Observational Semantics for Dynamic Logic with Binders

Abstract

The dynamic logic with binders $\mathcal {D}^{\downarrow }$ was recently introduced as a suitable formalism to support a rigorous stepwise development method for reactive software. The commitment of this logic concerning bisimulation equivalence is, however, not satisfactory: the model class semantics of specifications in $\mathcal {D}^{\downarrow }$ is not closed under bisimulation equivalence; there are $\mathcal {D}^{\downarrow }$-sentences that distinguish bisimulation equivalent models, i.e., $\mathcal {D}^{\downarrow }$ does not enjoy the modal invariance property. This paper improves on these limitations by providing an observational semantics for dynamic logic with binders. This involves the definition of a new model category and of a more relaxed satisfaction relation. We show that the new logic $\mathcal {D}^{\downarrow }_\sim$ enjoys modal invariance and even the Hennessy-Milner property. Moreover, the new model category provides a categorical characterisation of bisimulation equivalence by observational isomorphism. Finally, we consider abstractor semantics obtained by closing the model class of a specification $SP$ in $\mathcal {D}^{\downarrow }$ under bisimulation equivalence. We show that, under mild conditions, abstractor semantics of $SP$ in $\mathcal {D}^{\downarrow }$ is the same as observational semantics of $SP$ in $\mathcal {D}^{\downarrow }_\sim$.

A. Madeira—This work is financed by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within projects POCI-01-0145-FEDER-016692 and UID/MAT/04106/2013 and also with its Operational Programme NORTE 2020, trough the project NORTE-01-0145-FEDER-000037. The second author is also supported by the individual grant SFRH/BPD/103004/2014.