Decidability and Undecidability Results for Propositional Schemata

We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧n i=1 pi, where n is a parameter). The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus (called stab) allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability.