We discuss the syntax and semantics of relational Horn logic (RHL) and partial Horn logic (PHL). RHL is an extension of the Datalog programming language that allows introducing and equating variables in conclusions. PHL is a syntactic extension of RHL by partial functions and one of the many equivalent notions of essentially algebraic theory. Our main contribution is a new construction of free models. We associate to RHL and PHL sequents classifying morphisms, which enable us to characterize logical satisfaction using lifting properties. We then obtain free and weakly free models using the small object argument. The small object argument can be understood as an abstract generalization of Datalog evaluation. It underpins the implementation of the Eqlog Datalog engine, which computes free models of PHL theories.
翻译：我们讨论了“人际关系”逻辑(RHL)和“人际关系”部分逻辑(PHL)的语法和语义。“人际关系”是“数据”编程语言的延伸,允许在结论中引入和等同变量。“人际关系”是“人际关系逻辑(RHL)的综合延伸,通过部分功能和许多基本代数理论等同的概念之一,“RHL”是“RHL”的综合延伸。我们的主要贡献是自由模型的新构建。我们与“RHL”和“PHL”序列相联,通过提升属性,我们得以描述逻辑满意度。我们随后获得了使用小对象参数的自由和微弱自由模型。小对象参数可以被理解为“数据”评估的抽象概括化。它支持了“Eqlog Datalog”引擎的实施,该引擎将“PHL”理论的自由模型组成。
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