Compositional Verification: Decidability Issues Using Graph Substitutions

This paper deals with the compositional verification of sequential programs. This consists in deciding whether or not a given set of local structural properties of the functions of a program implies a given global behavioural property of the program. Here we consider properties expressed in monadic second-order logic dealing with the control flow of the program and the function calls occuring during its execution. This problem has been investigated in relation with the security of open multiapplication smart cards. We show that the compositionality is decidable for sequential programs whose control-flow graphs are of tree-width less than a fixed integer value, which includes in particular structured programs. Formally, we prove the decidability of MSO theories of families of hypergraphs obtained by uniform substitution of hyperedges by hypergraphs specified by MSO formulas.