Synthesis of Multi-valued Literal Using Lukasiewicz Logic

The synthesis of multi-valued combinational functions is well-studied topic, albeit less compared to the synthesis of two-valued logic families. The synthesis of multi-valued functions consists of bi-decomposition or functional decomposition of the given target function to obtain a multilevel network comprising of min and max gates. Synthesis tools, such as YADE and those based on Multiple-Valued Decision Diagrams make the implicit assumption regarding the availability of literals or CASE operator, while focusing on the optimization of the logic network solely based on the min and max gates. However, a literal cannot be assumed to exist as a primitive in a multi-valued logic system and therefore, renders it difficult for one to directly apply the existing synthesis flows in practical settings [1]. We address this important gap in MVL synthesis flows. Our target multivalued logic is £ukasiewicz logic, which supports implication and negation. We derive literals and CASE operators using these primitives, and propose a heuristic algorithm to synthesize it automatically. Our techniques are implemented as an extension in the YADE tool. Our experimental studies on a wide range of benchmarks reveal that an average overhead of 216% in terms of number implication gates, along with 55% increase in the number of levels is encountered, in contrast to a synthesis flow that assumes existence of literals.