A Decidable Class of Nested Iterated Schemata

Abstract

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. Schemata extend propositional logic with indexed propositions, e.g.P _i, P _i + 1, P ₁ or P _n, and with generalized connectives, e.g. $\bigwedge_{\rm i = 1}^n$, or $\bigvee_{\rm i = 1}^n$, where n is an (unbound) integer variable called a parameter. The expressive power of iterated schemata is strictly greater than propositional 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 [9]. But proving that a schema is unsatisfiable for every value of the parameter, is undecidable [1] so dpll ^⋆ does not terminate in general. Still, dpll ^⋆ terminates for schemata of a syntactic subclass called regularly nested.

This work has been partly funded by the project ASAP of the French Agence Nationale de la Recherche (ANR-09-BLAN-0407-01).