A Propositional Dynamic Logic for Instantial Neighborhood Semantics

Abstract

Abstract.We propose a new perspective on logics of computation by combining instantial neighborhood logic INL with bisimulation safe operations adapted from PDL. 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 INL. We also prove that our extended logic 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 IPDL, and establish its finite model property and decidability.

A New Game Equivalence, its Logic and Algebra

“…Combining Lemmas 6.7 and 6.6 with the easy observation that any consistent basic formula of depth ≤ k, variables in Q and atomic games among τ is provably entailed by some (Q, k)-description, 3 we obtain Theorem 6.1.…”

“…When added to our base logic BPL, these axioms lead to the following result. From another angle, [3] explores the present perspective in more depth, in the form of a PDL-like dynamic logic over instantial neighborhood models with a new computational interpretation. This logic includes the game operations defined here, except for unrestricted dual, but it adds further constructions of test, and especially, finite iteration, which calls for new axioms and techniques.…”

Pooling Modalities and Pointwise Intersection: Semantics, Expressivity, and Dynamics

“…Finally, as mentioned in the introduction, pointwise intersection is just one prototypical example of combinatorics over neighbourhood sets. Such operations form a larger family that comprises, for instance, neighbourhood operations motivated by PDL's non-deterministic choice or sequential composition [33,38]. It remains to be seen how the languages and logics introduced in [33,38] combine with pooling modalities.…”

“…Such operations form a larger family that comprises, for instance, neighbourhood operations motivated by PDL's non-deterministic choice or sequential composition [33,38]. It remains to be seen how the languages and logics introduced in [33,38] combine with pooling modalities. Relatedly, one may also generalize pointwise intersection to other boolean operations of union and negation, and investigate the expressivity and logical properties of ensuing systems.…”

“…In particular, the standard operations of non-deterministic choice, sequential composition, and iteration from PDL [23] have been generalized to various types of neighbourhood semantics and formal languages, cf. [33,38]. In [20,21], operations on monotonic neighbourhood models are studied from an abstract, algebraic viewpoint, giving rise to highly generic completeness results.…”