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A Propositional Dynamic Logic for Instantial Neighborhood Semantics

Abstract

We propose a new perspective on logics of computation by combining instantial neighborhood logic \(\mathsf {INL}\) with bisimulation safe operations adapted from \(\mathsf {PDL}\). \(\mathsf {INL}\) is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance for instantial neighborhood bisimulations, the appropriate bisimulation concept for \(\mathsf {INL}\). We also prove that our extended logic \(\mathsf {IPDL}\) is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for \(\mathsf {IPDL}\), and establish its finite model property and decidability.

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Acknowledgements

We thank the referee for valuable feedback and for spotting an error in the original version of the proof of Theorem 7. We also thank a number of colleagues for helpful discussions on earlier versions of the manuscript, where in particular we wish to mention Valentin Goranko, Helle Hansen, Tadeusz Litak and Lutz Schröder.

Author information

Correspondence to Sebastian Enqvist.

Additional information

Presented by Heinrich Wansing; Received December 16, 2017

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van Benthem, J., Bezhanishvili, N. & Enqvist, S. A Propositional Dynamic Logic for Instantial Neighborhood Semantics. Stud Logica 107, 719–751 (2019). https://doi.org/10.1007/s11225-018-9825-5

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Keywords

  • Dynamic logic
  • Game logic
  • Neighborhood models
  • Instantial neighborhood logic