← 研踪 PaperTrailarXiv:2610.35106 · v1
Unbelievably Complex
首次提交 2026-10-26 17:45
版本更新 2026-10-26 17:45
雷达观测 2026-10-29 00:36
摘要
We present the directed simplicial complex: this is an extension of a simplicial complex with a direction on its faces. This enables us to model a local notion of belief in a three-valued semantics, in which formulas may be true, false, or undefined. Of particular interest is a class of directed simplicial complexes called cups, in which any maximal set of vertices of different colors is a facet. Cups permit a more concise relational representation. We give examples and applications of our logical semantics. We then provide a sound and complete axiomatization for the class of directed simplicial complexes and extend it to an axiomatization for the class of cups. We include a detailed technical comparison with other works on belief on complexes. Our work generalizes a prior axiomatization for knowledge on (undirected) impure simplicial complexes.
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v1 · 2026-10-26 17:45 · 最新观测版本
Unbelievably Complex
We present the directed simplicial complex: this is an extension of a simplicial complex with a direction on its faces. This enables us to model a local notion of belief in a three-valued semantics, in which formulas may be true, false, or undefined. Of particular interest is a class of directed simplicial complexes called cups, in which any maximal set of vertices of different colors is a facet. Cups permit a more concise relational representation. We give examples and applications of our logical semantics. We then provide a sound and complete axiomatization for the class of directed simplicial complexes and extend it to an axiomatization for the class of cups. We include a detailed technical comparison with other works on belief on complexes. Our work generalizes a prior axiomatization for knowledge on (undirected) impure simplicial complexes.
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