Structural Analysis of Precomplete Classes and Closure Diagrams in Multi-Valued Logic

This paper develops a mathematical framework to ensure the functionality and efficiency of digital electronics built on multi-valued logic (MVL). We investigate the properties of precomplete classes and closure for families of MVL functions, focusing on the existence and structure of R-closed sets in \(P_k\) for \(k \geq ۴\). Our work provides a systematic analysis of logical bases and their representations through structured diagrams, with a particular focus on Boolean and MVL functions. We rigorously classify \(R_۲\)-complete families in \(P_۳\)-spaces and propose extensions for the minimal linear superposition operator. Furthermore, we highlight how MVL significantly enhances information density in memory storage systems, providing pivotal benefits for cutting-edge computational systems. These results bridge theoretical advancements and practical implementations in computer science and logic.

کلیدواژه ها:

Multi-Valued Logic (MVL) ، \mathcal{R}-closed sets ، Precomplete classes ، Logical bases ، Structured diagrams ، Digital electronics ، Boolean and MVL functions ، Memory density enhancement ، Minimal linear operators ، Computational logic applications