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Representing Knowledge and Querying Data Using Double-Functorial Semantics: A Flexible and Expressive Approach


Core Concepts
Double categories provide a flexible and expressive language to represent knowledge and query data, unifying the functional and relational models through double functors.
Abstract
The paper introduces double-categorical ologs as a knowledge representation framework, where the ontology or database schema is a small double category and instance data is a structure-preserving double functor. This approach recognizes both functions and relations as fundamental concepts, granting them first-class status. The key highlights are: Expressing facts: Double ologs can naturally express facts involving both functions and relations, including partial maps and implications, which are difficult to capture in functional or relational ologs. Creating types using tabulators: Tabulators in double ologs enable the creation of new types by reifying propositions and relations as subtypes of the original types. Forming conjunctions using local products: Local products in double ologs allow the expression of conjunctions of propositions, enabling complex queries to be formulated within the schema. Querying data: Queries can be formulated as abstract relations in the double olog schema and then evaluated by the double functor defining the instance. Basic relational algebra operations like permutations, projections, and joins appear directly as operations in the double category of relations. The paper provides an accessible, example-driven explanation of these features, demonstrating the expressive power and flexibility of double-categorical knowledge representation and data querying.
Stats
Each person may have expertise consisting of training or facility in a particular skill. Each person may or may not belong to a particular SGC team. The inner join of the expertise and membership relations gives the set of persons, their skills, and the teams they belong to.
Quotes
"Double categories provide a flexible and expressive language to represent knowledge and query data, unifying the functional and relational models through double functors." "Querying is double-functorial semantics."

Deeper Inquiries

How can double ologs be extended to handle more complex data structures, such as hierarchical or graph-like data

Double ologs can be extended to handle more complex data structures, such as hierarchical or graph-like data, by leveraging the rich structure and flexibility of double categories. In the context of hierarchical data, one approach could involve representing parent-child relationships as proarrows in the double category. For example, if we have a hierarchical structure where each node has a parent and possibly multiple children, we can define proarrows that capture these relationships. By extending the double olog with appropriate cells and relations, we can model and query hierarchical data effectively. When dealing with graph-like data, double ologs can represent nodes and edges as objects and arrows in the schema. By incorporating additional relations to capture the connectivity between nodes, double ologs can handle graph structures seamlessly. Queries on graph-like data can be formulated using the double-functorial semantics, allowing for expressive and efficient retrieval of information from complex interconnected datasets.

What are the computational and performance implications of using double-categorical semantics for querying large-scale databases

Using double-categorical semantics for querying large-scale databases can have both computational benefits and performance implications. On the computational side, the abstract structure of double categories allows for a unified and systematic approach to representing and querying data. This can lead to more efficient query processing, as the semantics of the double category provide a clear framework for formulating and evaluating queries. In terms of performance, the use of double-categorical semantics may introduce additional computational overhead compared to traditional database querying methods. The complexity of handling double functors and relations in large-scale databases could impact query execution times and resource utilization. However, the expressive power and flexibility offered by double-categorical semantics can outweigh these performance considerations, especially in scenarios where the structure of the data is intricate and requires sophisticated querying capabilities.

What are the potential applications of double-categorical knowledge representation beyond databases, such as in areas like ontology engineering or knowledge graph construction

Double-categorical knowledge representation has a wide range of potential applications beyond databases, particularly in areas like ontology engineering and knowledge graph construction. In ontology engineering, double ologs can provide a formal and structured way to represent complex relationships between entities, attributes, and concepts. By utilizing double functors and relations, ontologies can capture intricate domain knowledge and support reasoning and inference tasks effectively. In the context of knowledge graph construction, double-categorical semantics offer a robust framework for modeling interconnected data elements. Knowledge graphs can benefit from the expressive nature of double ologs to represent diverse types of information and their relationships. This can enhance the semantic richness of knowledge graphs, enabling more sophisticated queries and analyses across interconnected data points. Overall, the applications of double-categorical knowledge representation extend to various domains where complex data structures and relationships need to be captured and queried in a systematic and coherent manner.
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