Fundamenta Informaticae 118 (2012) 177–205 177 DOI 10.3233/FI-2012-709 IOS Press Counterpart Semantics for a Second-Order μ-Calculus ∗ Fabio Gadducci Department of Computer Science, University of Pisa Largo Bruno Pontecorvo 3, Pisa, Italy gadducci@di.unipi.it Alberto Lluch Lafuente † , Andrea Vandin IMT Institute for Advanced Studies Lucca Piazza San Ponziano 6, Lucca, Italy alberto.lluch@imtlucca.it; andrea.vandin@imtlucca.it Abstract. Quantified µ-calculi combine the fix-point and modal operators of temporal logics with (existential and universal) quantifiers, and they allow for reasoning about the possible behaviour of individual components within a software system. In this paper we introduce a novel approach to the semantics of such calculi: we consider a sort of labeled transition systems called counterpart models as semantic domain, where states are algebras and transitions are defined by counterpart relations (a family of partial homomorphisms) between states. Then, formulae are interpreted over sets of state assignments (families of partial substitutions, associating formula variables to state components). Our proposal allows us to model and reason about the creation and deletion of components, as well as the merging of components. Moreover, it avoids the limitations of existing approaches, usually enforcing restrictions of the transition relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of. The paper is rounded up with some considerations about expressiveness and decidability aspects. Keywords: Quantified µ-calculi, counterpart semantics, modal logics, graph transformation ∗ Research partly supported by the EU FP7-ICT IP ASCENS (IP 257414) and by the MIUR PRIN SisteR (PRIN 20088HXMYN). † Address for correspondence: IMT Institute for Advanced Studies Lucca, Piazza San Ponziano 6, Lucca, Italy