A dynamic logic with branching modalities

Abstract

We propose a dynamic logic ${\mathrm{DL}}_b$ called “dynamic logic with branching modalities”, which extends the temporal dynamic logic DLT with a “branching modality” for specifying safety properties of regular programs with tests (simply “regular programs”). Compared to the trace modality of DLT for while programs that do not abort, branching modality of ${\mathrm{DL}}_b$ does not exclude aborting traces introduced by regular programs, thus is able to capture a type of safety properties which are important for systems with failure behaviors. Moreover, it is congruent to the compositionality of regular programs so that the proof system naturally extended from that of DLT is proved to be complete for ${\mathrm{DL}}_b$. In this paper, we build the theory of ${\mathrm{DL}}_b$ on both propositional and first-ordered levels, defining two logics: propositional ${\mathrm{DL}}_b$ (${\mathrm{PDL}}_b$) and first-ordered ${\mathrm{DL}}_b$ (${\mathrm{FODL}}_b$). ${\mathrm{PDL}}_b$ forms the theoretical basis of ${\mathrm{DL}}_b$ while ${\mathrm{FODL}}_b$ is useful for practical verification. We propose the proof systems for ${\mathrm{PDL}}_b$ and ${\mathrm{FODL}}_b$, and analyze their decidability, soundness and (relative) completeness in a formal way, through comparing their expressiveness and deduction capabilities with propositional dynamic logic (PDL) and first-order dynamic logic (FODL) respectively. We show that ${\mathrm{FODL}}_b$ is actually an extension of DLT, and illustrate the motivations of using the branching modality through an example.