Hybrid fragments of Halpern–Shoham logic and their expressive power

Abstract

Halpern and Shoham modal logic of time intervals ($\mathsf{HS}$ in short) is an elegant and highly influential propositional interval-based modal logic. Its sub-propositional fragments, that is fragments obtained by restricting use of propositional connectives, and their hybrid extensions with nominals and satisfaction operators have been recently studied and successfully applied in real-world use cases. Detailed investigation of their decidability and computational complexity has been conducted, however, there has been significantly less research on their expressive power. In this paper we make a step towards filling this gap. In particular, we (1) compare classes of frames definable in full $\mathsf{HS}$ and in its hybrid extension, and (2) determine in which sub-propositional $\mathsf{HS}$-fragments we can express the difference operator, nominals, and satisfaction operators. The obtained results enable us to classify $\mathsf{HS}$, its sub-propositional fragments, and their hybrid extensions according to their expressive power.