$\mathcal{R}_1$/$\mathcal{R}_2$-closures in three-valued logic: precomplete classes and small bases of monotone functions, circuit synthesis, and applications to modelling information transmission systems

The first part of the talk studies enhanced closure operators $\mathcal{R}_1$ and $\mathcal{R}_2$ in the three-valued logic $P_3$, which allow a consistent description of equivalences between states in data transmission problems. It is shown that for $\mathcal{R}_1$ there exist exactly three precomplete classes, while for $\mathcal{R}_2$ there are five, including classes reflecting strict handling of erasures and constraints on device arity.
The second part presents a constructive approach to finite generation of monotone classes, including the use of selection functions and majority operations. Based on these results, small bases and normal forms are established, suitable for the synthesis of multi-valued logical circuits.
The obtained results provide a rigorous theoretical foundation for practical problems of QoS management and traffic aggregation in LTE/5G/6G scenarios in mobile networks, where robustness to signal erasures and the possibility of local recalibration of ternary levels are essential. The final part demonstrates how normal forms and small bases are used in the synthesis of multi-valued blocks for metric aggregation and prioritisation in mobile HSR scenarios. Fail-safe schemes with guaranteed transition to a safe state and their verifiable invariants for network controllers are presented.
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The talk is based on the materials of the author's PhD thesis