. Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established. 1 Introduction The logics that have appeared in artificial intelligence form a rich and varied collection. While classical (and maybe intuitionistic) logic su#ces for the formal development of mathematics, artificial intelligence has found uses for modal, temporal, relevant, and many-valued logics, among others. Indeed, I take it as a basic principle that an application should find (or create) an appropriate logic, if it needs one, rather than reshape the application to fit some narrow class of `established' logics. In this paper I want to enlarge the variety of logics...

) Glenn Bruns and Patrice Godefroid Bell Laboratories, Lucent Technologies fgrb,godg@bell-labs.com Abstract. We address the problem of relating the result of model checking a partial state space of a system to the properties actually possessed by the system. We represent incomplete state spaces as partial Kripke structures, and give a 3-valued interpretation to modal logic formulas on these structures. The third truth value ? means "unknown whether true or false". We define a preorder on partial Kripke structures that reflects their degree of completeness. We then provide a logical characterization of this preorder. This characterization thus relates properties of less complete structures to properties of more complete structures. We present similar results for labeled transition systems and show a connection to intuitionistic modal logic. We also present a 3-valued CTL model checking algorithm, which returns ? only when the partial state space lacks information needed ...

We discuss the problem of model checking temporal properties on partial Kripke structures, which were used in [BG99] to represent incomplete state spaces. We first extend the results of [BG99] by showing that the model-checking problem for any 3-valued temporal logic can be reduced to two model-checking problems for the corresponding 2-valued temporal logic. We then introduce a new semantics for 3-valued temporal logics that can give more definite answers than the previous one. With this semantics, the evaluation of a formula OE on a partial Kripke structure M returns the third truth value? (read "unknown") only if there exist Kripke structures M1 and M2 that both complete M and such that M1 satisfies OE while M2 violates OE, hence making the value of OE on M truly unknown. The partial Kripke structure M can thus be viewed as a partial solution to the satisfiability problem which reduces the solution space to complete Kripke structures that are more complete than M wit...

Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible --- in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal language, and on what sentences will they agree? This problem can be reformulated as one about many-valued Kripke models, allowing many-valued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the many-valued version and the multiple expert one will be formally established. Finally we will axiomatize many-valued modal logics, and sketch a proof of completeness.

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