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A modal walk through space. (English) Zbl 1185.03060
Summary: 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.

MSC:
03B80 Other applications of logic
03B45 Modal logic (including the logic of norms)
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
51N10 Affine analytic geometry
54A99 Generalities in topology
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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