Quantitative information architecture, granular computing and rough set models in the double-quantitative approximation space of precision and grade

• Space and plane mathematical forms of PG-Approx-Space are constructed.
• Semantics construction and semantics granules of PG-Approx-Space are established.
• Granular quantitative semantics, complete systems, optimal calculation are studied.
• Model expansion and model construction in PG-Approx-Space are studied.
• A basic and general exploration framework of double-quantification is established.

Because precision and grade act as fundamental quantitative information in approximation space, they are used in relative and absolute quantifications, respectively. At present, the double quantification regarding precision and grade is a novel and valuable subject, but quantitative information fusion has become a key problem. Thus, this paper constructs the double-quantitative approximation space of precision and grade (PG-Approx-Space) and tackles the fusion problem using normal logical operations. It further conducts double-quantification studies on granular computing and rough set models. (1) First, for quantitative information organization and storage, we construct space and plane forms of PG-Approx-Space using the Cartesian product, and for quantitative information extraction and fusion, we establish semantics construction and semantics granules of PG-Approx-Space. (2) Second, by granular computing, we investigate three primary granular issues: quantitative semantics, complete system and optimal calculation. Accordingly, six types of fundamental granules are proposed based on the semantic, microscopic and macroscopic descriptions; their semantics, forms, structures, calculations and relationships are studied, and the granular hierarchical structure is achieved. (3) Finally, we investigate rough set models in PG-Approx-Space. Accordingly, model regions are proposed by developing the classical regions, model expansion is systematically analyzed, some models are constructed as their structures are obtained, and a concrete model is provided. Based on the quantitative information architecture, this paper systematically conducts and investigates double quantification and establishes a fundamental and general exploration framework.