A Decidable Class of Nested Iterated Schemata (extended version)

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called iterated schemata, allow to express such patterns. i=1 or n i=1 (called iterations) where n is an (unbound) integer variable called a parameter. The expressive power of iterated schemata is strictly greater than propo-sitional logic: it is even out of the scope of first-order logic. We define a proof procedure, called dpll ⋆ , that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the dpll procedure [12]. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable [2] so dpll ⋆ does not terminate in general. Still, we prove that dpll ⋆ terminates for schemata of a syntactic subclass called regularly nested. This is the first non trivial class for which dpll ⋆ is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form n i=1 (n j=1. . .).