Possible Hyperworlds: representing Kripke models in RDF

Abstract

Modal logic extends propositional logic by adding modal operators, and enables us to reason with conceptssuch as necessity, possibility, knowledge, belief, obligation, or consequences of actions. The most common way tointerpret modal logic formulas is to apply the possible worlds semantics and to formally represent it with a Kripkemodel. Kripke models consist of a set of possible worlds, accessibility relations between possible worlds, and avaluation function that assigns to propositional atoms a truth value relative to those worlds. Existing work exploredthe connection between modal logic and Description Logics, including Modal Description Logics, and consider theinclusion of a Kripke model in their semantics. We are interested in considering a novel perspective, where weaim to represent in RDF a Kripke model, formulas from a modal logic language, and in which possible worldssuch formulas are verified. Such a representation could have several applications: 1. contributing to a standardexchange format for modal logic applications, including reasoners; 2. facilitating the Open Science initiative inthe modal logic research community; 3. supporting the representation and management of uncertainty or othermodalities; and 4. facilitating interoperability between heterogeneous systems. As a starting point, our intuitionis that a set of triples could be translated to a formula, and one could explicit in what possible worlds a formulais verified using metadata about this set of triples. Following that intuition, the main contribution of this articleis to provide an RDF representation of normal Kripke models, applying the linked data principles and standards,focusing on identifying possible worlds and expressing the (N+1)-ary accessibility relations between them. We callpossible hyperworld the result of this hypermedia approach to representing possible worlds. We discuss the differentmodeling alternatives for linking sets of triples to possible worlds, and for representing accessibility relations. Wemotivate one modeling choice, and compile our design rationale in a formal vocabulary for describing Kripke models:the Possible Worlds and Kripke Structures Ontology (PWKSO). Adopting a simple and generic usage pattern, weillustrate PWKSO with examples from epistemic multi-modal logic, dyadic deontic logic, and the dynamic logicof mental attitudes and joint actions (DL-MA). Perspectives and future work include 1. defining a translation,in the context of an RDF dataset, of sets of triples to modal logic formulas; 2. querying, instantiating, reasoningwith possible hyperworlds; 3. nesting possible worlds or Kripke structures as a way to model Dynamic EpistemicLogics (DEL) or Propositional Dynamic Logics (PDL); 4. potential applications that leverage existing modal logicformalisms for example to merge knowledge graphs, represent the knowledge or beliefs of agents, represent theeffects of actions or norms, or develop model checking applications.

Affiliated Institutions

• Institut de Biologie Valrose FR
• Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes FR
• Laboratoire d'Informatique, Signaux et Systèmes de Sophia Antipolis FR
• Mines Saint-Étienne FR
• Infection et inflammation
• Université Côte d'Azur FR

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