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This lecture introduces a new logic representation structure, B-Conditional Binary Decision Diagrams, enabling efficient logic synthesis by exploiting the functionality of double gate controllable polarity FETs. The instructor explains the motivation behind this novel representation form, its direct mapping onto the FETs, experimental results, and the advantages of using B-Conditional BDDs for logic circuits. By employing specific reduction rules, the lecture demonstrates the remarkable reduction in size achieved with these diagrams, showcasing their efficiency in representing functions like the majority and binary addition. The lecture also discusses the canonical nature of B-Conditional BDDs, their application in logic synthesis, and the significant improvements in device count reduction, logic depth, and speed compared to traditional synthesis methods.