The aim of this paper is to establish a new notion of equivalencebetween temporal models, so-called S-similarity, as the appropriate notionof bisimilarity for temporal logic with Since and Until. The main technicalresults of the paper provide semantical characterizations of the first-orderformulas that are equivalent to a temporal formula: Theorem 3.7 concernsthe equivalence of temporal and first-order formulas with respect to pointedtemporal models, whereas Theorem 4.4 takes the level of temporal modelsinto account.

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.

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History based models suggest a process-based approach to epistemic and temporal reasoning. In this work, we introduce preferences to history based models. Motivated by game theoretical observations, we discuss how preferences can dynamically be updated in history based models. Following, we consider arrow update logic and event calculus, and give history based models for these logics. This allows us to relate dynamic logics of history based models to a broader framework.

We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new fine-structure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.