designing and understanding complex logical systems. Abstract results about the semantics and proof-theory of fibred logics have been extensively developed, including general soundness and completeness preservation results. Decidability, however, a key ingredient for the automated support of the fibred logic, has not deserved similar attention. In this paper, we address the problem of deciding theoremhood in fibred logics without shared connectives. Namely, under this assumption, we provide a full characterization of the mixed patterns of reasoning that lead to theorems in the fibred logic, and use it to prove a general decidability preservation result. The complexity of the decision procedure we obtain is also analyzed. Mathematics Subject Classification (2000). 03B22: abstract deductive systems, 03B25: decidability of theories and sets of sentences, 03B62: combined logics.