In terms of Kripke semantics, S5 is characterized by models where the accessibility relation is an equivalence relation: it is reflexive, transitive, and symmetric. Alternatively, S5 can be characterized by models where the accessibility relation is "universal", that is, every world is accessible from any other. Although these characterizations produce different sets of models (since the former, but not the latter, allows for "closed" systems of worlds such that no world in one is accessible from any world in the other), they both model the theorems of S5.
Determining the satisfiability of an S5 formula is an NP-complete problem. The hardness proof is trivial, as S5 includes the propositional logic. Membership is proved by showing that any satisfiable formula has a Kripke model where the number of worlds is at most linear in the size of the formula.
Read more about this topic: S5 (modal Logic)
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Famous quotes containing the word kripke:
“Certainly the philosopher of possible worlds must take care that his technical apparatus not push him to ask questions whose meaningfulness is not supported by our original intuitions of possibility that gave the apparatus its point.”
—Saul Kripke (b. 1940)