We propose parametric constructive Kripke-semantics for multi-agentKD45-belief and S5-knowledge in terms of elementary set-theoretic constructionsof two basic functional building blocks, namely bias (or viewpoint) andvisibility, functioning also as the parameters of the doxastic and epistemicaccessibility relation. The doxastic accessibility relates two possible worldswhenever the application of the composition of bias with visibility to thefirst world is equal to the application of visibility to the second world. Theepistemic accessibility is the transitive closure of the union of our doxasticaccessibility and its converse. Therefrom, accessibility relations for commonand distributed belief and knowledge can be constructed in a standard way. As aresult, we obtain a general definition of knowledge in terms of belief thatenables us to view S5-knowledge as accurate (unbiased and thus true)KD45-belief, negation-complete belief and knowledge as exact KD45-belief andS5-knowledge, respectively, and perfect S5-knowledge as precise (exact andaccurate) KD45-belief, and all this generically for arbitrary functions of biasand visibility. Our results can be seen as a semantic complement to previousfoundational results by Halpern et al. about the (un)definability and(non-)reducibility of knowledge in terms of and to belief, respectively.
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