双范畴数据库模式中的量化
Quantification in Double-Categorical Database Schemas
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中文总结 AI 辅助
该研究探讨双范畴数据库模式的量化结构，将代换右伴随、制表符等结构与逻辑算子结合，应用于查询优化，得出Frobenius互反律等优化规则并研究否定查询。
中文摘要 AI 辅助
双范畴数据库模式通过代换的右伴随形式丰富了全称量化，这使得可以表述重要的查询——关系除法。研究表明，这类右伴随与合适的制表符共同解释了模态算子，提供了笛卡尔闭结构；当与全局余笛卡尔结构结合时，可解释一阶谓词逻辑及描述逻辑。这些结构被全程应用于查询与优化，例如，在任何具有合适结构的双数据库模式中，Frobenius互反律和Beck-Chevalley条件成立，且这些性质提供了下推优化规则；同时，基于余笛卡尔结构和局部蕴含结构，引入并研究了否定查询。
英文摘要
Double-categorical database schemas are enriched with universal quantification in the form of right adjoints to substitution. This allows phrasing of the important query, relational division. It is shown that such right adjoints together with suitable tabulators interpret modal operators, provide cartesian closed structure, and, when combined with global cocartesian structure, interpret first-order predicate logic and thus description logic. These structures are applied throughout to querying and optimization. It is seen, for example, that Frobenius reciprocity and Beck-Chevalley hold in any suitably structured double database schema and that these provide pushdown optimization rules. Such optimization rules are thus provably error-free as a property of the schema. Likewise, negation queries are derived from cocartesian and local implication structure.
